Preparing for the Regents Examination Geometry, AK

ANSWER KEY
Preparing for the
REGENTS EXAMINATION
GEOMETRY
AMSCO
AMSCO SCHOOL PUBLICATIONS, INC.
315 Hudson Street, New York, N.Y. 10013
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1 2 3 4 5 6 7 8 9 10 12 11 10 09 08 07

Contents
Chapter 1: Essentials of Geometry 1
Chapter 2: Logic 3
Chapter 3: Introduction to Geometric Proof 8
Chapter 4: Congruence of Lines, Angles, and Triangles 12
Chapter 5: Congruence Based on Triangles 18
Chapter 6: Transformations and the Coordinate Plane 28
Chapter 7: Polygon Sides and Angles 34
Chapter 8: Slopes and Equations of Lines 39
Chapter 9: Parallel Lines 44
Chapter 10: Quadrilaterals 56
Chapter 11: Geometry of Three Dimensions 68
Chapter 12: Ratios, Proportion, and Similarity 72
Chapter 13: Geometry of the Circle 79
Chapter 14: Locus and Constructions 87
Cumulative Reviews 92
Practice Geometry Regents Examinations 138

Essentials of Geometry
1-1 Undefined Terms
(page 2)
1 (1) finite
2 (2) infinite
3 (3) empty
4 A B
5 A given line lies on an infinite number of
planes.
6 a and b
m
1-2 Real Numbers and
Their Properties
(pages 4–5)
1 (4) 8
2 (4) 5
3 (3) 0n 0
4 (1) a(b c) ab ac
5 (1) additive inverse
6 Commutative
7 Multiplicative identity
8 Commutative
9 Additive identity
10 Associative
11 Associative
p
CHAPTER
1
12 Commutative
13 Distributive
14 Additive inverse
15 Distributive
16 Multiplicative inverse
17 1
18 0
19 0
20 Division by zero is undefined.
1-3 Lines and Line
Segments
(pages 6–7)
1 8
2 4
3 −−
BD and −−
CE
4 They must all lie on the same line.
5 They form a triangle.
6 Point S is the midpoint. It is halfway between
P and T.
7 Correct. The length of −−
AB is equal to the
length of −−−
CD .
8 Incorrect notation
9 Incorrect notation
10 Correct. −−
AB is congruent to −−−
CD .
11 −−−
CPD bisects −−−
APB . It divides it into two segments
with the same measure, 7.
12 MS 6
1-3 Lines and Line Segments 1

1-4 Angles
(pages 9–10)
1 (4) It is a portion of a line, beginning with
point M.
2 (1) DAE
3 (4) XYZ is a 180 angle.
4 (3) equal to 90
5 (2) three acute angles
6 Acute angles: JKM, MKN, NKO, OKL,
MKO
Obtuse angles: JKO, MKL
Right angles: JKN, NKL
Straight angle: JKL
7 mLKO 35; mMKN 30; mMKO 85
8
D
9
10
11
E F
R
V
H
B
12 a and b
E
A
G
B C
I
2 Chapter 1: Essentials of Geometry
I
T
1-5 Preliminary Angle and
Line Relationships
(pages 11–12)
1 a mDBC 90
b DBC ABD
2 a mONP 45
b ONP MNO
3 mCBD
4 mCBD
5 mABD mABC mCBD
90 45
135
6 l n
7 x 8
8 y 9.5
9 mCAT 28
10 mAFC 46
1-6 Triangles
(page 16)
1 False
2 True
3 False
4 False
5 True
6 The length of each leg of the isosceles
triangle is 18.
7 a The length of each side of the triangle
is 16.
b y 7
8 mA 40; mB 70; mC 70
9 Check students’ drawings of isosceles
triangle ABC with vertex at C.
10 Check students’ drawings of right triangle
PQR with hypotenuse ___
PQ .
11 Check students’ drawings of acute triangle
FGH that is scalene.
12 −−−
CD and −−
CE
Chapter Review (pages 16–17)
1 Point
2 Line, length
3 Plane, length and width
4 9
5 Only the square roots of perfect square numbers
are rational.

6 a −−
XZ −−
YZ
b ZXY and ZYX
7 Coordinate of B is 2.
8 mKJL 36
9 G is the midpoint of −−
FH .
10 Bisectors pass through the same point, the
midpoint of the line segment. Distinct parallel
lines have no points in common.
CHAPTER
2
2-1 Statements, Truth
Values, and Negations
(pages 20–21)
1 (2) a false closed sentence
2 (2) a false closed sentence
3 (4) not a statement
4 (1) a true closed sentence
5 (4) 4 3 is not greater than 2.
6 (2) The coat is not blue.
7 (2) false
8 (1) 3 6 7
9 Any open sentence
10 A triangle has three sides.
11 Statistics is the study of analyzing data.
12 Algebra is the study of operations on sets of
numbers.
13 Trigonometry is the study of triangles.
14 Geometry is the study of shapes and sizes.
15 “Chicago is not a city in Indiana.”
Original statement is false.
Negated statement is true.
16 “The sun does not rise in the west.”
Original statement is false.
Negated statement is true.
Logic
11 Straight angles: BCE and ACD
12 DEF
13 Right triangles: ABC and CED
14 CFE
15 −−
BD , −−
FD , −−
FE , and −−
BE are choices.
16 −−
BD
17 “January is not a winter month.”
Original statement is true.
Negated statement is false.
18 “The area of a rectangle is not length times
width.”
Original statement is true.
Negated statement is false.
19 “Each state does not have two senators.”
Original statement is true.
Negated statement is false.
20 “It is not the case that all real numbers are
rational.”
Original statement is false.
Negated statement is true.
2-2 Compound Statements
(pages 23–24)
1 (3) 3 7 10 or 4 7 3
2 (1) Albany is not the capital of New York
and is located on Long Island.
3 (2) 5 is not an odd number or 6 4 12.
4 (4) Daylight saving time ends in November
or the clock is not turned back one hour.
5 False
2-2 Compound Statments 3

6 True
7 True
8 True
9 True
10 Answers will vary. One possible answer is:
Every square has four sides or every triangle
has three sides. True
11 Answers will vary. One possible answer is:
The sum of the measures of the angles of
every triangle is 180 and every triangle has
four angles. False
2-3 Conditionals
(pages 26–27)
1 (2) You take this medicine.
2 (1) I will stay in this afternoon.
3 (1) true
4 (1) true
5 (1) true
6 Conditional
7 Conjunction
8 Conditional
9 Disjunction
10 Negation
11 Answers will vary. x 18 is one possible
answer.
12 x 2
13 Any positive number or zero
14 Any perfect square that is not even
15 Answers will vary. x √ 2 is one possible
answer.
16 True
17 False
18 The sun is
shining.
It is not
raining.
If the sun is
shining, it is
not raining.
Sunny T T T
Sun
showers
T F F
Cloudy F T T
Stormy F F T
4 Chapter 2: Logic
2-4 Converses, Inverses,
and Contrapositives
(pages 28–29)
1 (1) If you stop your car, then the traffic light
is red.
2 (3) If the sum of two numbers is not even,
then the numbers are not both even.
3 (1) If the merrygoround was not oiled, then
it will squeak.
4 (2) If we are on vacation, it is summer.
5 (4) If a triangle is not equilateral, then it
does not have three congruent angles.
6 (2) If I didn’t turn on the air conditioner,
then the house isn’t cool.
7 (2) disjunction, p ∨ q
8 Converse: If school is open, then it is September.
Inverse: If it is not September, then school is
not open.
Contrapositive: If school is not open, then it is
not September.
9 Converse: If the figure has four sides, then it
is a square.
Inverse: If the figure is not a square, then it
does not have four sides.
Contrapositive: If the figure does not have
four sides, then it is not a square.
10 Converse: If someone will trade desserts, then
I have cookies with my lunch.
Inverse: If I do not have cookies with my
lunch, then someone will not trade desserts.
Contrapositive: If someone will not trade
desserts, then I do not have cookies with
my lunch.
11 Converse: If my teacher will be able to read
my paper, then I typed it.
Inverse: If I do not type my paper, then my
teacher will not be able to read it.
Contrapositive: If my teacher will not be able
to read my paper, then I did not type it.
12 Converse: If I am late to school, then I didn’t
set my alarm.
Inverse: If I set my alarm, then I will not be
late to school.
Contrapositive: If I am not late to school, then
I set my alarm.

13 Converse: If I am not in the starting lineup,
then I will not go to practice.
Inverse: If I to go to practice, I will be in the
starting lineup.
Contrapositive: If I am in the starting lineup,
then I will go to practice.
14 Converse: If I am unhappy, then I did not
make honor roll.
Inverse: If I make honor roll, then I am happy.
Contrapositive: If I am happy, then I made
honor roll.
15 Converse: If I do not have money for the
movies, then I will go shopping.
Inverse: If I do not go shopping, then I will
have money for the movies.
Contrapositive: If I have money for the
movies, then I will not go shopping.
16 Converse: If I do not take a school bus, then I
live close to school.
Inverse: If I do not live close to school, then I
take a school bus.
Contrapositive: If I take a school bus, then I do
not live close to school.
17 Converse: If I don’t study Latin, then I will
study French.
Inverse: If I don’t study French, then I will
study Latin.
Contrapositive: If I study Latin, then I will not
study French.
18 a If the segments are congruent, then their
measures are equal. (True)
b If the measures of the segments are equal,
then the segments are congruent. (True)
2-5 Biconditionals
(pages 31–32)
1 (1) true for any value of x
2 (2) false for any value of x
3 (2) false
4 (1) If I am sleepy, then I did not get eight
hours of sleep and if I did not get eight hours
of sleep, then I am sleepy.
5 (3) Paul does not buy chicken or he does not
roast it.
6 (4) the conjunction of a conditional and its
converse
7 If I have money, then I earned it.
8 If tomorrow is Friday, then today is Thursday.
(True)
9 Converse: If a number is rational, then it is a
repeating decimal.
Biconditional: A number is rational if and only
if it is a repeating decimal.
The biconditional is false for integers and
terminating decimals.
10 If the sum of the measures of two angles is
not 45, then the two angles are not complementary.
Any pair of complementary angles makes
this statement and original false.
Any pair of angles with a sum of measures
not equal to 45 or 90 makes this statement
and the original true.
2-6 Laws of Logic
(page 36)
1 Margaret is a doctor. (Disjunctive Inference)
2 I will have straight teeth. (Detachment)
3 I did not save money. (Modus Tollens)
4 I helped my friend. (Modus Tollens)
5 It will not rain. (Detachment)
6 Joe is not a teacher. (Disjunctive Inference)
7 I will not eat dinner with Michael. (Modus
Tollens)
8 I will not play on Saturday. (Detachment)
9 If I practice the violin, my friend will be
jealous. (Chain Rule)
10 If I don’t help my brother, he cannot play
football. (Chain Rule)
2-7 Proof in Logic
(page 38)
1 1. p ∨ q
2. r → ~q
3. r
4. ~q Law of Detachment (2, 3)
5. p Law of Disjunctive Inference (1, 4)
2 1. a → b
2. c ∨ a
3. ~c
4. a Law of Disjunctive Inference (2, 3)
5. b Law of Detachment (1, 4)
2-7 Proof in Logic 5

3 1. e ∨ ~f
2. ~f → g
3. ~e
4. ~f Law of Disjunctive Inference (1, 3)
5. g Law of Detachment (2, 4)
4 1. a ∨ b
2. b → c
3. c → d
4. b → d Chain Rule (2, 3)
5. ~d
6. ~b Law of Modus Tollens (4, 5)
7. a Law of Disjunctive Inference (1, 6)
5 1. s ∨ t
2. s → r
3. r → q
4. s → q Chain Rule (2, 3)
5. ~q
6. ~s Law of Modus Tollens (4, 5)
7. t Law of Disjunctive Inference (1, 6)
8. t → v
9. v Law of Detachment (8, 7)
6 1. d → e
2. d ∨ f
3. h → ~e
4. h
5. ~e Law of Detachment (3, 6)
6. ~d Law of Modus Tollens (1, 5)
7. f Law of Disjunctive Inference (2, 6)
7 1. ~f → g
2. ~f ∨ j
3. g → ~h
4. j → k
5. h
6. ~g Law of Modus Tollens (3, 5)
7. f Law of Modus Tollens (1, 6)
8. j Law of Disjunctive Inference (2, 7)
9. k Law of Detachment (4, 8)
8 1. x → z
2. x → y
3. ~z
4. ~y Law of Modus Tollens (1, 3)
5. ~x Law of Modus Tollens (2, 4)
6. x ∨ t
7. t Law of Disjunctive Inference (5, 6)
Chapter Review (pages 38–40)
1 (1) true
2 (1) true
3 (2) false
6 Chapter 2: Logic
4 (2) false
5 (1) If I am late for school, I did not set my
alarm.
6 (2) If Marie is not bowling today, then today
is not Monday.
7 (1) The statement is always true but its converse
cannot be determined.
8 Hypothesis: The triangle has a 90 angle.
Conclusion: The square of the longest side of
a triangle is equal to the sum of the squares
of the other sides.
9 Hypothesis: A and B are alternate interior
angles.
Conclusion: They are congruent.
10 False
11 True
12 True
13 False
14 False
15 Inverse: If the altitude does not bisect the
base, the triangle is not isosceles.
Converse: If the triangle is isosceles, the altitude
bisects the base.
Contrapositive: If the triangle is not isosceles,
the altitude does not bisect the base.
Biconditional: The altitude bisects the base if
and only if it is isosceles.
16 True
17 False
18 p is true; q is false.
19 True
20 Converse: If a number is rational, then it is an
integer. (True)
Conditional statement is true.
Biconditional: A number is an integer if and
only if it is rational.
21 Converse: If a number is not prime, then a
number is a perfect square. (True)
Conditional statement is true.
Biconditional: A number is a perfect square if
and only if it is not prime.
22 Converse: If a parabola is tangent to the
x-axis, then its roots are equal. (True)
Conditional statement is true.
Biconditional: The roots of a quadratic equation
are equal if and only if the parabola is
tangent to the x-axis.

23 Converse: If a number is real, then it is
rational. (False)
Conditional statement is true.
24 Converse: If x 4 √ 2 , then x √ 32 . (True)
Conditional statement is true.
Biconditional: x √ 32 if and only if
x 4 √ 2 .
25 Converse: If a relation is a function, then it
passes the vertical line test. (True)
Conditional statement is true.
Biconditional: A relation passes the vertical
line test if and only if it is a function.
26 If I study, then I will not get poor grades.
27 If I get poor grades, then I did not study.
28 If I do not get poor grades, then I study.
29 If my baby cousin cries, then she gets a
bottle.
30 If the figure is a rectangle, then two adjacent
sides form a right angle.
31 If a triangle is equilateral, then it is equiangular.
32 I like math.
33 The class president is a girl. (Law of Disjunctive
Inference)
34 My teacher will be happy. (Law of Detachment)
35 I do not go to the park. (Law of Modus
Tollens)
36 If I learn to knit, I will save money. (Chain
Rule)
37 I eat lunch. (Law of Modus Tollens)
38 I feel good. (Law of Detachment)
39 Rich likes lacrosse. (Law of Disjunctive
Inference)
40 Marlene does not babysit. (Law of Modus
Tollens)
41 Jack is using his computer. (Law of Modus
Tollens)
42 If I read novels when my teacher assigns
them, I watch less television. (Chain Rule)
43 If I like school, I won’t get a job after school.
(Chain Rule)
44 1. a
2. p ∨ t
3. a → ~t
4. ~t Law of Detachment (3, 1)
5. p Law of Disjunctive Inference
(2, 4)
45 1. x → y
2. ~z ∨ x
3. z
4. x Law of Disjunctive Inference
(2, 3)
5. y Law of Detachment (1, 4)
46 1. c ∨ a
2. ~c
3. a Law of Disjunctive Inference
(1, 2)
4. a → b
5. b Law of Detachment (4, 3)
47 1. c → ~d
2. d
3. ~c Law of Modus Tollens (1, 2)
4. ~b → c
5. b Law of Modus Tollens (4, 3)
6. a ∨ ~b
7. a Law of Disjunctive Inference
(6, 5)
48 1. ~m ∨ n
2. ~m → r
3. r → ~q
4. ~m → ~q Chain Rule (2, 3)
5. q
6. m Law of Modus Tollens (4, 5)
7. n Law of Disjunctive Inference
(1, 6)
8. n → z
9. z Law of Detachment (8, 7)
Chapter Review 7

CHAPTER
3
3-1 Inductive Reasoning
(page 42)
1 For a polygon with n sides and vertices, from
each vertex, diagonals can be drawn to n 3
other vertices. This can happen n times. Each
diagonal is counted twice from a to b and
n(n 3)
from b to a. There are _
2 diagonals.
Test: Triangle 0
Quadrilateral
4(4 3)
_
2
2
5(5 3)
Pentagon _ 5
2
2 n 5
1 2 3 4 5
5 4 3 2 1
−−−−−−−−−−−
6 6 6 6 6 5(6) n(n 1) twice
the sum
The sum of the integers from 1 through
n(n 1)
n _
2 .
3 Answers will vary. 9, 15, 21 are some possible
answers.
4 Answers will vary. 3, 9, 15, 21 are some possible
answers.
3-2 Definitions and Logic
(page 43)
1 Line segments are congruent if and only if
they have the same measure.
2 A point of a line segment is the midpoint if
and only if it divides the line segment into
two congruent segments.
8 Chapter 3: Introduction to Geometric Proof
Introduction to
Geometric Proof
3 A line is an angle bisector if and only if
it divides an angle into two congruent
parts.
4 Lines are perpendicular if and only if the
lines form right angles.
5 A figure is a polygon if and only if it is a
closed figure in the plane formed by three
or more segments joined at their endpoints.
3-3 Deductive Reasoning
(page 44)
1 1. Given
2. Perpendicular lines form right angles.
3. Right triangles have one right angle.
2 EG divides FEH into two adjacent angles
with the same measure. FEG and HEG
are congruent because they have the same
measure. An angle bisector is the ray that
divides an angle into two adjacent congruent
angles.
3 1. Given
2. Line segments equal in measure are
congruent.
3. A midpoint divides a line segment into
two congruent segments.
4 Since −−
PR bisects SPQ, it divides the angle
into two congruent angles. SPR QPR.
If two angles are congruent, they have the
same measure.

3-4 Indirect Proof
(page 45)
1 Statements
1.
Reasons
−−−
CD and −−−
HK are
not congruent.
1. Assumption.
2. CD HK
3.
2. Given.
−−−
CD −−−
HK 3. Line segments
that are equal
in measure are
congruent.
Therefore, assumption is false.
2 Statements Reasons
1. ABC is not a 1. Assumption.
right angle.
2. −−
AB −−−
CD 2. Given.
3. ABC is a right 3. Perpendicular
angle.
lines form right
angles.
Therefore, the assumption is false.
3 Statements Reasons
1. ABC is not an
isosceles triangle.
1. Assumption.
2. A B 2. Given.
3. ABC is an isos- 3. An isosceles trianceles
triangle. gle contains two
congruent angles.
Therefore, the assumption is false.
4 Statements
1. DB is the angle bisector
of ADC.
Reasons
1. Assumption.
2. mADB
mBDC
2. Given.
3. DB is not the bi- 3. An angle bisector
sector of ADC. divides an angle
into two congruent
parts.
Therefore, the assumption is false.
3-5 Postulates, Theorems,
and Proof
(pages 47–49)
1 Yes
2 No
3 No
4 The symmetric property of equality
5 The reflexive property of congruence
6 The symmetric property and transitive
property of congruence
7 Statements Reasons
1. −−
PQ −−−
QR 1. Given.
2. PQR is a right
angle.
2. Perpendicular
lines are two lines
that intersect to
form right angles.
3. mPQR 90 3. A right angle is
an angle whose
degree measure
is 90.
4. −−
XY −−
YZ 4. Given.
5. XYZ is a right
angle.
5. Perpendicular
lines are two
lines that intersect
to form right
angles.
6. mXYZ 90 6. A right angle is
an angle whose
degree measure
is 90.
7. 90 mXYZ 7. Symmetric property
of equality.
8. mPQR
mXYZ
8. Transitive property
of equality.
8 Statements Reasons
1. AC is the angle 1. Given.
bisector of BAD.
2. BAC CAD 2. An angle bisector
divides an angle
into two congruent
parts.
3. mBAC
mCAD
4. AD is the angle
bisector of CAE.
3. If two angles are
congruent, they
have the same
measure.
4. Given.
5. CAD DAE 5. An angle bisector
divides an angle
into two congruent
parts.
3-5 Postulates, Theorems, and Proof 9

6. BAC DAE 6. Transitive
property.
7. mBAC
mDAE
7. If two angles are
congruent, their
measures are
equal.
9 Statements Reasons
1. 8 x y 1. Given.
2. y 3 2. Given.
3. 8 x 3 3. Substitution
property.
4. x 3 8 4. Symmetric
property.
10 Statements Reasons
1. M is the midpoint
of −−
AB .
1. Given.
2. −−−
AM −−−
MB 2. A midpoint
divides a line
segment into two
congruent line
segments.
3. −−−
MB −−
BC 3. Given.
4. −−−
AM −−
BC 4. Transitive
property.
3-6 Remaining Postulates
of Equality
(pages 51–52)
1 Partition postulate of equality
2 Division postulate of equality
3 Addition postulate
4 Subtraction postulate
5 Division postulate of equality
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
10 Chapter 3: Introduction to Geometric Proof
6 1. m1 m2
2. m3 m4
3. m1 m3 m2 m4
4. mDAB m1 m3
mBCD m2 m4
5. mDAB mBCD (Substitution
7 1.
postulate)
−−
AB −−
CB
2. −−−
AD −−
CE
3. AB CB
4. AD CE
5. AB AD CB CE
6. DB EB
7. −−
DB −−
EB
8 1. AB AC
2. AD
(Line segments
that are equal in
measure are congruent.)
1 _ AC
3
3. AE 1 _ AB
3
4. AD AE (Division
postulate)
9 1. mEAB mFBC
2. AG is the angle bisector of EAB.
3. BH is the angle bisector of FBC.
4. m1 1 _ mEAB
2
5. m2 1 _ mFBC
2
6. m1 m2
10 1. AB DE
2. AC 3AB
3. DF 3DE
(Division
postulate)
4. AC DF (Multiplication
postulate)
Chapter Review (page 52)
1 Reflexive property of equality
2 Transitive property of equality
3 Symmetric property of equality
4 mBAD 1 _ mBAC
2

Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
5 1. −−−−
CMD is a line segment.
2. −−−
CM −−−
MD
3. M is the mid- (Definition of a midpoint)
point of −−−
CD .
6 1. −−−−
ABCD is a line segment.
2. −−
AB −−−
CD
3. BC BC
4. AB CD
5. AB BC CD BC
6. AC BC
7. −−
AC −−
BD (Line segments that are
equal in measure are
congruent.)
7 1. −−
AB −−−
CD
2. AB CD
3. −−
AB and −−−
CD bisect each other at E.
4. AE 1 _ AB
2
5. CE 1 _ CD
2
6. AE CE
7. −−
AE −−
CE (Line segments that are
equal in measure are
congruent.)
8 1. −−
PQ −−−
QR
2. mPQR 90
3. mPQR mABC
4. mABC 90
5. −−
AB −−
BC (Perpendicular lines form
right angles.)
9 1. PQ XY
2. QR YZ
3. PQ QR XY YZ
4. PQ QR PR
5. XY YZ XZ
6. PR XZ
7. −−
PR −−
XZ (Line segments that are
equal in measure are
congruent.)
10 1. DAB and EBA are straight angles.
2. mDAB 180
3. mEBA 180
4. m1 m2
5. mDAB m1 mEBA m2
6. m3 m4 (Substitution postulate)
11 1. −−
AC −−
BC
2. AC BC
4. −−−
DC −−
EC
5. DC EC
6. AD DC BC EC
7. AD BE
8. −−−
AD −−
BE (Line segments that are
equal in measure are
congruent.)
12 1. −−−
AD −−
BC
2. AD BC
3. E is the midpoint of −−−
AD .
4. DE 1 _ AD
2
5. F is the midpoint of −−
BC .
6. BF 1 _ BC
2
7. BF DE
8. −−
BF −−
DE (Line segments that are
equal in measure are
congruent.)
Chapter Review 11

CHAPTER
4-1 Setting Up a Valid
Proof
(pages 56–57)
1 Given: ABC , BD AC
D
4
A C
B
Prove: ABD CBD
Statements Reasons
1. ABC
2. BD AC
1. Given.
2. Given.
3. ABD is a right
angle.
4. CBD is a right
angle.
3. Definition of perpendicular
lines.
4. Definition of perpendicular
lines.
5. ABD CBD 5. Congruent angles
are angles that
have the same
measure.
2 Given: PQRS , −−
PR −−
QS
P Q R S
Prove: −−
PQ −−
RS
12 Chapter 4: Congruence of Lines, Angles, and Triangles
Congruence of Lines,
Angles, and Triangles
Statements Reasons
1. PQRS
1. Given.
2. −−
PR −−
QS 2. Given.
3. −−−
QR −−−
QR 3. Reflexive
property of
congruence.
4. PR QR
QS QR
4. Subtraction
postulate
5. −−
PQ −−
RS 5. Substitution
postulate
3 Given: −−−
AD −−
BC
−−
EF bisects −−−
AD
−−
GF bisects −−
BC
Prove: −−
AE −−
BG
Statements Reasons
1. −−−
AD −−
BC 1. Given.
2. −−
EF bisects −−−
AD 2. Given.
3. −−
GF bisects −−
BC 3. Given.
4. −−
AE −−−
AD 4. Definition of a
bisector.
5. −−
BG −−
GC 5. Definition of a
bisector.
6. AE ED 6. Definition of
congruent
segments.
7. BG GC 7. Definition of
congruent
segments.
8. AE ED AD 8. Partition
postulate.

9. BG GC BC 9. Partition
postulate.
10. AE AE AD
or 2AE AD
10. Substitution
postulate.
11. BG BG BC or 11. Substitution
2BG BC
postulate.
12. AE 1 _ AD
2
12. Division
postulate.
13. BG 1 _ BC
2
13. Division
postulate.
14. AD BC 14. Definition of
congruent
segments.
15. AE 1 _ BC
2
15. Substitution
postulate.
16. 1 _ BC BG
2
16. Symmetric
property.
17. AE BG 17. Transitive
property.
18. −−
AE −−
BG 18. Segments with
equal measures
are congruent.
4 Given: URS UTS
−−
US bisects RUT
−−
US bisects RST
Prove: 1 2
Statements Reasons
1. URS UTS 1. Given.
2. −−
US bisects RUT 2. Given.
3. −−
US bisects RST 3. Given.
4. 1 TUS 4. Definition of an
angle bisector.
5. RSU 2 5. Definition of an
angle bisector.
6. m1 mTUS 6. Congruent
angles are equal
in measure.
7. mRSU m2 7. Congruent
angles are equal
in measure.
8. m1 mTUS 8. Partition
mRUT
postulate.
9. mRSU m2 9. Partition
mRST
postulate.
10. m1 m1
mRUT or
2(m1)
mRUT
11. m2 m2
mRST or
2(m2)
mRST
12. m1 1 _
10. Substitution
postulate.
11. Substitution
postulate.
mRUT 12. Division
2
postulate.
13. m2 1 _ mRST 13. Division
2
postulate.
14. mRUT
mRST
14. Congruent angles
are equal in
measure.
15. m1 1
_
2 mRST 15. Substitution
postulate.
16. 1 _ mRST 2 16. Symmetric
2
property.
17. m1 m2 17. Transitive
property.
18. 1 2 18. Congruent angles
are equal in
5 Given:
measure.
−−
HJ −−
FK
Prove: −−−
HK −−
FJ
Statements Reasons
1. −−
HJ −−
FK 1. Given.
2. HJ FK 2. Definition of congruent
segments.
3. JK JK 3. Reflexive
property.
4. HJ JK HK 4. Partition
postulate.
5. FK JK FJ 5. Partition
postulate.
6. HJ JK
FK JK
6. Addition
postulate.
7. HK FJ 7. Substitution
postulate.
8. −−−
HK −−
FJ 8. Segments with
equal measures
are congruent.
4-1 Setting Up a Valid Proof 13

6 Given: BA bisects CBE
1 2
2 4
Prove: 3 4
Narrative Proof: A bisector divides an angle
into two congruent angles, so 1 2.
Using the transitive property of congruence,
1 4. By substitution, 3 4.
7 4x 3 2x 21
x 12
mBCA 4x 3
4(12) 3
45
mBCE 2(45) 90
8 4x 4 2(3x 21)
x 23
AB 4x 4
4(23) 4
96
9 Since DE CE, then 2x y 5y, and
x 2y.
AC BD
6 5y (2x y) (x y)
6 5y 3x 2y
6 5y 3(2y) 2y
6 5 y 6y 2y
6 3y
2 y
x 2(2) 4
BD 2(4) 2 4 2 16
10 AB DE 8, so BD 4 and CD 2. Therefore,
CE 8 2 10.
4-2 Proving Theorems
About Angles
(pages 59–61)
1 If two angles are right angles, they both have
a measurement of 90. Angles with the same
measure are congruent.
2 If two angles are straight angles, they both
have a measurement of 180. Angles with the
same measure are congruent.
3 If 1 is a complement of A, then m1
90 mA. If 2 is a complement of A,
then m2 90 mA. Then m1 m2
by the symmetric property and transition,
and 1 2 by definition of congruence.
14 Chapter 4: Congruence of Lines, Angles, and Triangles
4 If 1 2, 3 is the complement of
1, and 4 is the complement of 2, then
m3 90 m1 and m4 90 m2.
Since m1 m2 by congruence, m3
m4 by symmetric property and transition,
and 3 4 by congruence.
5 If 1 is a supplement of A, then m1
180 mA. If 2 is a supplement of A,
then m2 180 mA. Then m1 m2
by the symmetric property and transition,
and 1 2 by definition of congruence.
6 If 1 2 and 3 is the supplement of
1 and 4 is the supplement of 2, then
m3 90 m1 and m4 90 m2.
Since m1 m2 by congruence, m3
m4 by symmetric property and transition,
and 3 4 by congruence.
7 By definition the sum of their measures is a
straight angle or 180.
8 The two angles are a linear pair so the sum
of their measures is 180. If they are congruent,
each measure is 90; they form right
angles and are therefore perpendicular.
9 Each angle forms a linear pair with the same
angle. They are supplementary to the same
angle and are therefore congruent.
10 m2 m3 because they are vertical angles.
By substitution, 1 is complementary
to 3. 1 4 because complements of the
same angle are congruent.
11 3 forms a linear pair with 1, 3 is supplementary
to 1. Since 1 2, 3 is
supplementary to 2. 4 forms a linear pair
with 2, 4 is supplementary to 2.
3 4 because supplements of the same
angle are congruent.
12 Since they form a linear pair, 2 is supplementary
to EDG. Since 1 and 2 have
the same measure (they are congruent), 1 is
supplementary to EGD.
13 Since 1 is complementary to 3,
m1 m3 90. Since ACB is a right
angle, m1 m2 90. By subtraction and
substitution, m2 m3, so they are congruent
angles.
14 By the transitive postulate, 1 3.
Because they are vertical angle pairs,
1 4, and 3 6. By substitution
and transition, 4 6.

15 x 2 5 180
x 155
16 2x x 180
x 60
17 x 85
18 (x 10) x 90
x 40
19 60 x x x 180
x 40
20 6x 2 0 4x 10
x 15
21 40 mz 180
mz 140
22 a mDOE 75
b mAOB 15
c mDOC 105
23 2x 1 5 4x 1
x 7
mPRT 2(7) 15 29
mRTQ 180 mPRT
180 29
151
24 3x 2 5 10x 4
x 3
mWVY 3(3) 25 34
mWVZ 180 mWVY
180 34
146
25 (5x 2y) (5x 2 y) 180
x 18
m AEC mBED
11y 5x 2y
1 1y 5(18) 2y
9 y 90
y 10
mAED mCEB 5x 2y
5(18) 2(10) 70
4-3 Congruent Polygons
(pages 64–66)
1 (3) IHG MNL
2 (4) a side of one is congruent to a side of the
other
3 (4) IV
4 (1) I
5 (2) II
6 a −−
AC
b BAC
c −−
BC
d CBA
7 a ACD BCD
b ADC CBA
8 a mE 44
b mB 110
c mC 26
9 Answers will vary. For instance, a 3-4-5
triangle has the same angle measure as a
6-8-10 triangle. They are not congruent.
10 Answers will vary.
11 Answers will vary.
12 a Vertical angles are congruent; the triangles
are congruent by SAS.
b AC BD
2 x x 5
x 5
AC BD 10
AE BE 11
CE DE 12
13 The triangles are congruent by SAS.
ACE BDE by corresponding parts of
congruent triangles are congruent. Therefore,
ACE BDE. So 5x 7x 18, and x 9.
mACE 45; mBDE 45; mBED 45.
mDBE 180 45 45 90. Therefore,
DBE is a right triangle.
14 x 2 4x x 14
x 2 5x 14 0
(x 7)(x 2) 0
x 7 and x 2
When x 7, AB AB 21
When x 2, AB AB 12
15 No, the angle must be included between the
pairs of sides.
(page 67)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
4-3 Congruent Polygons 15

1 1. −−−−
ABCD
2. 1 2
3. ___
AF −−
DE
4. −−
AC −−
BD
5. AC BD
6. BC BC (Reflexive property)
7. AC BC BD BC
8. AB CD
9. −−
AB −−−
CD
10. ABF DCE (SAS SAS)
2 1. −−−
ABC , −−−
DEF
2. 3 4
3. −−
BF −−
EC
4. 1 2
5. −−−
FBC −−−
CEF (If two angles are
congruent, their
supplements are
congruent.)
6. BCF EFL (ASA ASA)
3 1. −−
BE is a median to −−
FD .
2. −−
FE −−
DE
3. −−
BE −−
BE
4. −−−
AD −−
CF
5. −−
AB −−
CB
6. AD AB CF CB
7. −−
BD −−
BF
8. FBE DBE (SSS SSS)
4 1. −−
AE −−−
DC
2. −−
DE −−
DE
3. AE DE DC DE
4. −−−
AD −−
EC
5. 3 4
6. ADB and 3 are linear pairs.
7. ADB and 3 are supplements.
8. CEB and 4 are linear pairs.
9. CEB and 4 are supplements.
10. ADB CEB
11. 1 2
12. ADB CEB (ASA ASA)
5 1. D is the midpoint of −−
AB .
2. AD DB
3. −−−
AD −−
DB
4. −−
AC −−
BC
5. −−−
DC −−−
DC
6. ADC BDC (SSS SSS)
6 1. −−
AB −−
BC
2. −−
EF −−
AB
3. −−
EF −−
BC
16 Chapter 4: Congruence of Lines, Angles, and Triangles
4. −−
BE bisects −−
CF at D.
5. −−−
CD −−
DF
6. 1 2
7. BCD EFD (SAS SAS)
7 1. −−
AC −−
BD
2. −−
BC −−
BC
3. AC BC BD BC
4. −−
AB −−−
DC
5. 3 4
6. 3 and FBA are linear pairs.
7. 3 and FBA are supplements.
8. 4 and ECD are linear pairs.
9. 4 and ECD are supplements.
10. FBA ECD
11. 1 2
12. EDC FAB (ASA ASA)
8 1. EG is the perpendicular bisector of −−
AB .
2. −−
AF −−
BF
3. EFA is a right angle.
4. EFB is a right angle.
5. EFA EFB
6. mEFA m1 mEFB m1
7. 1 2
8. mEFA m1 mEFB m2
9. DFA CFB
10. −−
DF −−
CF
11. ADF BCF (SAS SAS)
Chapter Review (pages 68–69)
1 Given: ABC and DBE are vertical angles.
PQR ABC
Prove: PQR DBE
2 Given: AB and CD intersect at E.
AEC FGH
Prove: CEB is supplementary to FGH.
3 Given: AEB and CED are perpendicular lines.
Prove: AEC AED
4 Given: AEB and CED are perpendicular lines.
F is not on CD .
Prove: FE is not perpendicular to AB .
5 Since E is the midpoint of −−
AB , CD is a bisector
of −−
AB . Since AEC BEC and they are
a linear pair, the measure of each must be
180 _ 90.
2

6 1 3 because they are vertical angles.
4 2 because they are vertical angles. By
the substitution postulate, 2 3. By the
transitive postulate, 4 3, or 3 4
by the symmetric property.
7 Since −−
PQ bisects ABC, then 2 CBQ.
Since CBQ and 1 are vertical pairs,
CBQ 1. By the transitive postulate,
2 1, or 1 2 by the symmetric
property.
8 Since CDE is a right triangle, the two angles
that are not the right angle are complementary
because there sum is 180 (whole
triangle) 90 (right angle) 90. Therefore,
FED is complementary to FCD. By substitution,
EDF is complementary to FCD.
9 mAED mBED 180
4x 20 9x 4 8 180
x 16
mCEB mAED
4x 20
4(16) 20
84
10 mAED mCEB
5 x 7x 26
x 13
mAED 5(13) 65
mAED mDEB 180
65 mDEB 180
mDEB 115
11 mAED mCEB
1 2x 6x 10y
6x 10y
mAED mDEB 180
12x 6x 180
18 x 180
x 10
6x 10y
6 0 10y
y 6
mAEC mDEB
10y
10(6) 60
12 mCEB mDEB 180
9y 12y 1 2 180
y 8
mAEC mDEB
12(8) 12 108
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
13 1. −−
AC −−
DF
2. A D
3. C F
4. ABC DEF (ASA ASA)
14 1. −−
HE −−
FE
2. −−−
HG −−
FE
3. −−
HF −−
HF
4. EFH GHF (SSS SSS)
15 1. A, B, C, D lie on circle O.
2. −−−−
AOC ; −−−−
BOD
3. −−−
AO , −−
BO , −−−
CO , −−−
DO are radii.
4. −−−
AD −−−
CO
5. −−
BO −−−
DO
6. AOB COD (If two lines
intersect, the
vertical angles are
congruent.)
7. ABO CDO (SAS SAS)
16 1. −−
QT −−
RT
2. QT RT
3. −−
TV −−
TU
4. TV TU
5. QT TV RT TV
6. QT TV RT TU
7. −−−
QTV ; −−−
RTU
8. QU RU
9. −−−
QV −−−
RU
10. −−
PU −−
VS
11. −−
PU −−−
UV −−
VS −−−
UV
12. −−
PV −−
SU
13. −−
PQ −−
RS
14. PQU SRU (SSS SSS)
17 1. C is the midpoint of −−
AE .
2. −−
AC −−
CE
3. 1 2
4. 3 BCD
5. 4 BCD
6. 3 4
7. ABC EDC (ASA ASA)
Chapter Review 17

18 1. I is the midpoint of −−
EG .
2. −−
EI −−
IG
3. EI IG
4. −−
EG −−
EI −−
IG
5. EG EI IG
6. EG EI EI or EG 2EI
7. EH 2EI
8. EG EH
9. −−
EG −−
EH
10. FGE IEH
11. FEG HIE
12. FGE IEH (ASA ASA)
5-1 Line Segments
Associated With Triangles
(page 71)
1 One line
2 C
Median
CHAPTER
5
A E F D B
Angle Bisector
3 −−
AE and −−
BE
4 −−
AB and −−−
CD
5 ACF and BCF
6 ADC and BDC
Altitude
18 Chapter 5: Congruence Based on Triangles
19 1. AEB, CED are right angles.
2. AEB CED
3. A C
4. −−
AE −−
CE
5. ABC CDE (ASA ASA)
20 1. −−−
DC −−−
AD
2. −−
AB −−−
AD
3. −−−
DC −−
AB
4. −−
DF −−
BE
5. −−
EF −−
EF
6. DF EF BE EF
7. −−
DE −−
BF
8. 1 2
9. ABF DCE (SAS SAS)
Congruence Based on
Triangles
5-2 Using Congruent
Triangles to Prove Line
Segments Congruent and
Angles Congruent
(pages 72–73)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)

1 1. −−
AB −−−
CD
2. −−
BC −−−
DA
3. −−
AC −−
AC
4. ABC CDA (SSS SSS)
5. BAC DCA (Corresponding
parts of congruent
triangles are
congruent.)
2 1. −−
BA −−
BC
2. −−−
DA −−−
DC
3. −−
DB −−
DB
4. ABD CBD (SSS SSS)
5. ABC CBD (Corresponding
parts of congruent
triangles are
congruent.)
3 1. 1 3
2. 2 4
3. −−
AC −−
AC
4. DAC BCA (ASA ASA)
5. −−−
AD −−
CB (Corresponding
parts of congruent
triangles are
congruent.)
4 1. −−−
CD is the median drawn from C.
2. −−−
AD −−
DB
3. −−−
CD −−
AB
4. ADC is a right angle.
5. BCD is a right angle.
6. ADC BDC
7. −−−
CD −−−
CD
8. ADC BDC (SAS SAS)
9. −−
CA −−
CB
(Corresponding
parts of congruent
triangles are
congruent.)
5 RS RS
4x 1 3x 3
x 4
RT RT x 6 4 6 10
6 AD CB
2x 5 3x 7
x 12
AB x 10 12 10 22
CD AB 22
AD 2x 5 2(12) 5 29
CB 3x 7 3(12) 7 29
7 The triangle is isosceles and RS RT.
6x 4 3x 11
x 5
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
8 1. 1 3
2. 2 4
3. −−
AB −−−
CD
4. ABE CDE (ASA ASA)
5. −−
AE −−
CE
6. −−
BD bisects −−
AC .
7. −−
BE −−
DE
8. −−
AC bisects −−
BD . (Definition of a
bisector)
9 1. −−−
DA bisects BDF.
2. FDA BDA
3. 1 2
4. m1 mFDA m2 mFDA
5. m1 mFDA m2 mBDA
6. EDA CDA
7. −−−
CD −−
DE
8. −−−
AD −−−
AD
9. EDA CDA (SAS SAS)
10. −−
AE −−
AC (Corresponding
parts of congruent
triangles are
congruent.)
10 1. −−
BA is a median of CBF.
2. −−
CA −−
FA
3. −−−
CD −−
FE
4. −−
FC −−−
CD
5. −−
FC −−
EF
6. DCA is a right angle.
7. EFA is a right angle.
8. DCA EFA
9. DCA EFA (SAS SAS)
10. −−−
DA −−
EA (Corresponding
parts of congruent
triangles are
congruent.)
5-2 Using Congruent Triangles to Prove Line Segments Congruent and Angles Congruent 19

5-3 Isosceles and
Equilateral Triangles
(page 74)
1 Statements Reasons
1. Construct a
median from the
vertex formed
by the two congruent
sides of
the triangle, to
form RST with
median −−−
RU .
1. A median of a
triangle is a line
segment with one
endpoint at any
vertex of the
triangle, extending
to the
midpoint of the
opposite side.
2. −−
RS −−
RT 2. Definition of an
isosceles triangle.
3. −−
SU −−
TU 3. Definition of a
median.
4. −−−
RU −−−
RU 4. Reflexive
property of
congruence.
5. RSU RTU 5. SSS SSS.
6. RSU RTU 6. Corresponding
parts of congruent
triangles are
congruent.
2 Statements Reasons
1. Construct a
median from the
vertex formed
by the two congruent
sides of
the triangle, to
form RST with
median −−−
RU .
1. A median of a
triangle is a line
segment with one
endpoint at any
vertex of the triangle,
extending
to the midpoint of
the opposite side.
2. −−
RS −−
RT 2. Definition of an
isosceles triangle.
3. −−
SU −−
TU 3. Definition of a
median.
4. −−−
RU −−−
RU 4. Reflexive property
of congruence.
5. RSU RTU 5. SSS SSS.
20 Chapter 5: Congruence Based on Triangles
6. SRU TRU 6. Corresponding
parts of congruent
triangles are
congruent.
7. −−−
RU bisects SRT. 7. Definition of
angle bisector.
3 Statements Reasons
1. Construct a
median from the
vertex formed
by the two congruent
sides of
the triangle, to
form RST with
median −−−
RU .
1. A median of a
triangle is a line
segment with one
endpoint at any
vertex of the
triangle, extending
to the
midpoint of the
opposite side.
2. −−
RS −−
RT 2. Definition of an
isosceles triangle.
3. −−
SU −−
TU 3. Definition of a
median.
4. −−−
RU −−−
RU 4. Reflexive property
of congruence.
5. RSU RTU 5. SSS SSS.
6. SUR TUR 6. Corresponding
parts of congruent
triangles are
congruent.
7. SUR is a right
angle; TUR is a
right angle.
7. Adjacent congruent
angles are
supplementary.
8. −−−
RU −−
ST 8. Definition of perpendicular
lines.
4 Statements Reasons
1. Construct a
median from the
vertex formed
by the two congruent
sides of
the triangle, to
form ABC with
median −−−
AX .
1. A median of a
triangle is a line
segment with
one endpoint at
any vertex of
the triangle,
extending to the
midpoint of the
opposite side.
2. −−
AB −−
AC 2. Definition of
an equilateral
triangle.

3. −−
BX −−
XC 3. Definition of a
median.
4. −−−
AX −−−
AX 4. Reflexive
property of
congruence.
5. ABX ACX 5. SSS SSS.
6. ABX ACX 6. Corresponding
parts of congruent
triangles are
congruent.
7. Construct a second
median, −−
BY ,
from vertex B.
7. A median of a
triangle is a line
segment with
one endpoint
at any vertex
of the triangle,
extending to the
midpoint of the
opposite side.
8. −−
BA −−
BC 8. Definition of an
equilateral
triangle.
9. −−
AY −−
YC 9. Definition of a
median.
10. −−
BY −−
BY 10. Reflexive
property of
congruence.
11. BAY BCY 11. SSS SSS.
12. BAY BCY 12. Corresponding
angles of congruent
triangles are
congruent.
13. ABC is equiangular.
13. BCY is the
same as ACX,
and all three
angles of the triangle
are equal.
5 mA mC 3x 5
mA mB mC 180
(3x 5) (4x 10) (3x 5) 180
10 x 180
x 18
mC 3(18) 5 59
6 The measure of each angle is 60. So,
3x 6 60; x 22. And 3y 6 60;
y 18.
7 Show that RXZ TYZ SYX by
SAS SAS, and −−
XY −−
YZ −−
ZX because
corresponding parts of congruent triangles
are congruent.
8 Construct −−
AC and −−
BD , and label their intersection
E. ABE CBE. −−
AC and −−
BD are
perpendicular. AED and CDE are right
angles; therefore, AED CDE.
−−
AE −−
CE . −−
ED −−
ED . AED CED.
CAD ACD.
9 mA mB mC 180
(6x 12) (8x 8) (3x 6) 180
17x 1 0 180
x 10
mA 72; mB 72; mC 36
The triangle is isosceles.
10 mB 180 2x
5-4 Working With Two
Pairs of Congruent
Triangles
(pages 75–76)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
1 1. ABC EFG
2. −−
AC −−
EG
3. −−
BD is a median.
4. D is the midpoint (Definition of
of −−
AC . median)
5. DC 1 _ AC (Definition of mid-
2
point)
6. −−
FH is a median.
7. H is the midpoint of −−
EG .
8. HG 1 _ EG
2
9. AC EG
10. HG 1 _ AC
2
11. HG DC
5-4 Working With Two Pairs of Congruent Triangles 21

12. −−−
HG −−−
DC
13. −−
FG −−
BC
14. BCD FGH
15. BCD FGH (SAS SAS)
16. −−
BD −−
FH
2 a 1. −−
PR and −−
QS bisect each other at V.
2. −−
PV −−
RV
3. −−−
QV −−
SV
4. PVQ RVS (Vertical angles are
congruent.)
5. PQV RSV (SAS SAS)
b A continuation of part a.
6. UPV TRV (Corresponding
parts of congruent
triangles are
congruent.)
7. PVU RVT (Vertical angles are
congruent.)
8. PUV RTV
3 a 1.
(ASA ASA)
−−
AC −−−
AD
2. −−
BC −−
BD
3. −−
AB −−
AB
4. ABC ABD (SSS SSS)
5. BAC BAD (Corresponding
parts of congruent
triangles are
6.
congruent.)
−−
AE −−
AE
7. ACD ADE (SAS SAS)
b A continuation of part a.
8. 1 2 (Corresponding
parts of congruent
triangles are
4 1.
congruent.)
−−−
CD −−
CB
2. 1 2
3. −−
CF −−
CF
4. CFB CFD (SAS SAS)
5. CBF CDF
6. EDF ABF
7. BFA EFD
8.
(Corresponding
parts of congruent
triangles are
congruent.)
−−
DF −−
BF
9. BFA DFE (ASA ASA)
10. 3 4 (Corresponding
parts of congruent
triangles are
congruent.)
22 Chapter 5: Congruence Based on Triangles
5 1. −−
AC −−
BC
2. −−
AE −−
BE
3. −−
EC −−
EC
4. ACE BCE (SSS SSS)
5. ACE BCE
6. −−−
CD −−−
CD
7. ACD BCD (SAS SAS)
8. −−−
AD −−
BD
(Corresponding
parts of congruent
triangles are
congruent.)
6 a 1. −−
SP −−−
QR
2. 1 2
3. −−
AB bisects −−
QS at C.
4. −−
SC −−−
QC (Definition of
bisector)
5. −−
SQ −−
SQ
6. SQP QSR (SAS SAS)
7. SQP QSR (Corresponding
parts of congruent
triangles are
congruent.)
8. ACQ BCS
9. ACQ BCS (ASA ASA)
b Continuation of part a.
10. −−
AC −−
BC (Corresponding
parts of congruent
triangles are
congruent.)
5-5 Proving Overlapping
Triangles Congruent
(page 77)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
1 1. −−
AE −−
BE
2. EAB EBA (If two sides of
a triangle are congruent,
the angles
opposite these sides
are congruent.)

3. −−
AB −−
AB
4. −−
AC −−
BD
5. ABC BAD
2 1. 1 3
2. 2 2
3. 1 3 2 3
(SAS SAS)
4. ADB EDC
5. 4 5
6.
(Partition
postulate)
−−−
AD −−
ED
7. ADB EDC (ASA ASA)
3 1. −−
AE −−
BE
2. D is the midpoint of −−
AB .
3. −−−
AD −−
BD
4. −−
ED −−
ED
5. ADE BDE
4 1. −−
TP −−
TQ
2. TPQ TQP
3. SPQ RQP
4. SPT RQT
5. STP RTQ (Vertical angles are
congruent.)
6. STP RTQ
7.
(SAA SAA)
−−
SP −−
PQ
8. −−
PQ −−
PQ
9. SPQ RQP
5 1. −−
EF −−
GF
2. −−
DF −−
HF
3. DFH DFH
4. DFG EFH (SAS SAS)
5. 1 2 (Corresponding
parts of congruent
triangles are
congruent.)
6 1. −−−
AD and −−
FC bisect each other at G.
2. −−−
AG −−−
DG
3. −−
FG −−
CG
4. AGC DGF (Vertical angles are
congruent.)
5. AGC DGF (SAS SAS)
6. GDE GAC
7. AGB DGE
(Corresponding
parts of congruent
triangles are
congruent.)
8. AGB DGE (ASA ASA)
5-6 Perpendicular
Bisectors of a Line
Segment
(pages 79–80)
1 Given −−
AB and P and Q, such that PA PB
and QA QB, let −−
AB intersect −−
PQ at
point M. APQ BPQ. APM BPM.
APM BPM. −−−
AM −−−
BM . Therefore,
AMP BMP.
2 a) Given −−
AB and point P such that PA PB;
if two points are each equidistant from the
endpoints of a line segment, then the points
determine the perpendicular bisector of the
line segment. Select a second point that is
equidistant from both A and B, for example
the midpoint of −−
AB , M. Then −−−
PM is the perpendicular
bisector of −−
AB .
b) Given point P on the perpendicular bisector
of −−
AB , let point M be the midpoint of −−
AB .
PMA PMB.
3 The perpendicular bisector of −−
PQ
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
4 1. −−
JL −−
KL
2. −−
JM −−−
KM
3. JL KL (Congruent segments are
equal in length.)
4. JM KM
5. −−−
LM is the perpendicular bisector of −−
JK .
(Two points, each equidistant from the
endpoints of a line segment, determine
the perpendicular bisector of the line
segment.)
6. N is the midpoint of JK.
(A point on the perpendicular bisector
of a line segment is equidistant from the
endpoints of the line segment.)
7. −−
JN −−−
KN (Definition of a midpoint)
5-6 Perpendicular Bisectors of a Line Segment 23

5 1. −−
FG is the perpendicular bisector of −−
HI .
2. −−
JG is the perpendicular bisector of −−
HI .
(F, J, and G are collinear.)
3. −−
HJ −
IJ
4. −−
GJ −−
GJ
5. HJG and IJG are right angles.
(Definition of perpendicular bisector)
6. HJG IJG (Right angles are
congruent.)
7. HGJ IGJ (SAS SAS)
8. 1 2 (Corresponding parts of
congruent triangles are congruent.)
6 1. −−
AB and −−−
CD bisect each other.
2. −−
AB −−−
CD
3. −−
AC −−−
AD
4. −−
BC −−
BD
5. −−−
AD −−
BD
6. −−
AC −−
BC
7. ACBD is an equilateral quadrilateral.
(Definition of equilateral quadrilateral)
7 1. 1 2
2. 3 4
3. −−
XZ −−
XZ
4. XVZ XWZ (ASA ASA)
5. −−−
XV −−−
XW (Corresponding parts
of congruent triangles
are congruent.)
6. XV XW
7. −−−
XZY ; −−−−
VYW
8. −−
XY is the perpendicular bisector of −−−−
VYW .
(A point equidistant from the endpoints
of a line segment is on the perpendicular
bisector of the line segment.)
8 BP CP
4 y 4
y 1
AP CP
x y 4
x 1 4
x 5
9 AP BP
3x y x y
x y
BP CP
x y 4
x x 4
x 2
y 2
10 CG AG 10
24 Chapter 5: Congruence Based on Triangles
5-7 Constructions
(page 83)
1 (4) BAC and GHI
2 A B C
3
4
5
S
A B
V
E F
T
C D
R
1 2
3
6 If the legs of the compass form two sides of
a triangle, then the line segment connecting
the pencil to the point is the third side. If
the angle between the legs does not change
(and the legs remain the same length), then
any line segment connecting the legs is
congruent to the original because of SAS
congruence.
7
A
G
F
B
E
H C
D L
The first arc drawn in the construction creates
an isosceles triangle, BGH. The rest of
the construction copies the sides of BGH to
a new triangle. ED BH, FD GH, and
FE GB, since F and G lie on the intersections.
Therefore, BGH EFD by SSS
congruence and all corresponding angle
measures, including B and E, are equal.

8 Draw a point. Draw an arc centering on the
point. Draw two lines from the point intersecting
the arc. Connect the points of intersection.
This will form an isosceles triangle.
9
E
10
11
12
A P
C D
B
A
D
B C
A
T
C D
A B
B
M
C
13
14
15
16
A
A
B
A
A
D
D
C
D
D
B
C
C
B
C
B
5-7 Constructions 25

Chapter Review (pages 84–85)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable statements
to complete each proof. Depending on the
textbook used, the wording and format of reasons
may differ, so they have not been supplied for the
method of congruence applied in each problem.
(These solutions are intended to be used as a
guide—other possible solutions may vary.)
1 1. −−−
MN bisects PNQ.
2. RND RNQ
3. RNP RQN
4. −−−
RN −−−
RN
5. RPN RQN (ASA ASA)
2 1. −−
PQ −−
PR
2. −−
QT is a median.
3. −−
PT −−
RT
4. −−
RS is a median.
5. −−
PS −−
QS
6. PQT PRS (SSS SSS)
3 1. 3 4
2. −−
DE −−
DF
3. −−−
DC −−−
DC
4. DEC DFC (SAS SAS)
5. DCE DCF (Corresponding
parts of congruent
triangles are
congruent.)
6. EDC FDC
7. mCAD m1 mEDC mDCE
180
8. mCBD m2 mFDC mDCF
180
9. mCAD mCBD
10. CAD CBD
11. ABC is isosceles. (Definition of isosceles
triangle)
4 1. ABC is an equilateral triangle.
2. ___
AB ___
AC (Definition of equilateral
triangle)
3. DCB DBC
4. DB DC
5. ___
AD ___
AD
6. ABD ACD (SSS SSS)
7. BAD CAD (Corresponding
parts of congruent
triangles are
congruent.)
8. −−−
AD bisects BAC. (Definition of an
angle bisector)
26 Chapter 5: Congruence Based on Triangles
5 1. ____
TM ___
TA
2. MTA is isosceles. (Definition of isosceles
triangle)
3. ___
TH bisects MTA.
4. ___
TH is an altitude (In an isosceles
of MTA. triangle, the
bisector is the
altitude.)
6 1. ___
AB ___
AD
2. ___
CB ___
CD
3. ___
AC ___
AC
4. ABC ADC (SSS SSS)
5. BAE DAE (Corresponding
parts of congruent
triangles are
congruent)
6. ___
AE ___
AE
7. ABE ADE (ASA ASA)
8. ___
BE ___
DE
9. DAB is isoceles. (Definition of
isosceles triangle)
10. ___
AE is the median (Definition of
of DAB. median)
11. ___
AE is the altitude (In an isosceles triof
DAB. angle, the median
is the altitude.)
7 1. ___
AB ___
BD
2. A D
3. DBA CBE
4. EBD EBD
5. DBA EBD CBE EBD
6. EBA CBD
7. ABE DBC (ASA ASA)
8 1. ADE BDC
2. EDB EDB
3. ADE EDB BDC EDB
4. ADB EDC
5. DAE DEC
6. ___
DA ___
DE
7. DAB DEC (ASA ASA)
9 1. 1 2
2. 3 4
3. ___
DE ___
DF
4. DEG DFH (ASA ASA)
5. ___
GD ____
HD (Corresponding
parts of congruent
triangles are
congruent.)
6. EDF EDF

7. 1 EDF 2 EDF
8. GDF EDH
9. FGD EDH (ASA ASA)
10. 5 6 (Corresponding
parts of congruent
triangles are
congruent.)
10 1. ___
AC and ___
BD bisect each other at G.
2. ___
CG ____
AG
3. ___
BG ___
DG
4. AGB CGD (Vertical angles are
congruent.)
5. AGB CGD
6.
(ASA ASA)
___
AB ___
CD
7. GCD GAB
8. 1 2
(Corresponding
parts of congruent
triangles are
congruent.)
9. CED AFB
10.
(ASA ASA)
___
EC ___
FA (Corresponding
parts of congruent
triangles are
congruent.)
11 Assume that ABC and ABC are congruent
and that ___
BD and _____
BD are angle bisectors
of B and B, respectively.
ABD ABD. ___
AB ____
AB . A A.
ABD ABC. ___
BD _____
BD .
12 1. BIG is equilateral.
2. ___
IA ___
BC ___
GT
3. IA BC GT
4. __
IB ___
BG ___
GI
5. IB BG GI
6. IB IA AB; BG BC CG;
GI GT TI
7. IA AB BC CG GT TI
8. IA AB IA BC CG BC
GT TI GT
9. AB CG TI
10. ___
AB ___
CG __
TI
11. BIG IGB GBI
12. IAT BCA (ASA ASA)
GTC
13. ___
AT ___
CA ___
TC (Corresponding
parts of congruent
triangles are
congruent.)
14. CAT is (Definition of an
equilateral. equilateral triangle)
13 1. ___
RU
2. ___
RT ___
US
3. RT US
4. RT ST US ST
5. RS TU
6. ___
RS ___
TU
7. R U
8. VST WTS
9. mVST mWTS
10. VSR is the complement of VST.
11. WTU is the complement of WTS.
12. VSR WTU
13. RVS UWT (ASA ASA)
14 1. ____
MQ ____
NQ
2. ___
QP ____
QO
3. ___
PQ ____
MQ
4. MQP is a right angle.
5. ____
OQ ____
NQ
6. NQO is a right angle.
7. MQP NQO (Right angles are
congruent.)
8. PQO PQO
9. MQP PQO NQO PQO
10. MQO NQP
11. MQO NQP (SAS SAS)
15 1. ___
AC ___
BC
2. ACF BCG
3. DCF ECG
4. DCF ACF BCG ACF
5. DCF ACF BCG ECG
6. ACD BCE
7. CAF CBA
8. CAD CBE (ASA ASA)
9. ___
DC ___
EC
10. DCE is isosceles. ( Definition of an
isosceles triangle)
16 Draw a line longer than the sum of the
lengths of the two segments. Copy ___
AB onto
the new line. Place the compass vertex where
the arc swing intersects the line and mark off
the length of ___
CD . The line segment from the
original vertex to the final arc swing marks
off the new segment.
17 Bisect ___
AB and then bisect each half of the
original segment.
18 Use angle bisector procedure.
19 Bisect side ___
AB . Mark the point where the
bisector intersects the line M. Draw a line
from C to M.
Chapter Review 27

20 Use constructing congruent angles procedure.
21 Use angle bisector procedure on each side of
the triangle. The angle bisectors should meet
at a common point, P.
22 Use constructing a perpendicular bisector
procedure.
Transformations and
the Coordinate Plane
6-1 Cartesian Coordinate
System
(pages 91–92)
1 (2) 5
2 (2) 6
3 (4) 10
4 (2) 6
5 (1) √ 2
6 (2) √ 226
7 a A(5, 2), B(3, 3) b A(3, 4), B(0, 5)
c A(x, y), B(0, 0)
8 The length of base ___
AB is the difference
between the x-coordinates of A and B;
6 2 4. The height is the vertical distance
from C to ___
AB or the difference between the
y-coordinates; 4 1 3. Therefore,
A 1 _ bh
2 1 _ (4)(3) 6.
2
9 The length of the base ___
AB is the difference
between the x-coordinates of A and B;
1 (3) 4. The height is the vertical distance
from C to ___
AB or the difference between
the y-coordinates; 8 4 4. Therefore,
A 1 _ bh
2 1 _ (4)(4) 8.
2
28 Chapter 6: Transformations and the Coordinate Plane
23 Copy one side and the angles at each vertex.
Extend the rays until they meet.
24 Use the line bisector procedure. Mark the
intersection of the line and its bisector as the
midpoint.
CHAPTER
6
10 The length of the base −−
AB is the difference in
the y-coordinates of A and B; 6 2 4. The
height is the horizontal distance from C
to −−
AB or the difference between the
x-coordinates; 3 2 1. Therefore,
A 1 _ bh
2 1 _ (4)(1) 2.
2
11 The length of the base ___
AB is the difference
between the y-coordinates of A and B;
8 2 6. The height is the horizontal distance
from C to ___
AB or the difference between
the x-coordinates; 4 (2) 6. Therefore,
A 1 _ bh
2 1 _ (6)(6) 18.
2
12 D(x, y) (1, 2)
13 Draw a horizontal line from A to C. For
ABC, the length of the base ___
AC is the difference
between the x-coordinates of A and C;
3 (7) 10. The height is the vertical distance
from B to ___
AC or the difference between
the y-coordinates; 5 1 4. The area of
ABC is A 1 _ bh
2 1 _ (10)(4) 20. For
2
ADC, the base is 10 and the height is 4. The
area of ADC is A 1 _ bh
2 1 _ (10)(4) 20.
2
The area of quadrilateral ABCD is 20 20
40.

14 The distance from A to B is
√
[2 (7) ] 2 (5 1 ) 2
5 2 4 2 41 . The distance from B to C
is √
[3 (2) ] 2 (1 5 ) 2
5 2 4 2 41 . The distance from C to D
is
√
[(2) 3 ] 2 [(3) 1 ] 2
5 2 4 2 41 . The distance from D to A
is √
[(7) (2)] 2 [(1 (3)] 2
5 2 4 2 41 . The perimeter of ABCD
41 41 41 41 4 41 .
15 The length of the base ___
AC is the difference
between the x-coordinates of A and C;
6 (4) 10. The height is the vertical distance
from B to ___
AC or the difference between
the y-coordinates; 6 (6) 12. Therefore,
A 1 _ bh
2 1 _ (10)(12) 60.
2
16 Using the distance formula, AB BC 13
and CA 10. The perimeter of ABC
13 13 10 36.
17 Isosceles because PA AT 5
18 Scalene because AB √ 53 , CB √ 74 , and
AC √ 29
19 Scalene because MA √ 130 , AD √ 26 ,
and MD √ 104
20 Scalene because WI √ 26 , IT √ 125 , and
WT √ 109
21 All three sides measure 2, therefore the triangle
must be equilateral.
22 The radius is the distance from the center to
any point on the circle.
r √
(4 1 ) 2 (2 2 ) 2 5
Diameter 2(5) 10
23 The two diagonals are congruent because
AD CB 2.
24 A B 2 [(1) 3 ] 2 (2 2 ) 2 32
B C 2 (3 1 ) 2 (2 4 ) 2 8
A C 2 [(1) 1 ] 2 [(2) 4 ] 2 40.
32 8 40 or A B 2 B C 2 A C 2 .
6-2 Translations
(pages 94–95)
1 (1) (1, 2)
2 (2) (3, 2)
3 (4) T 8, 4
4 (4) 6
5 T 0, 2
6 T 3, 0
7 (4, 1)
8 (3, 7)
9 (2, 4)
10 (4, 3)
11 (4, 2)
12 (2, 2)
13 T 3, 2
14 T 2, 2
15 (3, 0)
16 (7, 3)
17 (4, 7)
18 K(8, 5) → (2, 9), E(10, 3) → (4, 1),
N(2, 2) → (8, 6)
19
y
20
D
10
9
8
7
D"
6
5
4
3 W"
D'
2
1
W
5 4 3 2 1
1
2
3
4
5
1 2 3 4
W'
5 6 7 8 9 10
E
E'
E"
a D’(1, 2), E’(6, 3), W’(2, 1)
b D(0, 6), E(7, 7), W(3, 3)
c T 3, 1
A'
W'
y
10
9
8
7
6
5
4
3
2
1
5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
S'
S(3, 4)
H'
A"
A(0, 1)
W"
W(1, 2)
S"
H(5, 1)
a W’(2, 0), A’(3, 3), S’(0, 6), H’(2, 3)
b W(5, 1), A(4, 2), S(7, 5), H(9, 2)
c T 4 , 1
H"
x
x
6-2 Translations 29

6-3 Line Reflections and
Symmetry
(pages 100–101)
1 (3) A
2 (3) V
3 (1) only vertical line symmetry
4 (2) 2
5 (3) (4, 7)
6 y-axis
7 both
8 x-axis
9 y-axis
10 P’(1, 8)
6
5
4
3
2
1
2
3
4
y
5
6
7
8
9
10
P(2, 4)
6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
1
11 P’(3, 7)
P(3, 3)
P'(3, 7)
P'(1, 8)
8
7
6
5
4
3
2
1
y
y 2
8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8
1
12 P’(4, 1)
2
3
4
5
6
7
8
8
7
6
5
4
3
2
1
y
y 2
8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
P'(4, 1)
y 2
P(4, 5)
30 Chapter 6: Transformations and the Coordinate Plane
x
x
x
13 (5, 2)
14 (2, 3)
15 (2, 5)
16 A’(2, 1), B’(3, 4), C’(4, 5)
C(4, 5)
8
7
6
5
4
3
2
1
y
8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8
1
A'(2, 1)
2
C'(4, 5)
3
4
5
6
7
8
B(3, 4)
A(2, 1)
B'(3, 4)
17 G’(0, 0), H’(4, 3), I’(1, 4), J’(5, 2)
5
I'(–1, 4)
4
H'(–4, 3)
3
I(1, 4)
J(–5, 2)
2
1
6
y
H(4, 3)
J'(5, 2)
6 5 4 3 21G
G'1
2 3 4 5 6
1
2
18 A’(3, 3), B’(5, 8), C’(1, 5)
y
10
9
8
7
6
B'(5, 8)
C'(1, 5 5)
4
3A'(3,
3)
B(8, 5)
2
1
6 5 4 3 2 1
1
2
3
4
A(3, 3)
1 2 3 4
C(5, 1)
5 6 7 8 9 10
x
y x
19 Q’(2, 4), Z’(3, 4), D’(1, 2)
5
Z'(3, 4)
4
Z(4, 3)
3
Q'(2, 4)
2
1
D'(1, 2)
6
y
8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8
1
D
Q(4, 2) (2, 1)
3
4
y x
x
x
x

20
y x 2
6
5
4
3
2
1
y
y x
65 4 3 2 1
1
2
3
4
5
6
1 2 3 4 5 6
x y 2
6-4 Point Reflection and
Symmetry
(pages 103–104)
1 (2) H
2 (1) only line symmetry
3 (3) both point and line symmetry
4 (3) (k, 2k)
5 line symmetry
6 both
7 point symmetry
8 both
9 (1, 3)
10 (3, 1)
11 (9, 0)
12 (2, 6)
13 (4, 4)
14 (8, 2)
15 (2, 4)
16 (2, 8)
17 (10, 10)
18 Point reflection is a transformation of a
figure into another figure.
Point symmetry is a quality of a figure. A figure
with point symmetry is not changed by a
reflection through that point. (It is rotated in
either direction 180 about a point.)
x
6-5 Rotations
(page 107)
1 (2) trapezoid
2 (1) regular hexagon
3 (2) regular pentagon
4 (2) rectangle
5 (1) equilateral triangle
6 (1)
7 Z
8 X
9 Y
10 Z
11 (1, 5)
12 (3, 3)
13 (2, 8)
14 (2, 3)
15 (2, 2)
16 (4, 4)
17 (6, 3)
18 (2, 6)
19 (5, 5)
20 W(3, 2), H(8, 2), Y(5, 10)
6-6 Dilations
(page 110)
1 (3) (4, 10)
2 (2) 1 _
2
3 (4) (x, y) → (2x, 2y)
4 (4) 20 square inches
5 (6, 9)
9
6 ( 3, _
2 ) or (3, 4.5)
7 (4, 6)
For problems 8–11, see figure below.
8 A(0, 8), B(2, 2), C(6, 4)
9 A(0, 3), B ( 3 _ ,
4 3 _
4 ) , C ( 2
1 _ , 1
4 1 _
2 )
10 A(0, 12), B(3, 3), C(9, 6)
6-6 Dilations 31

11 A(0, 8), B(2, 2), C(6, 4)
(8)
10
9
8
7
6
5
A
4
3
C
B
2
1
109 8 7 6 5 4 3 2 1
1
2
3
1 2
(9)
3 4 5 6 7 8 9 10
4
5
6
(11)
7
8
9
10
11
12
13
14
(10)
y
12 (1, √ 3 )
13 ( √ 2 _ ,
2 √ 2 _
2 )
14 (3a, 3b)
15 k 1 _ ; (2, 4) → (1, 2)
3
16 k 1 _ ; (2, 4) → (1, 2)
2
17 (3, 12)
18 G(1, 1), N(4, 1), A(4, 1), T(1, 1)
6-7 Properties Under
Transformations
(pages 111–113)
1 (2) dilation
2 (4) dilation
3 (2) translation
4 (2) r y x
5 (3) dilation D 1
6 r y-axis
7 T 5, 0
8 R 270 or R 90
9 T 4, 3
32 Chapter 6: Transformations and the Coordinate Plane
x
10
11
12
13
8
7
6
5
4
3
2
y
1
x
8 7 6 5 4 3 2 1
1
K'(2, 2)
1 2 3 4
K(2, 2)
5 6 7 8
M'(5, 2)
2
3
M(5, 2)
4
Z'(3, 4)
5
6
7
8
Z(3, 4)
direct isometry
y
8
7
6
5
4
3
K'(1, 1) 2
1
M'(2, 1)
x
8 7 6 5 4 3 2 1 1 2 3 4
1 Z'(0, 1)
5 6 7 8
2 K(2, 2)
3
M(5, 2)
4
5
6
7
8
Z(3, 4)
direct isometry
5
4
3
2
y
1
x
5 4 3 2 1
1
1 2 3 4 5 6 7 8 9 10 11 12
K(2, 2 2) M(5, 2)
3
4
5
K'(4, 4)
Z(3, 4)
W'(10, 4)
6
7
8
9
10
not an isometry
A'''
R"
A" R'''
C'''
C"
Z'(6, 8)
y
10
9
8
7
6
5
4
3
2
1
109 8 7 6 5 4 3 2 1
1
C
2
1 2 3 4
R
5 6 7 8 9 10
3
4
A
5
6
C'
R'
7
8
9
10
A'
a C(3, 5), A(4, 7), R(7, 4)
x

14
15
b C(3, 5), A(4, 7), R(7, 4)
c C(5, 3), A(7, 4), R(4, 7)
d c
D"
D'''
I"
K"
2
1
K'
11109 8 7 6 5 4 3 2 1
1
I'''
2
K'''
3
4
5
1 2 3 4 5 6 7 8
y
12
11
10
9
8
7
6
5
4
3
I
K
D
I'
D'
9 10 11
a K(6, 1), I(9, 2), D(10, 7)
b K(6, 1), I(9, 2), D(10, 7)
c K(1, 2), I(4, 1), D(5, 4)
10
9
8
7
6
A"
R'''
5
4
3
R"
A'''
2
1
J'''
J"
J' J
109 8 7 6 5 4 3 2 1
1
2
3
1 2 3 4
A'
R'
5 6 7
A
8 9 10
4
5
R
a J(1, 0), A(3, 1), R(2, 2)
b J(0, 2), A(2, 6), R(4, 4)
c J(2, 0), A(6, 2), R(4, 4)
6-8 Composition of
Transformations
(pages 117–119)
1 (3) R 200
2 (2) ___
AT
3 (1) (x, y)
4 (3) r y x D 3
5 (1) a direct isometry
6 (2) (x, y)
7 (4) (3, 8)
8 (4) D
9 (3) T 4, 0
10 (1) (x, y)
11 (3) orientation
12 N
y
x
x
13 A
14 R 90
15 Any combination of three rotations, the sum
of whose angles is 100
16 Any combination of two rotations, the sum
of whose angles is 180
17 Glide reflection
18 (6, 1)
19 (2, 3)
20 (2, 7)
21
y
22
10
9
8
7
P'
M
y x
6
5
M'
4
3
P
2
C
1
C'
109 8 7 6 5 4 3 2 1
1
C"
2
3
4
1 2 3 4 5 6 7 8 9 10
M"
5
6
7
P"
8
9
10
C(2, 1), M(7, 5), P(4, 8)
C(2, 1), M(7, 5), P(4, 8)
d r y x
Y'
10
9
8
7
6
5
Y
X"
X
4
3
2
X'
Z'
1
Z
109 8 7 6 5 4 3 2 1
1
1 2 3 4 5 6 7 8 9 10
2
3
4
5
Y"
y x
6
7
8
9
10
Z"
X(5, 3), Y(2, 6), Z(7, 1)
X(3, 5), Y(6, 2), Z(1, 7)
d (1) rotation
Chapter Review (pages 119–121)
1 (1) I
2 (1) WOW
3 (3) parallel to the y-axis
4 (3) (5, 2)
y
Chapter Review 33
x
x

5 (3) (x, y) → (x, 2y)
6 (1) translation
7 (1) rotation
8 (1) A
9 (4) D
10 (2) (5, 4)
11 (3) reflection in the line y x
12 (2) (x, y) → (4x, 2y)
13 (4) y 0
14 (3) r x 1 r y-axis
15 (1) _ 1
2
16 (4)
17 (3) (6, 1)
18 5
19 13
20 2 √ 10
21 13
22 3 √ 10
23 (1, 1)
24 (2, 6)
25 (3, 0)
26 (4, 1)
27 r y x (1, 8) (8, 1)
Polygon Sides and
Angles
7-1 Basic Inequality
Postulates
(pages 125–126)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
34 Chapter 7: Polygon Sides and Angles
28 (4, 9)
29 a (2, 2)
b (4, 0)
c (2, 2)
d (2, 2)
e (3, 2)
30 a and c
H
C
10
8
6
4
2
14 12 10 8 6 4 2
2 4 6 8 10 12
T"(8, 2)
14
2
b r y x
C"(2, 10)
4
6
8
10
y
T
C'(4, 3)
CHAPTER
7
T'(9, 1)
H'(10, 3)
in each problem. (These solutions are intended
to be used as a guide— other possible solutions
may vary.)
1 1. m1 m2
2. m2 m1
3. m2 m3
4. m3 m2
5. m3 m1 ( Transitive postulate of
inequality)
6. m1 m3
H"(10, 10)
x

2 1. PQ PS
2. ___
PQ −−
PS
3. 1 2
4. m1 m2
5. m1 m3
(Isosceles triangle
theorem)
6. m2 m3
3 1. CE BD
2. CF BF
3. CE CF FE
4. CF FE BD
5. BD BF FD
(Substitution
postulate of
inequality)
6. CF FE BF FD
7. BF FE BF FD
8. BF FE BF (Subtraction
BF FD BF postulate of
or FE FD
4 1.
inequality)
___
AE ___
EB
2. ABE BAE
3. mABE mBAE
4. mDAB mCBA
(Isosceles triangle
theorem)
5. mDAB mCAD mBAE
6. mCAD mBAE mCBA
7. mCBA mABE mDBC
8. mCAD mBAE
mABE mDBC
9. mCAD mABE
mABE mDBC
10. mCAD mABE (Subtraction
mABE mABE postulate of
mDBC mABE
or mCAD mDBC
5 1. C is the midpoint of
inequality)
−−
AB .
2. AC CB
3. AB AC CB
4. AB AC AC
or AB 2AC
5. AB _ AC
2
6. F is the midpoint of −−
DE .
7. DF DF
8. DE DF FE
9. DE DF DF
or DE 2DF
10. DE _ DF
2
11. AC DF
12. AB _ DF
2
13. AB _
DE
2 _
2
14. AB _ 2
DE
2 _
2 2 (Multiplication
or AB DE postulate of
equality.)
6 1. RP 3RS
2. RP _ 3RS
3 _
3
or RP _ RS
3
3. RQ 3RT
4. RQ _ 3RT
3 _
3
or RQ _ RT
3
5. RS RT
6. RP _ RT
3
7. RP _ RQ
3 _
3
8. RP _ RQ
3
3 _ 3 (Multiplication
3
or RP RQ postulate of
equality.)
7 1. I is the midpoint of −−
EH .
2. EH EI IH
3. EI IH
4. EH EI EI
EH 2EI
5. J is the midpoint of −−
EF .
6. EF EJ JF
7. EJ JF
8. EF EJ EJ
or EF 2EJ
9. EH EF
10. 2EI EF
11. 2EI 2EJ
12. EI EJ (Division postulate
of equality)
8 1. ___
BD bisects ABC.
2. ABD DBC (Definition of
bisector)
3. mABD mDBC
4. mABC mABD mDBC
5. mABC mABD mABD
or mABC 2mABD
6. −−
BD bisects ADC.
7. ADB BDC
8. mADB mBDC
9. mADC mADB mBDC
10. mADC mADB mADB
or mADC 2mADB
7-1 Basic Inequality Postulates 35

11. mABC mADC
12. 2mABD 2mADB
13. mABD mADB
(Division postulate of equality)
9 1. AE 1 _ AC
3
2. AE 3 1 _
AC 3 or 3AE AC
3
(Multiplication postulate of equality)
3. AD 1 _ AB
3
4. AD 3 1 _
AB 3
3
or 3AD AB
5. AC AB
6. 3AE 3AD (Substitution postulate)
7. AE AD (Division postulate of
inequality)
10 1. ___
AE is the median from A to ___
BC .
2. E is the midpoint of ___
BC .
(Definition of median)
3. BE EC (Definition of midpoint)
4. BC BE EC
5. BC BE BE
or BC 2BE
6. ___
CD is the median from C to ___
AB .
7. D is the midpoint of ___
AB .
(Definition of median)
8. AD DB
9. AB AD DB
10. AB AD AD
or AB 2AD
11. AB BC
12. 2AD 2BE
13. AD BE (Division postulate of
inequality)
7-2 The Triangle Inequality
Theorem
(page 128)
1 Yes
2 No
3 Yes
4 No
5 Yes
6 No
36 Chapter 7: Polygon Sides and Angles
7 Yes
8 Yes
9 Yes
10 Yes
11 1 S 9
12 6 S 10
13 2.5 S 5.5
14 0 S 12
15 1 1 _ S 3
2 1 _
2
7-3 The Exterior Angles
of a Triangle
(pages 130–132)
1 (2) isosceles
2 (4) right
3 a CAD
b ABC and ACB
c mCAD mABC; mCAD mACB
4 a True
b True
c True
d True
5 6 and 9
6 5, 9, 8, and 10
7 7 and 10
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
8 1. m1 m3 (Exterior angle
theorem)
2. m3 m5 (Exterior angle
theorem)
3. m1 m5 (Transitive postulate
of inequality)
9 1. ___
AB ___
CD
2. 1 is a right angle.
3. AED is a right angle.
4. AED 1 (Right angles are
congruent.)

5. mAED m1
6. mAED m2 (Exterior angle
theorem)
7. m1 m2
10 1. mDCB mCBA (Exterior angle
theorem)
2. mCBA mCBM mMBA
3. mCBA mMBA
4. mDCB mMBA (Transitive postulate
of inequality)
11 12x 8 (4x 6) (7x 6)
x 8
12 12x 3 2 (5x 5) (5x 5)
x 11
13 Let x be mB, then mA is 3x.
3x x 104
x 26
mA 3x 3(26) 78
14 If the vertex angle of the triangle measures
130, each base angle measures 25 and
each exterior angle at the base measures
180 25 155.
7-4 Inequalities Involving
Sides and Angles of
Triangles
(page 133)
1 ZXY
2 PRQ
3 ___
AC
4 GFH
5 ___
KL
6 ACB
7 ___
YZ
8 mDGE mDEF mGDE, so
mDGE mGDE and DE EG.
9 PQS QSP, but QSP SQR, so
PQS SQR
10 m4 m3 m1, so m4 m1
Chapter Review (pages 133–136)
1 (3) p m p n
2 (3) 28
3 (2) an obtuse angle
4 True. Transitive postulate of inequality
5 True. Additive postulate of inequality
6 True. Postulate of inequality
7 False
Note: Since there are many variations of proofs,
the following is simply one set of acceptable statements
to complete each proof. Depending on the
textbook used, the wording and format of reasons
may differ, so they have not been supplied for the
method of congruence applied in each problem.
(These solutions are intended to be used as a
guide—other possible solutions may vary.)
8 1. m2 m3
2. m1 m2 (Exterior angle
theorem)
3. m1 m3 (Transitive postulate
of inequality)
9 1. BEST is a parallelogram.
2. BT BE
3. ___
BT ___
ES (Definition of a
parallelogram)
4. BT ES
5. ES BE (Substitution
postulate of
inequality)
10 1. ___
GS ___
GD
2. NPS RSP (Isosceles triangle
theorem)
3. mNPS mRSP
4. mISR mIPN
5. mISR mRSP mIPN mRSP
(Addition postulate
of inequality)
6. mISP mRSP mIPN mNPS
7. mISP mIPS (Postulate of
inequality)
11 1. mBCE mDBC
2. mACE mABD
3. mBCE mACE (Addition
mDBC mABD postulate
inequality)
4. mACB mABC (Postulate of
inequality)
12 1. ___
AD ___
DC
2. ACD CAD (Isosceles triangle
theorem)
3. mACD mCAD
4. mDAB mDCB
5. mDAB mACD ( Subtraction
mDCB mACD postulate of
inequality)
Chapter Review 37

6. mDAB mCAD
mDCB mACD
7. mCAB mACB (Postulate of
inequality)
13 1. PQRS is a parallelogram.
2. SP RQ (Definition of
parallelogram)
3. ST TP RQ
4. ST TP QU RU
or ST RU QU TP
5. TP QU
6. 0 QU TP (Subtraction
postulate of
inequality)
7. 0 ST RU
8. ST RU (Addition
postulate of
inequality)
14 ___
ST
15 ___
AB
16 ___
AB
17 ___
BC
18 −−
DE
19 AD BD
20 mB 120
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
21 1. ___
AD bisects CAB.
2. ___
BE bisects CBA.
3. mDAB mEBA
4. mDAB 2 mEBA 2 (Multiplication
postulate of
inequality)
5. mCAB 2(mDAB) (Definition
of bisector)
6. mCBA 2(mEBA) (Definition
of bisector)
7. mCAB mCBA
38 Chapter 7: Polygon Sides and Angles
22 1. CF AE
2. CF 2 AE 2 (Multiplication postlate
of inequality)
3. F is the midpoint of ___
CD .
4. CF 2 CD (Definition of
midpoint)
5. CD AE 2
6. E is the midpoint of ___
AB .
7. AB AE 2 (Definition of
midpoint)
8. CD AB (Multiplication postulate
of inequality)
23 1. AC CB
2. AC _
2
CB
_
2
3. AD AC _
2
4. AD CB _
2
5. BE CB _
2
(Division postulate of
inequality)
(Definition of
midpoint)
(Definition of
midpoint)
6. AD BE
24 Let A be the acute angle and let B be
its supplement. mB 180 mA. Since
mA 90, mB 90 and is therefore
obtuse by the subtraction postulate of
inequality.
25 Let C be the complement of A, and
let D be the complement of B. mC
90 mA and mC mD by the subtraction
postulate of inequality.
26 m1 m3 because an exterior angle is
greater than either nonadjacent interior angle
and m1 m2. So m2 m3 by the
substitution postulate of inequality.
27 mABC mABD because a whole is
greater than its parts. mBAC mABC
because they are opposite congruent sides
of a triangle. So mBAC mABD by the
substitution postulate of inequality, and
DB DA because the greater side lies
opposite the greater angle.

28 mBDA mACB because a whole is
greater than its parts. mACB mBCA
because they are opposite congruent sides
of a triangle. So mBDA mBCA by the
substitution postulate of inequality, and
AB AD because the greater side lies
opposite the greater angle.
29 Draw diagonal −−
AC , forming two triangles.
In ABC, mBCA mBAC because
they are opposite congruent sides. In
ADC, mACD mCAD because the
greater angle is opposite the greater side.
mBCA mACD mBAC mCAD
or mBCD mBAD by the addition postulate
of inequality.
30 Look at the two right triangles formed with
common side −−
EG . mD mF by the subtraction
postulate of inequality, so DE EF
because the greater side lies opposite the
greater angle.
CHAPTER
8
8-1 The Slope of a Line
(pages 141–144)
1 (2) (0, 4)
2 (2) It has an x-intercept of 2.
3 (4) y 3
4 (3) a y-intercept of 5
5 (3)
6 (2) x 1
7 m 5 _ ; negative
3
8 m 2; positive
9 m 3 _ ; positive
4
31 m2 180 114 66
m1 180 50 66 64
32 x 4 5 105
x 60
33 y 18 0 110 70
x 7 0 130
x 60
34 True
35 False
36 True
37 False
38 False
Slopes and Equations
of Lines
10 Undefined
11 m 2; negative
12 m 3 _ ; positive
2
13 m 0; zero
14 m 1 _ ; positive
2
15 m 3; positive
16 a y 5
b y 1
c x 2
d x 10
e y 4.5
8-1 The Slope of a Line 39

g e
17 a _
f d
b Undefined
c a _
b
d 1
18 m 3 _
2
19 y 3
20 y 14
21 y 4
22 x 0
23 x 15
24 x 8
25 x 4
26 a m 1
b m 1
27 a–c Not collinear
28 m 4
29 15 feet
30 a m −−
1
AB _ , m
3 −−−
CD
b Not collinear
31 m −−
AB
2 _
1
_
, m
3 −−−
AD
4
_
11
, m
3 −− is undefined, m −−− 0. The
BC AD
triangle is a right triangle because −−
BC is
parallel to the x-axis and −−
CA is parallel to the
y-axis, making these sides perpendicular to
each other.
8-2 The Equation of a Line
(page 147)
1 a No
b Yes
c Yes
2 a y x 6; m 1, b 6
b y 3x 5; m 3, b 5
c y 3x; m 3, b 0
d y 2x 4; m 2, b 4
e y 2 _ x 4; m
3 2 _ , b 4
3
f y 3 _ x 3; m
2 3 _ , b 3
2
g y 3x 5; m 3, b 5
h y 1 _ x
2 9 _ ; m
2 1 _ , b
2 9 _
2
3 a x-intercept 7, y-intercept 7
b x-intercept 4, y-intercept 8
c x-intercept 3, y-intercept 6
40 Chapter 8: Slopes and Equations of Lines
d x-intercept 2, y-intercept 4
e x-intercept 0, y-intercept 0
f x-intercept 6, y-intercept 4
4 a y 2x 5
b y 1 _ x 2
2
c y 3
d y x 4
5 a y x 3
b y 2
c y 1 _ x 5
2
d y 5 _ x 3
2
e y 3 _ x 2
2
f y 3 _ x 5
5
6 a y 7 _ x 5
2
b y 2x 2
c y 5x 10
d y 3 _ x 3
2
e y 4x 13
f y 6 _ x 6
5
7 a y x 1
b y 2 _ x 2
3
c y 5
d y 2x 1
e y 2x 11
f y 1 _ x
3 1 _
3
8-3 The Slopes of Parallel
and Perpendicular Lines
(page 150)
1 (4) Line l has a negative slope.
2 (1) 5
3 a Slope parallel 9 _
7
Slope perpendicular 7 _
9
b Slope parallel 0
No slope for perpendicular line
c Slope parallel 7 _
8
Slope perpendicular 8 _
7

d Slope parallel 2 _
5
Slope perpendicular 5 _
e Slope parallel 9 _
10
Slope perpendicular 10 _
9
f Slope parallel 13
Slope perpendicular 1 _
13
4 a Perpendicular; slopes are negative
reciprocals
b Perpendicular; slopes are negative
reciprocals
c Neither; slopes are neither equal nor
negative reciprocals
d Neither; slopes are neither equal nor
negative reciprocals
e Perpendicular; lines with a slope of 0 are
perpendicular to lines with no slope
f Parallel; slopes are the same
5 a Yes
b Yes
c No
6 a y 1 _ x
4 17 _
4
b y 2x 7
c y 1 _ x 4
5
7 a Right triangle
b Not a right triangle
8-4 The Midpoint of a Line
Segment
(pages 153–154)
1 (2) (1.5, 1)
2 (4) (2.5, 2)
3 (2) (2, 0.1)
4 (1) (8, 0)
5 (4) (17, 20)
6 (4) (13, 13)
7 (5a, 5r)
8 (3x, 5y)
9 (10, 11)
2
10
y
8
7
6
5
4
3
2
G
1
8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
D
L
A
A rectangle is formed.
M GL (0, 1), M LA (2, 1), M DA (2, 3),
M GD (0, 3)
11 (0, 2)
12 a −−
BD is parallel to −−
EF because their slopes
are 3 _ .
4
b EF 1 _ BD
2
5 1 _ (10)
2
5 5
13 a m 5 _
2
b m 2 _
5
5
c M ( 4, _
2 )
d y 5 _ x
6 35 _
6
14 y 4x 9
15 y 1 _ x
5 4 _
5
8-5 Coordinate Proof
(pages 163–165)
1 The triangle is an isosceles triangle because
WI WN √ 40 .
2 Slopes of −−−
NA and −−−
AQ are negative reciprocals
proving these two lines form a right
angle by being perpendicular; therefore,
NAQ is a right triangle.
3 AM CM BM √ 26 ; the midpoint of the
hypotenuse is equidistant from all three
vertices.
4 The slopes are negative reciprocals so the
median is also the altitude.
x
8-5 Coordinate Proof 41

5 BIG is isosceles because it has two congruent
sides, BI IG √ 50 ; and BIG is
a right triangle because −−
BI is perpendicular
to −−
IG .
6 Two sides ( −−−
QR and −−
PS ) have the same slope
of 2 _ and the other two sides (
3 −−
OP and −−
RS )
have the same slope of 3. PQRS is a
parallelogram.
7 SAND is a parallelogram because the diagonals
bisect each other at the point (1.5, 0.5).
8 LEAP is a parallelogram because all of the
sides measure √ 50 and are congruent.
9 ABCD is a parallelogram because the diagonals
bisect each other at the point (2.5, 0.5).
10 a 6 √ 5 and 2 √ 13
b The diagonals meet at their midpoint
(1, 1).
c BETH is a parallelogram because the
diagonals bisect each other.
d BETH is not a rectangle because the
diagonals are not congruent.
11 a NICK is a parallelogram; both pairs of
opposite sides are parallel.
b NICK is not a rhombus; the diagonals are
not perpendicular.
12 The figure KATE is a square. KATE is a rectangle
because the diagonals bisect each other
and are the same length. KATE is a square
because two adjacent sides are congruent.
13 ABCD is a rhombus. ABCD is a parallelogram
because the diagonals bisect each other.
ABCD is a rhombus because two adjacent
sides are congruent.
14 a Slopes of −−
SU and −−
UE are negative reciprocals
proving these two lines form a right
angle by being perpendicular; therefore,
SUE is a right triangle.
b SU 5 and UE 10
15 a NORA is a parallelogram because the
diagonals bisect each other. NORA is a
rhombus because two adjacent sides are
congruent.
b NORA is not a square because it does not
contain a right angle.
42 Chapter 8: Slopes and Equations of Lines
16 a JACK is a trapezoid because it has only
one pair of opposite sides parallel;
slope of −−
JA slope of −−
CR 1.
b JACK is not an isosceles trapezoid because
the legs are not congruent.
17 MARY is a trapezoid because it has only
one pair of opposite sides parallel; slope of
−−−
MA slope of −−
RY 0. MARY is an isosceles
trapezoid because the legs are congruent;
AR YM 5.
18 C(a, 0)
19 P(b a, c)
20 E(a, 0), F(a, 2a), G(a, 2a)
21 C(a, b r)
22 M(a c, b)
23 T(a, 2a)
24 R(a b, c)
25 Yes
26 Slope of −−
AC is 1 and the slope of −−
BD is 1.
Since the slopes are negative reciprocals,
−−
AC −−
BD .
27 a Midpoint of −−
BC is (a, b).
28 a
b MA MB MC √ a 2 b 2
T(0, 0)
y
E(2x, 2y)
Q(4x, 0)
x
b A(x, y), B(3x, y), C(2x, 0)
c The length of median −−
TB is √ 9 x 2 y 2 .
The length of median −−
EC is 2y. The length
of median −−−
QA is √ 9 x 2 y 2 .
d TEQ is an isosceles triangle.
29 TA TR √
( c _
2 ) 2
d 2
30 −−
JA and −−
NE are parallel to the x-axis and
each slope is 0; thus they are parallel to each
other. Slope of −−−
AN slope of −−
EJ c _ ; thus
b
they are parallel. Therefore, JANE is a
parallelogram.
31 Midpoint of −−
AC midpoint of
−−
BD
a r ( _ ,
s
2 _
2 )
32 TA EM √
(a b) 2 c 2

8-6 Concurrence of the
Altitudes of a Triangle
(the Orthocenter)
(page 168)
1 (4) obtuse
2 (4) at one of the vertices of the triangle
Exercises 3–9: Check students’ graphs.
3 (2.2, 1.2)
5
4 ( _ , 4) 3
5 (1, 0)
6 (6, 6.5)
7 ( 2,
22 _
3 )
8 (1, 1)
9 (1, 0.5)
10 a Slope of −−
AB is 4 _ . Slope of
3 −−
BC is 3 _ . Since
4
the slopes are negative reciprocals, the
segments are perpendicular and B is the
right angle.
b −−
AB and −−
BC can each be altitudes. B is the
endpoint of the altitude drawn from B
to AC.
Chapter Review (pages 168–171)
1 (2) y 3 _ x 4
4
2 (2) (0, 3)
3 (4) (1, 9)
4 (2) y 5x 1
5 (1) y 3x 3
6 No. Sub-in: 3(5) 4(2) 0
7 x-intercept is (2, 0) and y-intercept is (0, 3).
Distance is √ 13 .
8 x-intercept is (15, 0) and y-intercept is (0, 3).
Distance is 15.3.
9 a 10
10 B(10, 10)
11 Slope of ___
OB is 1 _ . Slope of
2 ___
AC is 2. Their
product is ( 1 _
2 ) (2) 1.
12 a (0, 3) b (4, 0) c (8, 3)
13 a m 3 b x-intercept is (4, 0)
c y-intercept is (0, 12)
d (3, 3) is not on the line.
14 a neither b vertical c vertical
d neither e horizontal f vertical
15 a 1 _ b 2 c
2 3 _
4
d 2 _ e
3
2 _ f
3
a _
d
16 a no slope b 0 c 1
d 7 _ e
d b
2 _
c a
f
k _
h
17 a 7 b 2 c 14 d 2
e 6 f 5 g 4
18 a not collinear
b collinear
c collinear
d collinear
e collinear
f collinear
19 a y 3
b y 3 _ x
2 9 _
2
c y 1 _ x
3 5 _
3
d y 2 _ x
3
e y 3 _ x
2 19 _
2
20 a AB AC 2 √ 10
b Midpoint of ___
BC is (2, 1). Slope of the
median from A to ___
BC slope of altitude
from A to ___
BC 1.
21 a (3, 2) b (2, 3) c (6.5, 8.5)
22 Slope of ___
DR slope of ____
AW 1 _
. Slope of
3 ___
RA is not equal to the slope of ____
DW .
___
RA ____
DW 5
23 a NO 5 √ 2 and OP 2 √ 5
b Slope of ___
EO is 1 _ . Slope of
3 ___
PN is 3.
Slopes of perpendicular lines are negative
reciprocals of each other.
24 A(0, 3), B(4, 2), C(5, 4), and D(1, 3). ABCD
is a parallelogram because both pairs of opposite
sides are parallel. Slope of ___
AB slope
of ___
CD 1 _ . Slope of
4 ___
BC slope of ___
DA 6.
25 QRST is a parallelogram because both pairs
of opposite sides are parallel. Slope of
___
QR slope of ___
ST b _
a
. Slope of ___
RS slope
of ___
TQ 0.
26 a–b Check students’ graphs.
c (2, 10)
Chapter Review 43

___ 27 a M KA (5, 1), M ___
AT (4, 3), M ___
TK (2, 1)
___ b m KA 1, m ___
AT 0, m ___
TK 2
c Slope of line perpendicular to ___
KA is 1.
There is no slope for the line perpendicular
to ___
AT . Slope of the line perpendicular
to ___
TK is 1 _ .
2
d Perpendicular bisector of ___
KA : y x 6
Perpendicular bisector of ___
AT : x 4
Perpendicular bisector of ___
TK : y 1 _ x
2
e (4, 2)
Parallel Lines
9-2 Proving Lines Parallel
(pages 177–179)
1 none
2 a b, c d
3 none
4 b c
5 a b
6 a c
7 b c
8 l m
9 a and b
10 c and d
11 a and b
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
44 Chapter 9: Parallel Lines
28 a y 2x 5
b (6, 8)
c y 1 _ x 5
2
CHAPTER
9
12 1. 1 3
2. 2 4
3. 1 2 3 4 (Addition postulate
of equality)
4. ABG DEG
5. −−
ED −−
BA (If two lines are cut by a transversal
forming a pair of congruent
alternate interior angles,
the two lines are parallel.)
13 1. ___
AF ___
CD
2. −−
FC ___
FC
3. −−
AF −−
FC ___
CD ___
FC
4. ___
AC ___
DF
5. −−
BC ___
EF
6. ___
BC ___
AD
7. BCF is a right angle. (Definition of
right angle)
8. ___
EF ___
AD
9. EFD is a right angle.
10. BCF EFD (Right angles are
congruent)
11. ABC DEF (SAS SAS)

12. BAC EDF (CPCTC)
13. ___
ED ___
AD (Alternate interior angles are
congruent)
14 1. ___
BE bisects ABC.
2. 1 EBC
3. m1 mEBC
4. ___
CE bisects DCB.
5. 2 ECB
6. m2 mECB
7. m1 m2 90
8. mEBC mECB 90 (Substitution
postulate)
9. m1 m2 mEBC mECB
10. m1 mEBC 90
11. m2 mECB 90
12. ABC and DCB are right angles.
13. BA ___
BC
14. CD ___
BC
15. BA CD (Two lines perpendicular to the
same line are parallel.)
15 1. 1 3
2. m1 m3
3. 2 4
4. m2 m4
5. m1 m2 m3 m4 360
6. m1 m1 m2 m2 360
or 2(m1) 2(m2) 360
7. m1 m2 180 (Division
postulate)
8. −−−
AD −−
BC (Interior angles on the same
side of the transversal are
supplementary.)
9. m1 m1 + m4 m4 360
or 2(m1) 2(m4) 360
10. m1 m4 180
11. ___
AB ___
CD
16 1. D is the midpoint of ___
CF and of ___
BE .
2. ___
CD ___
DF
3. ___
BD ___
DE
4. ___
FC ___
FC
5. CBD FDE (Vertical
angles are
congruent.)
6. CBD FDE (SAS SAS)
7. EFD DCB (CPCTC)
8. ___
AC ___
FE (Alternate interior angles are
congruent.)
17 1. 1 2
2. AFC DCF (Supplementary angles
of congruent angles are
congruent.)
3. ___
EF ___
CB
4. ___
FC ___
FC
5. ___
EF ___
FC = ___
BC ___
FC
6. ___
EC ___
BF
7. ___
AF ___
CD
8. AFB DCE (SAS SAS)
9. ABF DEC (CPCTC)
10. ___
AB ___
ED (Alternate interior angles
are congruent.)
18 1. ___
AE ___
FC
2. ___
EF ___
EF
3. ___
AE ___
EF ___
FC ___
EF
4. ___
AF ___
CE
5. ___
DE ___
AC
6. DEA is a right angle.
7. ___
BF ___
AC
8. BFC is a right angle.
9. DEA BFC (Right angles are
congruent.)
10. ___
DE ___
BF
11. AFB CED (SAS SAS)
12. ______
AEFC
13. BAF DCE (CPCTC)
AB ___
DC
14. ___
9-3 Properties of Parallel
Lines
(pages 182–185)
1 (1) same-side exterior angles
2 (2) 2, 3, 6, 7, 10, 11, 14, 15
3 m1 m4 m6 m7 60;
m2 m3 m5 120
4 m1 m4 m5 m8 135;
m2 m3 m6 m7 45
5 ma md mg 65;
mb mc me mf 125
6 w y 70; x z 110
7 a 8x 6x 3 0 180
x 15
b 2x 1 0 5x 47
x 19
m3 2(19) 10 48
9-3 Properties of Parallel Lines 45

8 106
9 mA 75, mC 67
10 57
11 60
12 1. Perpendicular lines form right angles.
2. Right angles measure 90.
3. When two parallel lines are cut by a
transversal, corresponding angles are
congruent.
4. Congruent angles have equal measure.
5. Transitive property of congruence (3, 5)
6. If an angle measures 90, it is a right
angle.
7. If two lines intersect to form a right angle,
they are perpendicular.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
13 1. ___
AB
2. 2 is supplement of 1.
3. 2 is supplement of 3.
4. 1 3 (If two angles are supplements
of the same angle, then they
are congruent.)
5. ___
BC ___
AD (When two lines are cut by
a transversal creating corresponding
angles, the lines are
congruent.)
14 1. ABC
2. ___
AB ___
BC
3. BAC BCA (Isosceles triangle
theorem)
4. _____
FHD ___
BC
5. FDA BCA (Corresponding
exterior angles are
congruent.)
6. _____
EHG ___
AB
7. BAC GEC (Corresponding
exterior angles are
congruent.)
8. GEC FDA
9. ___
HE ____
HD
10. EHD is isosceles. (Definition of
isosceles triangle)
46 Chapter 9: Parallel Lines
15 1. ___
DE ___
AC
2. 1 2 (Corresponding exterior
angles are congruent.)
3. 3 4
4. 2 3
5. 1 4 (Transitive property of
congruence)
16 1. ___
AC intersects ___
BD at E.
2. ___
AE ___
ED
3. A D
4. ___
AD ___
BC
5. A C (Alternate interior angles are
congruent.)
6. B D (Alternate interior angles are
congruent.)
7. B C (Transitive property of
congruence)
8. ___
AE ___
CE (Isosceles triangle
theorem)
9. ___
BE ___
DE
10. ___
AC ___
BD (Addition postulate)
17 1. n m
2. ABE and BAD are supplementary.
(Two interior angles on the
same side of the transversal
are supplementary.)
3. mABE mBAD 180
4. 1 _ mABE
2 1 _ mBAD 90
2
(Division postulate)
5. ___
BC bisects ABE.
6. mABC 1 _ mABE
2
(Definition of angle bisector)
7. ___
AC bisects BAD.
8. mACB 1 _ mBAD
2
( Definition of angle bisector)
9. mABC mACB 90
10. mBCA mABC mACB 180
11. mBCA 90
12. BCA is a right angle.
13. ___
BC ___
AC (Perpendicular lines form
right angles.)
18 1. r m and a b
2. 4 and 3 are supplementary.
3. 2 and 3 are supplementary.
4. (a) 4 2 (Supplements of the same
angle are congruent.)
5. 4 and 5 are supplementary.

6. 2 and 1 are supplementary.
7. (b) 5 1 (Supplements of the same
angle are congruent.)
19 a (x 12) (3x) 180
x 42
m1 m3 m5 54
m2 m4 m6 126
b (2x) (2x 20) 180
x 50
m1 m3 m5 100
m2 m4 m6 80
c (7x 65) (5x 5)
x 30
m2 m5 145
m1 m3 m4 m6 35
20 a 5
b 7
c 6 and 8
d m5 60
e m4 90
21 Since the corresponding angles are congruent,
the halves of each are congruent. They
form new congruent corresponding angles
cut by the same transversal. The bisectors are
therefore parallel.
22 Draw a diagonal line, forming two congruent
triangles by SAS. The other two sides of
the quadrilateral and remaining angles are
congruent by CPCTC. Therefore, since the
diagonal is a transversal for these sides, the
sides are parallel.
9-4 Parallel Lines in the
Coordinate Plane
(pages 187–188)
1 a 4
b 7 _
3
c 2 _
5
d x _
a
2 a 1
b 3 _
5
c 2 _
5
3 a 1 _
4
b 7
c 1
d 3 _
2
4 a perpendicular
b parallel
c perpendicular
d parallel
e perpendicular
f neither
5 a 1
b 5 _
2
c 3 _
10
d 7 _
9
e 4 _
3
f 3 _
2
6 a 7
b 1
c 11
d no slope
e 0
f 1 _
2
___
7 a Slope AB 7, slope ___
1
CD _ ,
7 ___
AB ___
CD
___ b Slope AB 2, slope ___
1
CD _ ,
2 ___
AB ___
CD
8 y 5x 3
9 y = 1 _ x 3
5
10 y 3 _ x
2 7 _
2
11 y 1 _ x 3
3
12 y 4
13 y 2x 6
14 y 2 _ x 9
3
15 k = 1
16 (7, 5)
___ 17 Slope
4
AB _
3
___ , slope CD
3 _ . The slopes are
4
negative reciprocals, therefore −−
AB and −−−
CD
are perpendicular and ABC is a right
triangle.
18 Opposite sides have the same slope.
___ m PQ m ___
1
RS _
___ and m PS 3 m ___ 5 QR
19 H(5, 6)
20 a D(4, 3)
9-4 Parallel Lines in the Coordinate Plane 47

9-5 The Sum of the
Measures of the Angles
of a Triangle
(pages 193–195)
1 (4) scalene
2 (3) obtuse
3 (3) 120
4 (2) right
5 (3) 112
6 a base angles are 30, vertex angle is 120
b base angles are 60, vertex angle is 60
c base angles are 52, vertex angle is 76
d base angles are 36, vertex angle is 108
e base angles are 20, vertex angle is 140
7 a base angles are 50, exterior angle is 130
b base angles are 40, exterior angle is 140
c base angles are 54, exterior angle is 126
d base angles are 60, exterior angle is 120
e base angles are 80, exterior angle is 100
8 base angles are 74, vertex angle is 32
9 mP 27, mQ 45, mR 108
10 base angles are 28, vertex angle is 124.
11 exterior angle is 30, base angles are 15,
vertex angle is 150
12 18, 54, 108
13 99, 45, 36
14 mc 35
15 mB 78
16 mx 150
17 mx 30
18 m1 150
19 mD 20
20 md 125
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
21 1. A C
2. ___
BD bisects ABC.
3. ABD CBD
4. ___
BD ___
BD
5. ABD CBD (AAS AAS)
48 Chapter 9: Parallel Lines
6. ADB CDB (CPCTC)
7. ADB is supplementary to CDB.
8. ADB and CDB are right angles.
9. −−
BD −−
AC (Segments forming right
angles are perpendicular.)
22 1. A C
2. ___
BD ___
AC
3. BDA and BDC are right angles.
4. BDA BDC
5. ___
BD ___
BD
6. BDA DBC (AAS AAS)
7. ABD CBD (CPCTC)
8. ___
BD bisects ABC. (Definition of angle
bisector)
23 Both ABC and DEC share C. Since the
remaining angles are the same as well, 1
and B are congruent corresponding angles.
Therefore, ___
BA ___
ED .
24 Compare ABC and DEC. Both are right
triangles with one pair of congruent acute
angles. Therefore, the other pair of acute
angles are also congruent, B CED. But
CED 1, vertical pairs. Then B 1
by the transitive postulate of congruence.
9-6 Proving Triangles
Congruent by Angle,
Angle, Side
(pages 198–200)
1 b and f
2 a Not sufficient. If the third side is congruent,
SSS. If included angles are congruent,
SAS.
b Sufficient, SSS
c Sufficient, hypotenuse-angle
d Not sufficient. If either pair of corresponding
angles are congruent, AAS.
e Sufficient, AAS
f Not sufficient. If any pair of corresponding
sides are congruent, AAS.
g Sufficient, hypotenuse-leg
h Sufficient, SAS

i Sufficient, SSS
j Sufficient, AAS
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
3 1. ___
AB ___
EF
2. ABC FED
3. ___
BC ___
DE
4. 1 2
5. I II (ASA ASA)
4 1. 1 4
2. B D
3. ___
AC ___
AC
4. I II (AAS AAS)
5 1. ___
AB ___
DE
2. DEC BAC (Alternate interior
angles are congruent.)
3. ABC EDC (Alternate interior
angles are congruent.)
4. C is the midpoint of ___
BD .
5. ___
BC ___
CD (Definition of
midpoint)
6. ABC EDC (AAS AAS)
6 1. B D
2. BEC DEA
3. ___
BC ___
AD
4. AED CEB (AAS AAS)
5. ___
AE ___
CE (CPCTC)
7 1. A E
2. ___
BC ___
DC
3. ___
AC ___
CE
4. ___
AC ___
BC (Subtraction
___
CE ___
DC postulate)
or ___
AB ___
DE
5. ___
BG ___
AE
6. BGA is a right (Definition of
angle. right angle)
7. ___
DF ___
AE
8. ____
DFE is a right angle.
9. BGA DFE (Right angles are
congruent.)
10. BGA DFE (AAS AAS)
BG ___
DF (CPCTC)
11. ___
8 1. ___
AB ___
EF
2. ABC FED
3. A F
4. ___
AC ___
DF
5. I II (AAS AAS)
9 1. ___
AB ___
CD
2. ___
CD ___
AB
3. CDA is a right angle.
4. ___
AE ___
CB
5. AEC is a right angle.
6. BAC BCA (Isosceles triangle
theorem)
7. ___
AC ___
AC
8. CDA AEC (AAS AAS)
9. ___
CD ___
AE (CPCTC)
10 1. E is the midpoint of ___
AC .
2. ___
AE ___
CE
3. ___
AF ___
BD
4. AFE is a right angle.
5. ___
CD ___
BD
6. CDE is a right angle.
7. AFE CDE
8. CED AEF (Vertical angles are
congruent.)
9. AFE CDE (AAS AAS)
10. ___
AF ___
CD (CPCTC)
11 1. ___
QR ___
SR
2. RQS RSQ (Isosceles triangle
theorem)
3. 1 2
4. ___
QS ___
QS
5. QTS SMQ (AAS AAS)
6. ____
QM ___
ST (CPCTC)
12 1. ___
BA ___
CD
2. ___
BA ___
AD
3. BAQ is a right angle.
4. ___
CD ___
AD (Two lines perpendicular
to the same line
are parallel.)
5. CDP is a right angle.
6. BAQ CDP
7. B C
8. ___
AP ____
QD
9. ___
PQ ___
PQ
10. ___
AD ___
PD (Addition postulate)
11. BAQ CDP (AAS AAS)
BA ___
CD (CPCTC)
12. ___
9-6 Proving Triangles Congruent by Angle, Angle, Side 49

9-7 The Converse of the
Isosceles Triangle Theorem
(pages 202–203)
1 2(3x 4) 2x 4 180
x 22
mA mC 70
mB 40
2 2(3x 3) 4x 16 180
x 19
mA mB mC 60
3 2x 180 82 82
x 8
4 (2x 14) (3x 2) (5x 8) 180
x 16
2x 14 2(16) 14 46
3x 2 3(16) 2 46
5x 8 5(16) 8 88
Since two angles are equal in measure,
ADC is isosceles.
5 mA mB 49
mx 180 49 49
my 49
mz 180 82 98
6 mQ 58
mx mz 64
my 116
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
7 1. A C
2. ___
BC ___
AB
3. ___
AD ___
EC
4. ABD CBE (SAS SAS)
8 1. ___
BD ___
AC
2. 1 ACB
3. 2 BAC
4. 1 2
5. ACB BAC (Substitution
postulate)
6. ABC is isosceles. (Definition of
isosceles triangle)
50 Chapter 9: Parallel Lines
9 1. 1 2
2. 3 4 (Supplements of
congruent angles are
congruent.)
3. ___
AB ___
BC
4. ABC is isosceles. (Definition of
isosceles triangle)
10 1. ___
BD ___
AE
2. ___
AC ___
CE
3. A E
4. A B (Two parallel lines are cut
by a transversal then the
corresponding angles are
congruent.)
5. E D (Two parallel lines are cut
by a transversal then the
corresponding angles are
congruent.)
6. B D (Transitive postulate of
congruence)
7. ___
BC ___
DC (Congruent angles imply
congruent sides.)
11 1. ___
BC ___
BD
2. BCD BDC
3. BDA BDE (Linear pairs of
congruent angles)
4. BAC BED (ASA ASA)
5. ___
AB ___
BE (CPCTC)
12 1. ___
AB ___
EB
2. BAE BEA
3. ___
BD ___
AE
4. BAE 1
5. BAE 2
6. 1 2 (Substitution
postulate)
13 1. ___
PQ ___
SR
2. ___
PQ ___
SR
3. QPR SRP (Alternate interior
angles are
congruent.)
4. ___
PR bisects QPS.
5. QRP RPS (Definition of
bisector)
6. SRP RPS (Transitive postulate
of congruence)
7. ___
PS ___
SR (Converse of the
isosceles triangle
theorem)

14 1. ___
DF ___
FE
2. 2 3
3. 1 4
4. ADF CEF (Linear pairs of congruent
angles)
5. ADF CEF (ASA ASA)
6. A C (CPCTC)
7. ABC is an isosceles triangle. (Definition
of an isosceles triangle)
15 1. ______
AEDC
2. 1 2
3. ___
BE ___
BD
4. AEB CDB
5. ___
AE ___
DC
6. ABE CDB (ASA ASA)
7. A C (CPCTC)
8. ABC is an isosceles triangle. (Definition
of an isosceles triangle)
9-8 Proving Right Triangles
Congruent by Hypotenuse-
Leg; Concurrence of Angle
Bisectors of a Triangle
(pages 207–208)
1 (a)
2 (d)
3 (c)
4 (b)
5 (a)
6 m1 24, m2 52, m3 104,
m4 52, m5 14, m6 114,
m7 14, m8 24, m9 142
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
7 1. ___
SQ ___
PR
2. ___
SA ___
PB
3. SQP and SAP are right angles.
4. SQP and SAP are right triangles.
SP ___
SP
5. ___
6. ___
SQ ___
SA
7. SQP SAP (HL HL)
8. SPQ APS (CPCTC)
9. ___
PT bisects RPB. (Definition of
angle bisector)
8 1. ___
AE ___
BC
2. ___
CD ___
AB
3. CDA and CEA are right angles.
4. CDA and CEA are right triangles.
5. ___
AC ___
AC
6. ___
AE ___
CD
7. ACE CAD (HL HL)
9 1. ___
AE ___
BC
2. ___
CD ___
AB
3. AEB and CDB are right angles.
4. AEB and CDB are right triangles.
5. ___
DB ___
EB
6. DBE DBE
7. AEB CDB (Leg–acute angle)
8. ___
AE ___
CD (CPCTC)
10 1. ___
BD ___
AC
2. ___
QS ___
PR
3. BDA and QSP are right angles.
4. BDA and QSP are right triangles.
5. ABC PQR
6. ___
AB ___
PQ (CPCTC)
7. A P (CPCTC)
8. BDA QSP (Hypotenuse–
acute angle)
9. ABD PQS (CPCTC)
10. mABD mPQS
11. ___
BD bisects ABC.
12. 1 _ mABC mABD mPQS
2
13. 1 _ mABC
2 1 _ mPQR (Division
2
postulate)
14. mPQS 1 _ mPQR (Transitive
2
postulate)
15. ___
QS bisects PQR. (Definition of
angle bisector)
11 1. ___
AB ___
CF , ___
DE ___
CF
2. ABF and CED are right angles.
3. ABF and CED are right triangles.
4. ___
CD ___
AF
5. ___
BE ___
BE
6. ___
CE ___
FB
7. ABF CED (HL HL)
AB ___
DE
8. ___
9-8 Proving Right Triangles Congruent by Hypotenuse-Leg; Concurrence of Angle Bisectors of a Triangle 51

12 1. ___
CE ___
BA , ___
BD ___
AC
2. CEA and BDA are right angles.
3. CEA and BDA are right triangles.
4. ___
AB ___
AC
5. ABC ABC
6. CEA BDA (Hypotenuse–acute
angle)
7. ___
CE ___
BD (CPCTC)
13 1. ___
AC ___
BF , ___
ED ___
BF
2. ACB and FDE are right angles.
3. ACB and FDE are right triangles.
4. ___
AC ___
ED
5. ___
BA ___
EF
6. ACB FDE (HL HL)
7. A E (CPCTC)
9-9 Interior and Exterior Angles of Polygons
(pages 210–212)
1 Polygon Number
of Sides
Number
of Triangles
Sum of
Interior
Angles
180(n 2)
Triangle 3 1 180(1) 180 180
_
3
Quadrilateral 4 2 180(2) 360 360
_
4
Pentagon 5 3 180(3) 540 540
_
5
Hexagon 6 4 180(4) 720 720
_
6
Heptagon 7 5 180(5) 900 900
_
7
Octagon 8 6 180(6) 1,080 1,080
_
8
Nonagon 9 7 180(7) 1,260 1,260
_
9
Decagon 10 8 180(8) 1,440 1,440
_
10
24-gon 24 22 180(22) 3,960 3,960
_
24
n-gon n n 2 180(n 2)
52 Chapter 9: Parallel Lines
Measure of
Each Exterior
180(n 2)
Angle _
n
Measure of
Each Interior
Angle
360
60 _ 120
3
360
90 _ 90
4
360
108 _ 72
5
360
120 _ 60
6
360
128.57 _ 51.43
7
360
135 _ 45
8
360
140 _ 40
9
360
144 _ 36
10
360
165 _ 15
24
180(n 2) _
n 360 _
n

2 18,000 180(n 2)
n 2 100
n 102
3 Diagonals n 3
a 0 b 1 c 2
d 4 e 8
4 Triangles n 2
a 7 b 10 c 15 d 98
5 180n
a 720 b 900 c 1,160 d 3,420
6 sum _
180
a 1,000 sides b 100 sides
7 360 _
n
a 40 b 36 c 10 d 5
360
8 __
# of degrees
a 12 b 10 c 6 d 8
180(n 2)
9 # of degrees _
n
a 18 b 360 c 8 d 6
10 sum 180(n 2)
a 12 b 24 c 50 d 1,000
11 Each exterior angle is 30. 360 _ 12 sides
30
12 180(n 2) 720
180n 360 720
180n 1,080
n 6
180(n 2)
13 _ 8 360
n
_
n
180(n 2) 2,880
n 2 16
n 18
14 m1 70, m2 75, m3 105,
m4 145
15 m1 90, m2 110, m3 70,
m4 35, m5 95, m6 85
16 m1 m2 m6 120
m3 m4 m5 60
17 180(n 2) 5(360)
n 12
18 (5x 10) (6x 25) (6x 25)
(5x 10) (3x 5) 180(3)
25x 65 540
x 19
Interior angles: 105, 139, 139, 105, 52
19 x + 3x + 4x + 4x + 6x 360
18x 360
x 20
Exterior angles: 20, 60, 80, 80, 120
Interior angles: 160, 120, 100, 100, 60
20 (3x 4) (7x 7) (6x 5) (5x 8)
(3x 2) 5x 360
29x 348
x 12
Exterior angles: 40, 91, 67, 68, 34, 60
Interior angles: 140, 89, 113, 112, 146, 120
Chapter Review (pages 212–216)
1 a no slope
b 1 _
4
c 15 _
7
d 5 _
2
e 3 _
5
f 0
2 a 2
b 1
c no slope
d 0
e 7 _
5
f no slope
3
___ 3 m AC _ 5
___ , m BC
5 _ . Slopes are negative
3
reciprocals.
___ 4 m SL m ___
BC m ___
PS m ___ 1. Opposite sides
LA
are parallel (slopes are equal) and slopes of
consecutive sides are negative
reciprocals (sides are perpendicular).
___ 5 m PL m ___
AN 1 and m ___
PS m ___ 4.
LA
Opposite sides are parallel (slopes are
equal).
6 For parallel lines m and n cut by transversal
a, m6 m10. So 2y 3y 10, and
y 38.
m1 m6 m9 m14 76
m2 m5 m10 m13 104
For parallel lines m and n cut by transversal
b, m4 m12. So 2y 15 3y 7,
and y 22.
m4 m7 m12 m15 59
m3 m8 m11 m16 121
7 m1 42, m2 48, m3 42, m4 42
8 mx 85
9 mx 60
10 mx 68
11 130
Chapter Review 53

12 a mx 45, my 45
b mx 98, my 82
c mx 60, my 70
d mx 65, my 52
e mx 67, my 78
f mx 15, my 55
g mx 55, my 62.5
13 a 5 b 20 c 35
d 594 e
n 3 _
2
14 102 sides
15 8 sides
16 14 sides
17 6 sides
18 a mx 85
b my 55
19 mD 45
20 m1 m2 25
21 m1 140
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
22 1. 6 4
2. x y
3. x z
4. y z (Transitive postulate)
23 1. 2 4
2. 1 3
3. 1 2 3 4
4. m r (Corresponding angles are
congruent.)
24 1. ___
QR ___
PS
2. QPS QRS
3. RPS QRP
4. QPS RPS QRS QRP
or QRP SRP
5. ___
QP ___
RS (Corresponding angles are
congruent.)
25 1. ___
BC bisects ABD.
2. 2 3
3. 1 2
4. 1 3 (Transitive postulate)
5. 5 1
54 Chapter 9: Parallel Lines
6. 3 5
7. a b (Alternate interior angles are
congruent.)
26 1. ___
AC bisects BCD.
2. 3 4
3. ___
AB ___
BC
4. ABC is a right angle.
5. ___
CD ___
AD
6. CDA is a right angle.
7. ___
AC ___
AC
8. I II (Hypotenuse–acute
angle)
27 1. ABCD
2. ___
CF ___
DE
3. ACF BDE
4. ___
CF ___
DE
5. ___
AB ___
CD
6. ___
AB ___
BC ___
CD ___
BC
7. ___
AC ___
BD
8. ACF BDE (SAS SAS)
9. ___
AF ___
BE
28 1. ___
BE bisects ABC.
2. DBE EBA
3. ___
DE ___
BA
4. DEB EBA (Alternate interior
angles are congruent.)
5. DEB DBE (Transitive postulate)
6. ___
DB ___
DE
7. BDE is isosceles. (Definition of
isosceles triangle)
29 1. ___
AF
2. ___
AB ___
CD
3. ___
AC ___
EF
4. ___
AC ___
CE ___
EF ___
CE
5. ___
AE ___
CF
6. ___
AB ___
CD
7. BAE DCF (Corresponding angles
are congruent.)
8. BAE DCF (SAS SAS)
9. ___
BE ___
DF (CPCTC)
30 1. ___
AB ___
CB
2. A C
3. B B
4. ABE CBD (ASA ASA)
5. ___
AE ___
CD
31 1. ___
BA ___
AE
2. BAE is a right angle.
3. 1 2
4. B D

5. ___
AE ___
AE
6. ABE EDA (AAS AAS)
7. BAE DEA (CPCTC)
8. DAE is a right angle.
9. ___
DE ___
AE (Perpendicular lines
intersect to form right
32 1.
angles.)
___
BA ___
DE
2. CAE CEA
3. ___
AE ___
AE
4. ABE EDA (SAS SAS)
5. BEA DAE
33 1. 1 2
2. 3 4
(CPCTC)
3. 5 6 (Linear pairs of
congruent angles
4.
are congruent.)
___
AD ___
EC
5. ___
DE ___
DE
6. ___
AE ___
DC (Addition postulate)
7. ABE CBD (ASA ASA)
8. ABE CBD
34 1.
(CPCTC)
___
RS ___
ST ___
TR
2. 1 2
3. ___
RP ___
TP (Converse of isosceles
4.
triangle theorem)
___
SP ___
SP
5. RPS TPS (SSS SSS)
6. RSP TSP
7.
(CPCTC)
___
SP bisects RST. (Definition of an
angle bisector)
35 1. ___
PT ___
QR , ___
RS ___
PQ
2. RSP and PTR (Definition of right
are right angles. angles)
3. RSP and PTR are right triangles.
4. ___
PT ___
RS
5. ___
PR ___
PR
6. RSP PTR (HL HL)
7. SRP TRP (CPCTC)
8. PQR is isosceles. (Converse of
isosceles triangle
theorem)
36 1. ___
AS ___
PR , ___
BT ___
PR
2. ASP and BTR are right angles.
3. PAS and RBT are right triangles.
4. 1 2
5. PAS RBT (Linear pairs of
congruent angles
are congruent.)
6. ___
AS ___
BT
7. PAS RBT (Leg–acute angle)
8. APS BRT (CPCTC)
9. PQR is isosceles. (Definition of an
isosceles triangle)
37 True
38 True
39 False
40 True
41 True
42 False
43 True
44 False
45 True
46 False
47 False
48 True
49 True
50 True
Chapter Review 55

Quadrilaterals
10-2 The Parallelogram
(pages 221–223)
1 mP 48, mQ mS 132
2 mR 30, mQ mS 150
3 mQ mR 180
(3x 5) (4x 25) 180
7 x 210
x 30
mS mQ 85
mP mR 95
4 mP mQ 180
(7x 12) (2x 3) 180
9 x 189
x 21
mR 135
mS 45
5 3x 5 1 _
2 (5x 3)
6x 1 0 5x 3
x 7
PR 5(7) 3 32
QS 3(7) 5 16
6 PRS PRQ; QPS SRP; QMP
SMR; QMR SMP
7 x (5x 6) 180
x 31
The angles are 31 and 149.
8 (3x 24) x 180
x 39
The angles are 39 and 141.
9 x 3
10 4
11 mETS mBTE mTEB 42
56 Chapter 10: Quadrilaterals
CHAPTER
10
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
12 1. ABCD and DAPQ are parallelograms.
2. −−−
AD ___
BC (Opposite sides of
a parallelogram are
congruent.)
3. −−−
AD −−
PQ (Opposite sides of
a parallelogram are
congruent.)
4. −−
BC −−
PQ (Transitive property)
13 1. ABCD is a parallelogram.
2. −−
BC −−−
AD
3. GBF GDE
4. BGF DGE
5. BG −−−
GD (Diagonals of a parallelogram
bisect
each other.)
6. BFG DEG (ASA ASA)
7. −−
FG −−
GE
8. G is the midpoint ( Definition of
of −−
FE . midpoint)
14 1. −−−
MX −−
QS
2. −−
TX −−−
QM
3. QMXT is a parallelogram.
4. M is the midpoint of −−−
QR .
5. QM −−−
MR
6. QM −−
TX (Opposite sides of
a parallelogram are
congruent.)

7. −−−
MR −−
TX (Transitive
postulate)
8. XTS MXT (Alternate
interior angles)
9. MXT MQT (Opposite
angles)
10. MQT RMX (Corresponding
angles)
11. XTS RMX (Transitive
postulate)
12. XST RXM (Corresponding
angles are
congruent)
13. MRX TXS (AAS AAS)
15 1. Parallelogram ABCD
2. X is the midpoint of −−
BC .
3. XC 1 _ BC ( Definition of a
2
midpoint)
4. Y is the midpoint of −−−
AD .
5. AY 1 _ AD
2
6. −−−
AD −−
BC
7. AD BC
8. 1 _ AD
2 1 _ BC
2
9. AY XC
10. −−
AY −−
XC
11. AMY CMX (Vertical angles
are congruent.)
12. MAY MCX (Alternate interior
angles are
congruent.)
13. MAY MCX (AAS AAS)
14. −−−
XM −−−
YM (CPCTC)
15. a M is the midpoint (Definition of
of −−
XY . midpoint)
16. −−−
AM −−−
MC (CPCTC)
17. b M is the midpoint (Definition of
of −−
AC . midpoint)
16 1. −−
AC is a diagonal in parallelogram ABCD.
2. −−
AF −−
CE
3. −−
FE −−
FE
4. −−
AF −−
FE −−
CE −−
FE
5. −−
AE −−
CF
6. DAC BCD (Definition of a
parallelogram)
7. BAF DCE (Alternate interior
angles are
congruent.)
8. EAD FBC (Subtraction
postulate)
9. −−−
AD −−
BC (Opposite sides of a
parallelogram are
congruent.)
10. ADE BCF (SAS SAS)
11. AED CFB (CPCTC)
12. −−
DE −−
BF (Alternate interior
angles are congruent.)
17 1. ABCD is a parallelogram.
2. −−
DE −−
AF
3. −−
CF −−
AF
4. DAE and CFB are right angles.
5. DAE CFB
6. −−
CA −−−
AD
7. −−−−
AEBF
8. −−
EB −−
EB
9. −−
AE −−
BF (Subtraction
postulate)
10. DEA CFB (SAS SAS)
11. −−
DE −−
CF (CPCTC)
18 1. Parallelogram ABCD
2. H is the midpoint of −−
AB .
3. AH 1 _ AB
2
4. F is the midpoint of −−−
DC .
5. FC 1 _ DC
2
6. −−
AB −−−
DC (Opposite sides of a
parallelogram are
congruent.)
7. AB DC
8. 1 _ AB
2 1 _ DC
2
9. AH FC
10. −−−
AH −−
FC
11. −−−
HG −−
AC , −−
FE −−
AC
12. HGA and FEC are right angles.
13. HFA FEC
14. −−
AB −−−
DC
15. CAB ACD
16. GAH ECF (AAS AAS)
17. −−−
HG −−
FE
19 1. Parallelogram ABCD
2. −−
AR −−−
CM
3. −−
AR −−−
MR −−−
CM −−−
MR
4. −−−
AM −−
CR
5. −−−
AD −−
BC
6. −−−
AD −−
BC
10-2 The Parralallogram 57

7. MAD RCB
8. MAD RCB (SAS SAS)
9. −−
BR −−−
DM (CPCTC)
20 1. Parallelogram ABCD
2. −−−
QD bisects D.
3. mCDQ 1 _ mCDP
2
4. −−
PB bisects B.
5. mABP 1 _ mABQ
2
6. CDP ABQ
7. mCDP mABQ
8. 1 _ mCDP
2 1 _ mABQ
2
9. mCDQ mABP
10. CDQ ABP
11. DCQ BAP
12. ABP CDQ (SAS SAS)
13. a −−
AP −−−
CQ (CPCTC)
14. −−
BC −−−
AD
15. b −−
BQ −−
PD (Subtraction
postulate)
21 1. Parallelogram PQRS
2. PS PQ
3. mx mQSP
4. −−−
QR −−
PS
5. y QSP (Alternate
interior angles
are congruent.)
6. my mQSP
7. mx > my (Substitution
postulate)
22 1. Parallelogram MARC
2. AR MA
3. mAMR mARM
4. ARM CMR (Alternate
interior angles
are congruent.)
5. mARM mCMR
6. mAMR mCMR (Substitution
postulate)
7. AMR is not congruent to CMR.
58 Chapter 10: Quadrilaterals
10-3 Proving That a
Quadrilateral Is a
Parallelogram
(pages 224–225)
1 Check students’ answers. The following is
one possible solution.
Slope of −−
AB slope of −−−
CD 3. Slope of
−−−
AD slope of −−
BC 0. Both pairs of opposite
sides are parallel.
2 a PR 5
b ( 7 _ , 8) or (3.5, 8)
2
3 a Check students’ answers. The following is
one possible solution.
DR AB 10. Slope of −−−
DR slope
of −−
AB 0. One pair of opposite sides are
both congruent and parallel.
b The length of the altitude from B to −−−
DR
is 5.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
4 1. Parallelogram LOVE
2. AOV BEL
3. OAV EBL
4. −−−
OV −−
EL
5. OAV EBL (AAS AAS)
5 BF 1 _ CE (segment connecting midpoints of
2
sides of a triangle) and CD 1 _ CE (defin-
2
tion of midpoint). BF CD. DF 1 _ AC and
2
BC 1 _ AC. BF CD and DF BC. Opposite
2
sides of a parallelogram have equal measure
so they are congruent.
6 1. −−
KL and −−
EU bisect each other at M.
2. −−−
KM −−−
LM
3. −−−
EM −−−
MU
4. EMK UML
5. EMK UML (SAS SAS)
6. −−
EK −−
LU (CPCTC)
7. K is the midpoint of −−
JE .

8. EK 1 _ EJ
2
9. L is the midpoint of −−−
UN .
10. UL 1 _ UN
2
11. EK UL
12. 1 _ EJ
2 1 _ UN
2
13. EJ UN
14. −−
EJ −−−
UN
15. −−
EU −−
EU
16. LUM KEM
17. EUN UEJ
18. −−
EN −−
UJ
19. JUNE is a ( Both pairs
parallelogram. of opposite sides
are congruent.)
7 1. ABCD is a parallelogram.
2. −−
RC
3. −−−
DQ −−
RC , −−
AR −−
RC
4. DQC and ARB are right angles.
5. DQC ARB
6. RBA QCD (Corresponding
angles are
congruent.)
7. −−
AB −−−
DC (Opposite sides of
a parallelogram are
congruent.)
8. AB DC
9. ARB DQC (AAS AAS)
10. −−
AR −−−
DQ (CPCTC)
11. AR DQ
12. −−
AR −−−
DQ Segments perpendicular
to the same
segment are
parallel.)
13. ARQD is a ( One pair of opparallelogram.
posite sides is both
congruent and
parallel.)
8 1. −−
PE bisects −−
HL at M.
2. −−−
HM −−−
ML
3. EPL PEH
4. HME LMP
5. HME LMP (AAS AAS)
6. −−
HE −−
LP
7. −−
HE −−
LP
8. HELP is a ( One pair of opparallelogram.
posite sides is both
congruent and
parallel.)
9 1. Parallelogram ABCD
2. −−−
AD −−
BC
3. −−−
AM −−−
NC
4. M is midpoint of −−−
AD .
5. AM 1 _ AD
2
6. N is the midpoint of −−
BC .
7. NC 1 _ BC
2
8. −−−
AD −−
BC
9. AD BC
10. 1 _ AD
2 1 _ BC
2
11. AM NC
12. −−−
AM −−−
NC
13. ANCM is a ( One pair of opparallelogram.
posite sides is both
congruent and
10 1. BR and DM
2. 2 3
3.
parallel.)
−−
BC −−−
AD
4. 1 4
5. BAD DCB (Supplements of
congruent angles
6.
are congruent.)
−−
BD −−
BD
7. BCD DAB
8.
(AAS AAS)
−−
BC −−−
AD
9. ABCD is a ( One pair of opparallelogram.
posite sides is both
congruent and
parallel.)
11 1. −−
QS bisects −−
PR (at M).
2. −−−
QM −−−
MS
3. 1 2
4. QMR SMP
5. QMR SMP
6.
(AAS AAS)
−−−
QR −−
SP
7. −−
PR −−
PR
8. PSR RQP
9.
(SAS SAS)
−−
QP −−
SP
10. PQRS is a ( One pair of opparallelogram.
posite sides is both
congruent and
12 1.
parallel.)
−−−
QV bisects −−
RT .
2. −−
RS −−
ST
3. −−−
QR −−
PV
4. RQS TVS (Alternate interior
angles)
10-3 Proving That a Quadrilateral Is a Paralallogram 59

5. RSQ TSV (Vertical angles)
6. QRS VTS (AAS AAS)
7. −−−
QR −−
PT (CPCTC)
8. −−
PT −−
TV
9. −−−
QR −−
PT
10. −−−
QR −−
PT
11. PQRT is a ( One pair of opparallelogram.
posite sides is both
congruent and
parallel.)
13 1. Parallelogram ABCD
2. −−−
DC −−
AB (Opposite sides of
a parallelogram are
congruent.)
3. −−
DF −−
BE
4. −−−−
DFEB
5. CDF ABE (Alternate interior
angles are congruent.)
6. DFC BEA (SAS SAS)
7. −−
CF −−
AE (CPCTC)
8. −−−
AD −−
CB (Opposite sides of
a parallelogram are
congruent.)
9. ADF EBA
10. AFD CEB (SAS SAS)
11. −−
AF −−
CE (CPCTC)
12. AECF is a (Both pairs
parallelogram. of opposite sides are
congruent.)
14 1. −−
KJ is a diagonal in parallelogram KBJD.
2. −−
KA −−
JC
3. BJC AKD
4. −−
BJ −−−
KD
5. BJC KAD (SAS SAS)
6. −−
BC −−−
AD (CPCTC)
7. −−
BK −−
JD
8. CJD BKA
9. ABK CDJ (SAS SAS)
10. −−
AB −−−
CD (CPCTC)
11. ABCD is a ( Both pairs
parallelogram. of opposite sides are
congruent.)
15 1. −−
BD is a diagonal in parallelogram
ABCD.
2. BEC AFD
3. EBC ADF (Alternate interior
angles are congruent.)
4. −−
BC AD (Opposites sides of
a parallelogram are
congruent.)
60 Chapter 10: Quadrilaterals
5. CBE AFD
6.
(AAS AAS)
−−
AF −−
EC
7.
(CPCTC)
−−
DF −−
EB
8. ABE FDC
9.
(CPCTC)
−−
AB −−−
DC
10. ABE CDE
11.
(SAS SAS)
−−
EA −−
CF (CPCTC)
10-4 Rectangles
(pages 227–228)
1 (1) are congruent
2 AR DR
3(4x 3) 10x 1
x 5
AR CR DR BR 51
3 AR BR
2(x 6) 3x 20
x 8
AR CR 4
BD 8
4 DR CR
4(3x 10) 3(x 2) 12
x 46 _
9
AR 46 _
9
AC BD 128 _
9
5 AC BD
3(2x 5) 1 _ (4x 4)
4 2 _ (12x 3) 5x
3
x 2
AC 24
DR 12
6 2x 3 0 36
x 3
7 PR QS
4x 3 6x 7
x 5
PR 4(5) 3 23
QS 6(5) 7 23
√ PR QS √ 23 23 23
8 mADB mDAC 90 49 41
9 (5, 6)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended

to be used as a guide—other possible solutions
may vary.)
10 1. Rectangle ABCD
2. ABCD is a parallelogram.
3. −−
BC −−−
AD
4. −−
AB −−−
CD (Opposite sides of
a parallelogram are
5.
congruent.)
−−
BD −−
CA (Diagonals of a
rectangle are
congruent.)
6. CDA BAD (SSS SSS)
7. CAD BDA
11 1. Rectangle PQRS
(CPCTC)
2. PQRS is a parallelogram.
3. −−
PQ −−
PQ
4. −−−
QR −−
PS
5. SQP PQR (SSS SSS)
6. 1 2 (CPCTC)
12 1. Rectangle ABCD
2. ABP and NCD are right angles.
3. ABP and NCD are right triangles.
4. −−
AP −−−
DN
5. −−−
DC −−
AB (Opposite sides of
a parallelogram are
congruent.)
6. ABP DCN (HL HL)
7. DNC APB (CPCTC)
8. PAD APB
9. NDA APB
(Alternate interior
angles are
congruent.)
10. PAD NDA (Transitive
11.
postulate)
−−
AE −−
DE (Definition of an
isosceles triangle)
13 By the addition postulate of inequality, −−
AY
is not congruent to −−
TX , and AMY is not
congruent to THX. Therefore, −−−
MY is not
congruent to −−−
HX .
14 1. Rectangle ABCD
2. N is the midpoint of −−−
CD .
3. CN DN
4. −−−
CN −−−
DN
5. BCN NDA (A rectangle is
6.
equiangular.)
−−
BC −−−
AD (Opposite sides of
a parallelogram are
7.
congruent.)
−−
BN −−−
AN (CPCTC)
15 1. −−
AB −−−
CD
2. BJH and AJH are linear angles.
3. mBJH mAJH 180
4. 1 _ mBJH
2 1 _ mAJH 90
2
(Division postulate)
5. −−
JG bisects BJH.
6. mHJG 1 _ mBJH
2
7. −−
EJ bisects AJH.
8. mEJH 1 _ mAJH
2
9. mHJG mEJH 90 ( Substitution)
10. mEJG 90
11. EJG is a right angle.
12. CHJ BJH
13. EHJ GJH
14. AJH DHJ
15. HJE JHG
16. −−
HJ −−
HJ
17. EJH GHJ
18. −−
EJ −−−
GH
19. −−
EH −−
JG
20. EJGH is a parallelogram.
21. EJGH is a rectangle. (Definition of a
rectangle)
16 1. Rectangle PQRS
2. −−
PA −−
CS
3. −−
PA −−
AC −−
CS −−
AC
4. −−
PC −−
AS
5. −−
QP −−
RS
6. QPS RSP
7. QPC RSA (SAS SAS)
8. a 1 2 (CPCTC)
9. PQR and SRQ are right angles.
10. 3 is complementary to 1.
11. 4 is complementary to 2.
12. b 3 4 (Complements of
congruent angles
are congruent.)
13. c −−
QB −−
RB (Definition of an
isosceles triangle)
10-5 Rhombuses
(pages 229–231)
1 mB mD 105; mC 75.
AB CD 7
2 mADB 66
10-5 Rhombuses 61

3 4x 2 3x 3
x 5
RS 18
4 Perimeter of MABC 8
5 mADC 110
6 2x 2 x 8
x 10
CD 18
7 mADC 94
8 a Midpoint of −−
AC midpoint of −−
BD (6, 5)
b Slope of −−
AC 1, slope of −−
BD 1.
Slopes are negative reciprocals, diagonals
are perpendicular.
9 a (9, 5)
b PQ PS 5 √ 2
c Slope of −−
PR 1 _ , slope of
3 −−
QS 3.
Slopes are negative reciprocals, diagonals
are perpendicular.
10 a Slope of −−−
AD slope of −−
BC k _
x
.
AD BC √ x 2 k 2 . ABCD is a parallelogram.
(One pair of opposite sides have
the same length and are parallel.)
b Slope of −−
CA k _
−−
, slope of BD
x k k _
x k ,
not negative reciprocals. The diagonals are
not perpendicular.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
11 −−−
QR −−
AC , −−
PS −−
AC , −−−
QR −−
PS . Likewise,
−−
QP −−
BD , −−
RS −−
BD , and −−
QP −−
RS . Thus, PQRS
is a parallelogram. Since −−
AC −−
BD , −−−
QR
and −−
PS are perpendicular to −−
QP and −−
RS .
Thus, PQRS has at least one right angle, and
PQRS is a rectangle.
12 1. Rhombus ABCD
2. −−
ED
3. −−
BF −−
AC
4. −−
EF −−
AC
5. F bisects −−
AC .
6. EA EC
62 Chapter 10: Quadrilaterals
7. −−
EA −−
EC
8. ACE is isosceles. (Definition of an
isosceles triangle)
13 Prove that −−
DF −−
AB and −−
DE −−
BF by using
the midpoints and alternate interior angles.
Thus, EBFD is a parallelogram. By the division
postulate, −−
EB −−
FB . EBFD is a rhombus
because a rhombus is a parallelogram with
two congruent consecutive sides.
14 1. Parallelogram ABCD
2. −−
AB −−−
CD
3. BAD 2
4. mBAD m2
(Corresponding angles
are congruent.)
5. BAD 1 CAD
6. mBAD m1 (Whole is greater
than a part.)
7. m2 m1 (Substitution
postulate)
15 1. Rhombus PQRS
2. −−
QS −−
PR
3. −−−
QC −−
CS
4. B is the midpoint of −−−
QC .
5. D is the midpoint of −−
CS .
6. −−
BC −−−
CD
7. ___
AC ___
AC
8. ABC ADC
9.
(SAS SAS)
−−
AB −−−
AD
(CPCTC)
10. BAD is isosceles. (Definition of an
isosceles triangle)
16 If a quadrilateral is equilateral, it is a parallelogram
with a pair of consecutive congruent
sides.
17 The diagonal creates two congruent triangles
(ASA ASA) and corresponding congruent
sides are consecutive.
18 1. −−−
CD BE
2. −−−
CD −−
BA
3. −−−
AD BF
4. −−−
AD −−
BC
5. ABCD is a ( Definition of a
parallelogram. parallelogram)
6. BAD BCD (Opposite angles
7. BG bisects FBE.
are congruent.)
8. CBD ABD
9. ABCD is a rhombus. (Diagonal −−
BD
bisects opposite
angles.)

10-6 Squares
(pages 232–233)
1 (4) A rectangle is a square.
2 (3) sides and angles are congruent
3 (4) A trapezoid is a parallelogram.
4 (3) −−
AC −−−
DC
5 (1) congruent and bisect the angles to which
they are drawn
6 (2) x √ 2
7 Slope of −−−
DA slope of −−
VE 3 _ . Slope of
4 −−
ED slope of −−
AV 4 _ . A parallelogram
3
has two pairs of opposite sides that are parallel.
A rectangle is a parallelogram in which
consecutive sides have slopes that are negative
reciprocals. Slope of −−−
DV 1 _ . Slope
7
of −−
AE 7. Slopes of the diagonals are negative
reciprocals, thus they are perpendicular.
Therefore, DAVE is a square.
8 Slope of −−−
MA slope of −−
TH 4 _ . Slope of
3 −−−
HM slope of −−
AT 3 _ . A parallelogram has
4
two pairs of opposite sides that are parallel.
A rectangle is a parallelogram in which consecutive
sides have slopes that are negative
reciprocals. Slope of −−−
MT 1 _ . Slope of
7 −−−
AH 7. Slopes of the diagonals are negative
reciprocals, thus they are perpendicular.
Therefore, MATH is a square.
9 The diagonals bisect the vertex angles, creating
four congruent isosceles triangles. The
bisected angles (the base angles of the triangles)
measure 45. The vertex angles thus
measure 90. The diagonals cross, forming
90 angles.
10 a Slope of −−
PQ slope of −−
RS 0. −−−
QR and −−
SP
have no slope. A parallelogram has two
pairs of opposite sides that are parallel.
A rectangle is a parallelogram in which
consecutive sides are perpendicular.
b PQ RS 20. QR PS 7. Since all
sides are not congruent to each other,
PQRS is not a square.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
11 a Given ABCD is a square. BFE is a right
angle because −−
EF −−
BD . mEBF 45
because diagonal −−
BD bisects right angle
ABC. mFEB 45. Therefore, −−
BF −−
EF .
b 1. ABCD is a square.
2. −−
EF −−
BD
3. EFD is a right angle.
4. DAE is a right ( Definition of
angle. a square)
5. DAE and DFE are right triangles.
6. (Draw −−
DE ). −−
DE −−
DE
7. DAE DFE (HL HL)
8. −−
EF −−
EA (CPCTC)
12 1. Rhombus ABCD with diagonals −−
AC
and −−
BD intersecting at E.
2. 1 2
3. −−
BE −−
CE
4. E bisects −−
AC and −−
BD .
5. −−
BD −−
AC
6. ABCD is a rectangle.
7. ABCD has all right angles.
8. ABCD is a square. (A square is a
rhombus in which
all angles are right
angles.)
10-7 Trapezoids
(pages 236–238)
1 (2) They are congruent.
2 (3) ADC ABC
3 a x z 70; y 110
b x 73; y z 107
c x 40; y 108; z 32
d x 70; y 44; z 66
e x 130; y 20; z 30
f x 82; y z 41
4 Slope of −−−
MA slope of −−
TH 1 _ . These legs
2
are parallel. Slope of −−
AT 3 _ HM has
4 . −−−
no slope. These legs are not parallel.
AT HM 10. Therefore, MATH is an
isosceles trapezoid.
10-7 Trapezoids 63

Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
5 1. ABCD is an isosceles trapezoid.
2. BAD CDA
3. −−
BC −−−
AD
4. NBC BAD
5. NCB CDA
(Corresponding
angles are
congruent.)
6. NBC NCB (Substitution
postulate)
7. NBC is isosceles. (Base angles of an
isosceles triangle
are congruent.)
6 1. Isosceles trapezoid ABCD
2. −−
AB −−−
DC
3. BAD CDA (Base angles of an
isosceles trapezoid
4.
are congruent.)
−−−
AD −−−
AD
5. ADB DAC
7 1. Trapezoid ABCE
2.
(SAS SAS)
−−
BD −−−
AD
3. −−
AB −−
CE
4. A B
5. CED A
6. ECD B (Corresponding
angles are
congruent.)
7. CED ECD (Substitution
8.
postulate)
−−−
CD −−
ED
9. −−
BD −−−
CD ( Subtraction
−−−
AD −−
ED
10.
postulate)
−−
BC −−
AE
11. ABCE is an (Nonparallel sides
isosceles trapezoid. of an isosceles
trapezoid are
congruent.)
8 1. Quadrilateral PQRS
2. −−−
QAB , −−−
RAS , and −−−
PSB
3. −−
QB bisects −−
RS .
4. −−
RA −−
AS
64 Chapter 10: Quadrilaterals
5. −−−
PSB −−−
QR
6. RAQ BAS (Vertical angles are
congruent.)
7. QRA BSA (Alternate interior
angles are
congruent.)
8. QRA BSA
9.
(ASA ASA)
−−−
QA −−
AB
9 1. Isosceles trapezoid ABCD
2. A D
3. −−−
AD −−
BC
4. −−
AB −−−
DC
5. E is the midpoint of −−−
AD .
6. −−
AE −−
ED
7. ABE DCE
8. a
(SAS SAS)
−−
BE −−
CE
9. BE CE
(CPCTC)
10. BCE is isosceles. (Definition of isos-
11. b
celes triangle)
−−
EH −−
BC (The median from
the vertex angle of
an isosceles triangle
is perpendicular
to the base.)
10 1. Isosceles trapezoid PQRS
2. −−
PQ −−
RS
3. −−−
QR −−
PS
4. P S
5. −−
PS −−
PS
6. PQS SRP (SAS SAS)
7. PQS SRP (CPCTC)
8. QAP BAS (Vertical angles are
congruent.)
9. PAQ SAR
11 1. Trapezoid ABCD
2.
(AAS AAS)
−−
BR −−−
AD
3. −−−
CM −−−
AD
4. ARB and DMC are right angles.
5. ARB and DMC are right triangles.
6. −−
AB −−−
CD
7. BAR CDM
8. ARB DMC (Hypotenuse–acute
angle)
9. 1 2 (CPCTC)
12 1. Trapezoid PQRS
2. Q R
3. −−
PA −−−
QR , −−
SE −−−
QR
4. PAQ and SER are right angles.
5. PAQ and SER are right triangles.

6. AP ES (Distances between
parallel lines are
7.
equal.)
−−
AP −−
ES
8. PAQ SER
9.
(Leg–Acute Angle)
−−
QP −−
RS (CPCTC)
10-8 Kites
(page 239)
1 A kite is a quadrilateral with only two pairs
of adjacent congruent sides. A rhombus has
four congruent sides.
2 True
3 True
4 False
5 True
6 True
7 False
8 False
9 True
10 a −−
AC
b 1 8; 2 7; 3 6; 4 5
11 a 90 b 45 c 17
d m4 25 and mKLM 50
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
12 1. −−
QS is the perpendicular bisector of −−
PR .
2. −−
PT −−
TR
3. QTP QTR
4. −−
QT −−
QT
5. QPT QRT (SAS SAS)
6. STP STR
7. −−
TS −−
TS
8. STP STR (SAS SAS)
9. −−
QP −−−
QR (CPCTC)
10. −−
SP −−
SR (CPCTC)
11. −−
QT −−
ST
12. PQRS is not ( Definition of
a rhombus. a rhombus)
13. PQRS is a kite. (Definition of a
kite)
10-9 Areas of Polygons
(pages 243–245)
1 4
2 16
3 9, 12, 15
4 Length is 4 ft and width is 5 ft.
5 a h 10 x
b A x(10 x)
6 a x 2 b 9 x 2 c x 2 4x 4
d x 2 4x 4 e 4 x 2 4x 1
7 a 25 _
2
b 18
49
c _
2
d 1
8 11
e 9
9 a 12 b 45 c 20
d 18 √ 3 e 45 _
2
10 √ 36 144 6 √ 5
11 a BD 10
12 72 √ 3
b 120
13 s 6
14 a 13 b 120 c 120 _
13
15 a 21x b 10 x 2 c 4x 12
d 18x 12 e 2 x 2 16 a x 4
b 4(4) 4 12
17 8
18 27 √ 3
19 30 √ 2
20 10 and 20
6x
21 a 96
22 72
23 100
24 a 6
b 260 c 60
b SMP and AML are similar. Let x be the
perpendicular drawn from M to −−
AP .
Then x _ 20
6 x _ , and x 5.
4
25 a _ x 10
12 x _
20
b x 4
c Area of RCS 20. Area of QCT 80.
4(8 11)
26 a A _
2 38 b 9 ( 5, _
2 )
27 a Slope of −−
JE 3 _ , slope of
4 −−
EN 4 _ . The
3
slopes of the two legs are negative reciprocals,
therefore perpendicular, forming a
right angle.
b A 1 _ (JE)(EN) 25
2
10-9 Areas of Polygons 65

28 Enclose PAT in a large rectangle and subtract
the excess areas.
a Area of PAT 120 60 24 6 12
18
b AT 10
c 3.6
29 a SA SM √ 52 ; sides are congruent.
Slope of −−
SA 3 _ , slope of
2 −−−
SM 2 _ ;
3
slopes are negative reciprocals, therefore
perpendicular and forming a right angle.
b A 1 _ (SA)(SM) 26
2
c √ 26
30 Subdivide pentagon SIMON into two triangles
and a trapezoid and take the sum of all
the areas.
Area 72
31 Divide pentagon JANET into two triangles
and take the sum of the two areas.
Area 64
32 Enclose quadrilateral RYAN in a large rectangle
and subtract the excess areas.
Area 72 13 _ 6 12 12 35.5
2
33 −−
NI and −−
CK have zero slope, the two sides are
parallel. −−−
NK −−
IC m √ 10 . Therefore, NICK
is an isosceles trapezoid.
34 Slope of −−
JO slope of −−−
HN 1 _ ; the sides are
3
parallel. Slope of −−
NJ 3. The slopes are
negative reciprocals, therefore perpendicular,
forming a right angle. Slope of −−−
OH is not
equal to the slope of −−
NJ , therefore, only one
pair of sides are parallel. JOHN is a right
trapezoid.
35 a 38 b 28 c 28 d (5, 4)
Chapter Review (pages 245–248)
1 (4) The diagonals of a parallelogram bisect
each other.
2 (4) mB mC 360
3 (1) rhombus
4 (1) a rhombus
5 (3) A rhombus is a square.
6 (4) parallelogram
7 (3) 3
8 (4) −−
QS and −−
PR bisect each other.
9 (1) the diagonals are congruent
10 a square b rhombus
c parallelogram d rectangle
66 Chapter 10: Quadrilaterals
11 3x 4 0 x 50
x 5
12 5
13 5 x 90
x 18
14 (3x 20) (7x 4 0) 180
x 20
15 2
16 mDAE 90 75 15
17 (4, 3)
18 3x 1 5 7x 55
x 10
19 a mACD mCAB 30
b rectangle
180 75
20 mAEK _ 52.5
2
21 5 ft
22 (1, 1)
23 3 √ 2
24 mA 45
25 h 8
26 Rhombus. Since the triangles are congruent,
the opposite angles that are originally vertex
angles are congruent. By the addition postulate,
the opposite angles formed by joining
the triangles at their bases are congruent.
Thus the quadrilateral is a parallelogram.
All sides are congruent. The parallelogram is
a rhombus.
27 PS QR
2x 3 x 2
x 5
PS QR SR PQ 7
28 a (2x 8) 3(x 3 4) 180
x 14
mABC mCDA 144
mDAB mBCD 36
b AE EC
4 y 6y 36
y 18
BE ED
3x 1 x 13
x 7
AC 144, BD 40
29 mABP mBAP 90
(5x 10) (2x 4) 90
x 12
mDCB 140, mAPB 90
30 a mR 150 b mRAD 150
c mGAD 135 d mD 30

31 a 20 √ 3 35
b 20
c 200 √ 3 346.4
32 a AB BC 5 √ 2 . ABC is isosceles because
there is one pair of congruent sides.
b Slope of −−
AB 1. Slope of −−
BC 1. Slopes
are negative reciprocals, thus −−
AB −−
BC .
c Area of ABC 25
33 −−
DE −−
AV because the slope of −−
DE slope
of −−
AV 1 _ .
3 −−−
DA is not parallel to −−
VE because
the slope of −−−
DA is not equal to the slope
of −−
VE . DA VE 5; nonparallel sides are
congruent. Therefore, DAVE is an isosceles
trapezoid.
34 Slope of −−
PQ slope of −−
RS 3 _ . Slope of
4 −−
PS slope of −−−
RQ 4 _ . Slope of
3 −−
PR 1 _ .
7
Slope of −−
QS 7. PQRS is a parallelogram
because opposite sides have the same slope,
and a rhombus because the diagonals are
perpendicular—slopes are negative reciprocals,
and a square because consecutive sides
are perpendicular—slopes are negative reciprocals.
35 Slope of −−
AB slope of −−−
CD 0. Slope of
−−
BC slope of −−−
AD 4 _ . Slope of
3 −−
AC 2.
Slope of −−
BD 1 _ . ABCD is a parallelogram
2
because opposite sides have the same slope,
and a rhombus because the diagonals are
perpendicular—slopes are negative
reciprocals.
36 Slope of −−
LE slope of −−
AF 4 _ . Slope
5
of −−
EA slope of −−
FL 4 _ . Slope of
5 −−
LA is
udefined. Slope of −−
EF 0. LEAF is a parallelogram
because opposite sides have the same
slope, and a rhombus because the diagonals
are perpendicular.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
37 1. Quadrilateral TRIP
2. −−
TR −−
RI
3. m1 m2
4. 1 2
5. −−
RA −−
RA
6. a RAT RAI (SAS SAS)
7. −−
TA −−
AI (CPCTC)
8. TAP IAP (Supplements of
congruent angles
are congruent.)
9. −−
AP −−
AP
10. TAP IAP (SAS SAS)
11. b 3 4 (CPCTC)
38 1. Parallelogram PQRS
2. Diagonal −−
QS bisects PQR.
3. PQS RQS
4. RQS QSP (Alternate interior
angles are congruent.)
5. PQS QSR (Alternate interior
angles are congruent.)
6. RQS QSR (Transitive postulate)
7. QSP QSR
8. −−
PS −−
PQ
9. PS PQ
10. PQRS is a rhombus. (A rhombus is a
parallelogram with
two congruent
consecutive sides.)
39 1. ABED is a rhombus.
2. −−
BD intersects −−−−
AOEC at O.
3. BOE and DOE are right angles.
4. BOE and DOE are right triangles.
5. −−
OE −−
OE
6. −−
BE −−
DE
7. BOE DOE (HL HL)
8. −−
BO −−−
OD (CPCTC)
9. BOC and DOC are right triangles.
10. −−−
OC −−−
OC
11. BOC DOC (Leg-leg)
12. a −−
BC −−−
DC (CPCTC)
13. −−
AB −−−
AD
14. −−
AC −−
AC
15. ABC ADC (SSS SSS)
16. b ABC ADC (CPCTC)
40 Mark the point of intersection of the diagonals
E. The diagonals bisect each other,
thus AE DE. By the triangle inequality
theorem, mADE mDCE and
mADC mDCB by the multiplication
postulate of inequality.
Chapter Review 67

Geometry of Three
Dimensions
11-1 Points, Lines, and
Planes
(pages 251–252)
1 False
2 False
3 True
4 True
5 True
6 True
7 True
8 False
9 True
10 False
11 False
12 H
13 E
14 C
15 G
16 C
17 G
18 E
19 D
20 a −−
AB −−
FG , −−
AB −−−
HC , −−
AB −−
ED ,
−−−
AH −−−
DG , −−−
AH −−
CB , −−−
AH −−
EF , −−
FA −−
GB ,
−−
FA −−−
DC , −−
FA −−
EH
b There are many possible answers. The following
are just a few.
−−
AB & −−
EF , −−
AB & −−−
DG , −−
BC & −−
EF , −−
BC &
−−
ED , −−−
DC & −−−
AH , −−−
DC & −−
EF .
21 Infinitely many
22 Infinitely many
23 Infinitely many
24 One
25 One
26 Infinitely many
68 Chapter 11: Geometry of Three Dimensions
27 Infinitely many
28 One
29 Infinitely many
30 Infinitely many
31 One
32 One
33 One
34 None
35 One
CHAPTER
11
11-2 Perpendicular Lines,
Planes, and Dihedral
Angles
(pages 257–258)
1 False
2 True
3 True
4 False
5 True
6 True
7 False
8 True
9 One
10 Infinitely many
11 Four
12 Infinitely many
13 One
14 One
15 a ABCD b GFE
16 a BCDE and GCDE
b PRS, SRT, ADE, and EDF
17 The plane that bisects the dihedral angle
18 −−−
QR , −−
QS , −−−
QU , −−−
WP , −−
TP , −−
VP

Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
19 1. BC plane M
2. BCA and BCD are right angles.
3. BCA BCD
4. −−
AB −−
DB
5. −−
BC −−
BC
6. BCA BCD (HL HL)
7. BAC BDC (CPCTC)
11-3 Parallel Lines and
Planes
(pages 259–260)
1 Parallel
2 Perpendicular
3 Perpendicular
4 Parallel
5 Parallel
6 Parallel
7 Parallel
8 Infinitely many
9 None
10 One
11 Line a is parallel to plane M. Line b is contained
in plane R. Line b is in plane M. Lines
a and b are coplanar and do not intersect.
Therefore, a b.
12 True
13 True
14 True
15 False
16 False
17 True
18 False
19 False
20 True
21 True
22 True
23 True
24 True
25 True
26 False
27 False
28 False
29 False
30 True
11-4 Surface Area of
a Prism
(pages 264–265)
1 (3) 80
2 a 104 b 108 c 224
3 4 faces, 7 edges, 4 vertices
4 5 faces
5 Right prism
6 a 404 cm 2 b 223 ft 2
7 208
8 242 square inches
9 588
10 435 in. 2
11 Lateral area: 255 cm 2 ; total area:
25 √ 3
255 _ 2
c m
2
12 Lateral area: 45 cm 2 ; total area:
9 √ 3
45 _ 2
c m
2
13 144 32 √ 3 cm 2
14 3,880 in. 2
15 Lateral area: 72; total area: 72 12 √ 3
16 Lateral area: 510 square inches; total area:
510 75 √ 3 square inches
17 3 √ 3 in.
18 d 2 h 2 (AB) 2 , but (AB) 2 l 2 w 2 . By substitution,
d 2 h 2 l 2 w 2
11-5 Symmetry Planes
(pages 267–268)
1 (1) A
2 (2) E
3 (1) F
Exercises 4–7: Check students’ sketches.
8 7 symmetry planes
9 Check students’ sketches.
10 112.5 √ 3
11-5 Symmetry Planes 69

11-6 Volume of a Prism
(pages 269–270)
1 288
2 15,625 m 3
3 a Volume: 105 cm 3 ; surface area: 142 m 2
b Volume: 3,600 in. 3 ; surface area: 1,500 in. 2
c Volume: 48 ft 3 ; surface area: 88 ft 2
4 147
5 a 2 b 6 c 4
d 5 e 3
6 14
7 1 _
2
8 Volume: 1,920 ft 3 ; surface area: 992 ft 2
9 1
10 Volume: 2,197 ft 3 ; diagonal: 13 √ 3 ft
11 30 inches
12 480 cm 3
13 9
14 2.7 ft
15 x √ 3
16 81 √ 3
17 Volume: 343; surface area: 294
18 24 √ 3 in. 3
19 Volume is ten times as large.
20 a Each solid has a volume of 360 √ 3 cubic
units.
b triangular prism 360 square units,
hexagonal prism 120 √ 6
21 Diagonal, d √
l 2 w 2 h 2 , so d 2
l 2 w 2 h 2 . Multiply by four, so that
4 d 2 4 l 2 4 w 2 4 h 2 .
11-7 Cylinders
(pages 273–274)
1 (3) 6 inches
2 h 11
3 V _
9
4 r 1 _
4
5 h 1 _
6 a 112 cm 2 b 136.5 cm 2 c 196 cm 3
7 a 96 in. 2 b 168 in. 2 c 288 in. 2
8 a 24 in. 2 b 26 in. 2 c 12 in. 2
9 a 6 m 2 b 14 m 2 c 6 in. 2
10 a r 3 b r 4
70 Chapter 11: Geometry of Three Dimensions
11 A cylinder with a radius of 8 and a height of
12 has a volume of 768. A cylinder with a
radius of 12 and a height of 8 has a volume
of 1,152. Their difference is 384.
12 38 pounds
13 r 3.4
14 h 29.4
15 a 64 b The volume is twice as
large; 128
c 32 d The lateral area stays
the same; 32.
16 a Doubled b Volume is 1 _ of the
4
volume original.
17 Yes. The volume of the glass with a diameter
of 2.4 is 1.44. Double the volume is 2.88.
The volume of the second glass is 2.89,
which is greater than twice the volume of the
first glass.
18 False
19 True
20 Lateral surface area: 960 in. 2 ; volume:
3,840 in. 3
21 h 6,930 _
36
385
_
2
22 16 : 64 or : 4
23 a V 256 b V 512
c V 256 512
11-8 Pyramids
(pages 278–279)
1 234 cm 2
2 240 cm 2
3 1,152 in. 2
4 4,848 cm 3
5 57.2 in. 3
6 7.5 ft 3
7 48 √ 3
8 h 300 ft
9 h 6
10 279 ft 2
11 864 in. 3
12 120
13 a 180 25 √ 3 b 340
c 360 150 √ 3
14 9 √ 3
15 Volume of pyramid is 80. Volume of prism
is 240.

16 240
17 144 √ 3 cm 2
18 432 √ 3 ft 3
11-9 Cones
(pages 283–284)
1 Lateral area: 60; total area: 96;
volume: 96
2 Lateral area: 20; total area: 36;
volume: 16
3 Lateral area: 136; total area: 200; volume:
320
4 Lateral area: 80; total area: 105
5 Lateral area: 135 cm 2 ; volume: 324 cm 3
6 Lateral area: 260 in. 2 ; total area: 360 in. 2 ;
volume: 800 in. 3
7 r 3
8 V 36
9 V 16
10 V 12
11 r 10
12 1 _
3 base height
13 equal
14 three times
15 one-third
16 No, because the slant height is always
greater than the radius.
17 4 : 1
18 h 8
19 28.5 in. 3
20 Volume: 3,056 _
3 1,018.67 in. 3 ; lateral area:
300 in. 2
21 144 √ 3
22 60
11-10 Spheres
(pages 287–288)
1 a r 10.5 ft b r 6.2 ft c r 2.9 ft
2 a r 5 mi b r 2 √ 5 4.5 mi
c r 4 mi d r 1.6 mi
3 a 288 in. 3 b 36 ft 3
c 972,000 mi 3 d 1 _ yd
6 3
e 4 _ 3
m
81
4 r 7 in.
5 Surface area: 100 cm 2 ; volume: 500 _ 3
cm
3
6 Surface area: 192 in. 2 ; volume: 256 √ 3 in. 3
7 Volume: 1,679,616,000 cubic miles; surface
area: 4,665,600 square miles
8 63,361,600 square miles
9 1,000,000 times
10 r 5
11 r 3
12 9 : 25 or 9 _
25
13 a 2 : 5 or 2 _
5
14 a 2 _
b
3
4 _
9
15 a m 2 : n 2 b m 3 : n 3
16 64 _
27
17 108 ounces
18 1,458 lb.
b 4 : 25 or 4 _
19 a r √ 2 _
b r 4 _
20 r _ √ 24
√
21 a 1 _ ft
6 3 b 4 _ ft
3 3 c 4.5 ft 3
22 64
23 h 4r
24 : 6 or _
6
25 2 : 3 or 2 _
3
26 They are equal.
Lateral area of cylinder 16 in. 2
Surface area of sphere 16 in. 2
25
Chapter Review (pages 289–290)
1 (4) 216 in. 2
2 False
3 False
4 True
5 True
6 False
7 False
8 False
9 False
10 True
11 True
12 a 8 vertices, 6 faces, 12 edges
b 6 vertices, 5 faces, 9 edges
c 8 vertices, 6 faces, 12 edges
d 10 vertices, 7 faces, 15 edges
13 a Surface area: 216, volume: 432
b Surface area: 90, volume: 100
c Surface area: 52 ft 2 , volume: 24 ft 3
Chapter Review 71

14 47.5 in. 2
15 h 4 in.
16 25 cm 3
17 The cylinder formed by spinning the rectangle
about the shorter side is 96 cm 3 greater
in volume.
r 6 and h 8, V 288 cm 3
r 8 and h 6, V 384 cm 3
18 36 9 √ 3 in. 2
19 36 √ 3 in. 3
20 144 √ 3 cm 3
21 480
22 156 in. 2
23 √ 65 in.
Ratios, Proportion,
and Similarity
12-1 Ratio and Proportion
(pages 294–295)
1 No
2 Yes
3 No
4 Yes
5 a 3 _
2
b 2 _
5
c 3 _
5
d 5 _
3
6 77
7 15
8 9
9 15
10 5
72 Chapter 12: Ratios, Proportion, and Similiarity
24 100 ft 3
25 60 ft 2
26 Lateral area: 15 in. 2 ; volume: 12 in. 3
27 r 3 in.; volume: 36 in. 3
28 Surface area: 40,000 cm 2
Volume: 4,000,000 _
3 cm 3 or 1,333,333 1 _ cm
3 3
29 Surface area: 4 ft 2 ; volume: 4 _ ft
3 3
30 r 8 _
√ meters
31 5 _ inches or 2.5 inches
2
32 40.57 41 scoops
11 44
12 15
13 39
14 mt _
a
15 2ma _
r
16 4ar _
t
17 9
18 8
19 15
20 4 √ 3
21 3 √ 11
22 5 √ 3
23 6 √ 2
24 4 √ 6
25 1 _
10
26 1 _ or _ √ 3
3 √ 3 9
CHAPTER
12

27 40, 50
28 70, 110
29 20, 35
30 16, 20, 24
31 length 28, width 49
32 45, 135
33 a 8 b 6 c 10
34 a 20 b 9
12-2 Proportions Involving
Line Segments
(pages 298–300)
1 a–d are all true
2 2 _
5 4 _ Proportions are not true, lines not
8
parallel.
3 6
4 28
5 5.4
6 6
7 24
8 5
9 ac _
b
10 np
_
m
11 6
12 20
c(a b)
13 _
a
14 No
15 No
16 No
17 No
18 Yes
19 12
20 1 : 3 or 1 _
3
21 11 cm
22 x 6, AB 47, DE 17, AC 34, EC 9,
BE 9, AD 23.5, DB 23.5
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
23 a 1. ABC
2. P is the midpoint of −−
AB .
3. Q is the midpoint of −−
BC .
4. −−
PQ −−
AC (A line joining the midpoints
of two sides of a triangle is
parallel to the third side.)
5. −−
PQ ___
AR
6. R is the midpoint of −−
AC .
7. −−
AB −−−
RQ
8. −−
AP −−−
RQ
9. PQRA is a parallelogram. (Definition
of a parallelogram)
b 30
24 1. ABC, ADC
2. −− wx −−
AC (A line dividing two
sides of a triangle proportionally
is parallel to
the third side.)
3. −− zy −−
AC
4. −− wx −− zy
5. BAD, BCD
6. −− wz −−
BD
7. −− xy −−
BD
8. −− wz −− xy
9. wxyz is a ( Definition of a
parallelogram. parallelogram)
25 48 √ 3
12-3 Similar Polygons
(pages 302–303)
1 1 : 1
2 4, 4.5, 5
3 1 : 4
4 45, 55
5 a 2 : 3
b 4.5, 7.5
c 18
6 9, 13.5, 18
7 5a, 5b, 5c
8 All corresponding angles are congruent, all
corresponding sides are in proportion, 1 : 2.
9 w _ ,
3 x _ ,
3 y
_ ,
3 z _
3
10 32, 40
11 mR mQ 110
12 mI 40; mK 160
13 12
12-3 Similar Polygons 73

14 a 3 : 2 b QR 10, RS 20,
ST 12, PT 22
15 a True b False c True
d True e False f True
g False h False
16 If the vertex angles are congruent, then the
base angles must also be congruent. Similarly,
if the base angles are congruent, then
the vertex angles are congruent. Triangles are
similar by (AA) or (AAA).
12-4 Proving Triangles
Similar
(pages 307–308)
1 (2) similar
2 Vertical angles are congruent. (AA)
3 Not similar
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
4 a 1. −−
AE −−
BC
2. BCD DAE ( Alternate interior
angles)
3. −−
AC and −−
BE intersect at D.
4. BDC ADE (Vertical angles)
5. ADE CDB (AA)
b AD 30
5 All corresponding sides are in
proportion; AB _
BE
CD _
AE
CE _
4
DE _ . Triangles
5
are similar.
6 4 _ and
6 5 _ , corresponding sides are not in
3
proportion. Triangles are not similar.
7 6 _ and
7 4 _ , corresponding sides are not in
5
proportion. Triangles are not similar.
8 1. −−−
CD −−
AB
2. CDA and CDB are right angles.
3. CDA CDB
4. CAD CBD
5. CAD CBD (AA)
74 Chapter 12: Ratios, Proportion, and Similiarity
9 1. −−
AB −−
DE
2. EDF BAC (Corresponding
angles are
congruent.)
3. −−
BC −−
EF
4. EFD BCA
5. ABC DEF (AA)
10 1. −−
BE −−
AC and −−−
CD −−
AB
2. HEC and HDB are right angles.
3. HEC HDB
4. DHB CHE (Vertical angles are
congruent.)
5. EHC DHB (AA)
11 1. −−
DE −−
BC and −−
DF −−
AB
2. DEC and DFA are right angles.
3. DEC DFA
4. Parallelogram ABCD
5. FAD ECD
6. AFD CED (AA)
12 1. DBE ADF
2. BAC BDE (Corresponding
angles)
3. B B
4. DBE ABC (AA)
13 1. Parallelogram ABCE
2. −−−
AED
3. −−
BC −−
AE
4. −−
BC −−−
AD
5. CBF ADB (Alternate interior
angles)
6. BAD FCB
7. BAD FCB (AA)
14 1. ABC PQR
2. ABC PQR
3. −−
BD bisects ABC.
4. −−
QS bisects PQR.
5. ABD PQS
6. BAD QPS
7. ABD PQS (AA)
15 1. Parallelogram ABCD
2. −−−
AD −−
BC
3. GAE BCG (Alternate interior
angles are
congruent.)
4. −−
AC bisects HAE.
5. HAG GAE
6. HAG BCG
7. −−−
AH −−
AE

8. GEA GHA
9. GEA GBC
10. GHA GBC
11. HAG BCG
12-5 Dilations
(pages 310–311)
1 a 10 _
80
b
3
_ c 9
3
2 a BC 10, PA 2.2, BA 11, PS 9.6
b 10 _ 5
8 _
4
3 (21, 9)
4 (6, 12)
5 (9, 0)
6 (27, 3)
7 (12, 12)
8 (10, 36)
9 (20, 12)
10 (2.5, 1)
11 (4, 2)
12 (3, 0)
13 (3, 3.5)
14 (2 √ 2 , 2.5)
15 (16, 8)
16 (10, 2)
17 (15, 0)
18 (3, 15)
19 (3, 2)
20 (4, 8)
21 D 2 r x-axis
22 D 3 r y-axis
23 r x-axis D 1 _
2
24 r x-axis D 1 _
4
25 a A(0, 0), B(12, 0), C(15, 6), D(3, 6)
b Slope AB 0; slope BC 2; slope CD 0;
slope DA 2
c Midpoint M (2.5, 1) and midpoint
M (7.5, 3). Yes, M is the image result
of D 3 operating on point M. Midpoints are
preserved under dilation.
12-6 Proving Proportional
Relationships Among
Segments Related to
Triangles
(pages 315–316)
1 3 : 4 and 3 : 4
2 5 cm
3 4 : 9
4 11 in.
5 AD 6, DC 8
6 18
7 15
8 4 : 7
9 AC 23, PR 8
10 RQ 6, BC 12.2, AC 20.6
11 13
12 19 _ or 4.75
4
13 22
14 12
15 a 4 : 7 b 16 and 20
c 84 and 48 d 48 _
4
84 _
7
16 a BQ 8, QR 5, PR 8 b 36
12-7 Using Similar Triangles
to Prove Proportions
or to Prove a Product
(pages 319–321)
1 128
2 40
3 108
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
4 a 1. −−
ED −−
AC and −−
AB −−
CB
2. EDA and ABC are right angles.
3. EDA ABC
12-7 Using Similar Triangles to Prove Proportions or to Prove a Product 75

4. A A
5. ADE ABC
b
(AA)
AC _
AB
AE _
AD
c 32
5 a Each smaller triangle is similar to the
larger triangle so they are similar to each
other.
b Corresponding sides of similar triangles
are in proportion.
c Product of means product of extremes.
d BD 6
6 1. MAH is a right angle.
2. −−
UT −−−
AH
3. UTH is a right angle.
4. MAH UTH
5. H H
6. MAH UTH (AA)
7. MA _
AH
UT _
TH (Corresponding
sides of similar
triangles are in
7 1.
proportion.)
−−
AB −−
DE
2. CAB CED (Alternate interior
angles are
congruent.)
3. ACB ECD (Vertical angles are
congruent.)
4. ACB ECD (AA)
5. AB _
ED
AC _
EC
6. AB _ AC
ED _
EC (Corresponding
sides of similar
triangles are in
proportion.)
8 1. EBA CDA
2. A A
3. EBA CDA
4. BCF DEF
5. −−
AC −−
AE
6. DEC BCE (Isosceles triangle
theorem)
7. DEC DEF (Subtraction
BCE BCF
8. FEC FCE
9. CBD EDB
postulate)
10. CBD EDB
11.
(AA)
BC _
BE
DE _
DC
76 Chapter 12: Ratios, Proportion, and Similiarity
12. BC DC (The product of the
BE DE means equals the
product of the
extremes.)
9 1. EBC CDE
2. BFC DFE (Vertical angles are
congruent.)
3. DEF BCF (AA)
4. FB _
FD
FC _
FE
5. FB _ FC
FD _ (Corresponding sides
FE
of similar triangles are
in proportion.)
10 1. −−
AE −−
BC
2. ADC and EDC are right angles.
3. ADC EDD
4. DAC DEC
5. DEC DAC (AA)
6. EC _ AC
_
ED
AD
7. EC AD (The product of the
AC ED means equals the
product of the
extremes.)
11 1. Isosceles triangle WXZ with −−−
WX −−−
WZ
2. X Z
3. YTW YRW
4. XTY ZRY (Supplements of congruent
angles are
congruent.)
5. XTY ZRY (AA)
6. YT _
YR
XY _
ZY
7. YT _
XY
YR _ (Corresponding sides
ZY
of similar triangles are
in proportion.)
12 1. Rectangle DEFG
2. DGB and EFC are right angles.
3. DGB EFC
4. Isosceles triangle ADE
5. BDG CEF (Supplements of congruent
angles are
congruent.)
6. BDG CEF (AA)
7. DG _ BG
EF _ (Corresponding sides
CF
of similar triangles are
in proportion.)

12-8 Proportions in a
Right Triangle
(pages 323–324)
1 (1) 10
2 (3) √ 21
3 (3) 4
4 (1) √ 77
5 a False b True c False
d False e True f True
g True
6 2 √ 15
7 5
8 6
9 √ 55
h False
10 x 2 √ 5 , y 4 √ 5 , z 4
11 x 2 √ 5 , y 4 √ 5 , z 8
12 x 3 √ 6 , y 6 √ 3 , z 6
13 x 4 √ 3 , y 8 √ 3 , z 12
14 x 20, y 8 √ 6 , z 40 √ 6
15 x 9, y 6.75, z 18.75
16 x 4 √ 5 , y 10
17 x 3 √ 10 , y 3 √ 6 , z 3 √ 15
18 x 12, y 4 √ 5 , z 6 √ 5
19 x 9, y 6, z 6 √ 3
20 x 25, y 17.5, z 29.2
21 x 2.4, y 3.2, z 1.8
22 a x 21 b (10) 2 23 13
x(x 21) c 4
24 a LR 2 √ 26 1, LM 4 √ 26
b 9 and 10
12-9 Pythagorean Theorem
and Special Triangles
(pages 327–329)
1 (2) 10 √ 2
2 (3) 25 feet
3 (1) √ 13
4 (1) 50 miles
5 (4) √ 41
6 (1) 24 feet
7 (2) 13
8 (4) {10, 24, 26}
9 14
10 b, c, d, e, f, g
11 a 3 √ 2 b √ 3 c
d 9 e 5 _
12
f
√
5
√
14
12 6
13 8
14 240 cm 2
15 32 in.
16 x 10 √ 5 , 2x 20 √ 5
17 24
18 5
19 19.2 in.
20 Perimeter is 56. Area is 192.
21 3
22 10
23 3 √ 3
24 BD 2 a 2 b 2
BD 2 d 2 c 2
a 2 b 2 d 2 c 2
a 2 d 2 b 2 c 2
25 AB 2 BC 2 AC 2
AC 2 AD 2 CD 2
AB 2 BC 2 AD 2 CD 2
Special Triangles
(pages 332–334)
1 (1) 6 √ 2 ft and 6 √ 2 ft
2 (3) 1 : √ 3 : 2
3 (4) 6 √ 2
4 (1) 1
5 (2) 6 √ 3
6 (1) 30
7 (3) 3.75
8 (2) 2 √ 6
9 (3) 60
10 a BC 7, AC 7 √ 3
b BC 2.5, AC 2.5 √ 3
c BC 3 √ 3 , AC 9
d AC 9 √ 3 , AB 18
e AC 12, AB 8 √ 3
f BC 7, AB 14
g BC 2.75 √ 3 , AC 8.25
h BC 4 √ 3 , AB 8 √ 3
11 a AC 4, AB 4 √ 2
b BC 10, AB 10 √ 2
c AC 8, BC 8
d AC 1.5, BC 1.5
e AC 6 √ 2 , BC 6 √ 2
f BC 3 √ 2 , AB 6
g BC 10 √ 2 , AB 20
h AC 7.5 √ 2 , BC 7.5 √ 2
i BC 4 √ 3 , AB 4 √ 6
j AC 4 √ 6 , BC 4 √ 6
12-9 Pythagorean Theorem and Special Triangles 77

12 7 √ 2
13 4
14 5 √ 3
15 x 6 √ 3 , y 12, z 6 √ 2
16 a 6 by 6 √ 3
b 12 12 √ 3
c 36 √ 3
17 9 √ 2
18 46
19 AB 16 √ 3 , AC 24, DC 8, DB 16
20 2.28
21 a 8
b 2
12-10 Perimeters, Areas,
and Volumes of Similar
Figures (Polygons and
Solids)
(pages 336–337)
1 (1)1 : 3
2 (3) 3 : 5
3 (1) 28
4 a 9 : 1 b 16 : 81 c 81 : 4
d 25 : 1 e 2 : 3
5 a 2 : 3 b 6 : 5 c 1 : 7
d 2 : 5
6 1 : 9
7 343
8 13.5
9 25 : 1
10 539 ft
e 9 : 11
2
11 1 : 25
12 27 : 64
13 48
14 27 ft 3
15 27.7 in.
16 506.25 kg
17 a 1 : 9 b 1 : 3
18 a Yes b 2 : 3 c 4 : 9 d 8 : 27
e Ratio of surface areas: 312 _
4
702 _
9
Ratio of volumes: 360 _ 8
1,215 _
27
19 72
20 a 2.4 b 27 : 125
78 Chapter 12: Ratios, Proportion, and Similiarity
Chapter Review (pages 338–341)
1 (1) 4 : 25
2 (4) 5 : 14
3 (3) 15
4 (2) x a _
a
5 (3) √ 3
6 (4) 64 pounds
7 a 6 b 6 c an _
8 a 5 : 6 b 2 : 9
m
c n : m d c : (a b)
e p : (m n) f (m n) : (a b)
g (n b) : (a m)
9 a 6 b 8a c 10 _
10 30
11 11.25
12 6 : 35
13 6
14 6
15 14
3
16
ac bc _
a
17
8 _ 5
8 x _ 8
, EC
x 4 _ , AC
3 20 _
3
18 a 3
19 54
b Expansion
20 102
21 Yes, the ratio of similitude is 3 _ .
2
22 Yes, the ratio of similitude is 1 _
23 mQ 125, PQ 6 _ , QR 2
5
24 2
25 13
26 BD 12, AB 4 √ 13 , BC 6 √ 13
27 20, 21, 29
28 a 5 √ 2
b 50
29 34
30 x 40 √ 3 , y 80
31 x 8 √ 3 , y 8
32 x 6 √ 3 , y 9
33 √ 2
34 126
35 75 ft 2
36 8 _
27
37 80 in. 3
38 81
39 45
40 18
41 72 √ 3 in. 2
42 Area 60, perimeter 24 10 √ 2
2 .

Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
43 1. −−
TS −−
QP
2. QRP TSR (Corresponding
angles are
congruent.)
3. PQR STR
4. R R
5. PQR STR (AAA AAA)
44 1. −−
AE −−
BC , −−
BD −−
AC
2. FEB and BDC are right angles.
3. FEB BDC
4. FBE FBE
5. BFE BCD (AA)
45 1. −−
AB −−−
CD
2. ABE DCE (Alternate interior
angles are
congruent.)
Geometry of the Circle
13-1 Arcs and Angles
(pages 345–346)
1 a 36 b 51 c 90
d 180 e 2r
2 a BC , CD , AD , BD , AB
b ABC , ADC
c ABD , BDA , ADB
3. BAE CDE
4. AEB CED (Vertical angles are
congruent.)
5. a ABE DCE (AAA AAA)
6. BE _ EC
AE _
ED
7. b BE _
AE
EC _
ED (Corresponding
sides of similar
triangles are in
proportion.)
8. c BE ED (The product of
EC AE means equals
the product of
46 1.
extremes.)
−−
BC −−
AF
2. ECA and FDA are right angles.
3. ECA FDA
4. A A
5. ADF ACB
6. BA _ =
AF _
BC
DF
7. BA DF (The product of
AF DC means equals
the product of
extremes.)
CHAPTER
13
3 RB 105, BS 75, AS 105, AR 75
4 a 155 b 25 c 155 d 335
5 a 296 b 244 c 296 d 244
6 a 73 b 132 c 155 d 180
e 155
i 287
f 228 g 253 h 205
7 a 20 b 60 c 120 d 110
e 7 f 60 g 70 h 120
i 110 j 230 k 240 l 290
13-1 Arcs and Angles 79

13-2 Arcs and Chords
(pages 350–351)
1 a 16 b 7.5 c m _
d 3 √ 6 e 5 √ 2
2
2 a True
3 109
b True c False
4 a 90 b 72 c 60 d 36
5 a 24 b 22.5 c 20
d 18 e 15
6 a 10 b 5 √ 2 c 5
7 a 6
8 6 in.
9 17
b 6 √ 2 c 12 d 6 √ 2
10 a 15
11 6.
12 8 √ 2
13 x √ 2
14 5
15 25
b 7.5
16 a 30 b 60 c 60
d 60 e 60 f 300 g 150
17 a 30 b 60 c 10
d 5 e 5 √ 3
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
18 1. Radii −−−
OK and −−−
OM
2. AOL DOL (Central angles are
congruent.)
3. −−
LA −−−
OK , −−
LD −−−
OM
4. LAO and LDO are right angles.
5. LAO and LDO are right triangles.
6. LAO LDO
7. −−
LO −−
LO
8. LAO LDO (Leg-angle)
19 1. ABD CBD
2. −−−
OA −−
OB
3. BAO ABD
4.
(All radii of the
same circle are
congruent.)
−−−
OC −−
OB
5. BCO CBD
80 Chapter 13: Geometry of the Circle
6. AOB COB
7. AOD COD
8. AD CD (Congruent inscribed
angles intercept congruent
arcs.)
20 1. −−−
OA , −−
OB , and −−−
OC are radii of circle O.
2. BAC BCA
3. AB BC (Congruent inscribed
angles intercept congruent
arcs.)
4. AOB COB (Central angles are
equal to their arcs.)
13-3 Inscribed Angles
and Their Measure
(pages 354–357)
1 a 25 b 55 c 125
d 60.4 e 4
2 a 52 b 124 c 15
d 180 e (16x)
3 mx 70, my 35; 70 35 105
4 a 50 b 130 c 56
d 25
5 53
6 70
7 80
8 35
9 140
e 62 f 25
10 mC 75, mM 82
11 85.5
12 m 1 120, m2 45, m3 75, m4 60
13 m1 57, m2 33, m3 90, m 4 114
14 m 1 74, m 2 74, m3 37, m4 106
15 a 140 b 80 c 40
d 70 e 70
16 m1 57, m2 48, m3 57, m4 48
17 a 80 b 188 c 46 d 94
18 a 64 b 108 c 140 d 70
e 24 f 54 g 94
19 a 86 b 102 c 88
d 56 e 116
20 a 44 b 88 c 44 d 44
e 92 f 88
21 a 60 b 60 c 60 d 30
e 60 f 120 g 30 h 60
i 120 j 60 k 90 l 30

22 Label arcs with degree measures that sum
to 360, as with x, 2x, 3x, 4x. Then find the
values of the inscribed angles.
23 Parallel lines cut congruent arcs x and y.
Therefore,
2x 2y 360
x y 180
Each inscribed angle is equal to one-half the
intercepted arc: 90.
13-4 Tangents and Secants
(pages 363–365)
1 a
b
c
d
2 a 12 b 3
3 a 3.25 b 1
4 a disjoint b externally tangent
c intersecting twice
d internally tangent
e disjoint internally
f concentric
5 a Sketch of two circles externally disjoint
b Sketch of two intersecting circles, not
tangent
c Sketch of externally tangent circles
d Sketch of internally tangent circles
6 a 1 b 3
c 2 d 4
7 a 160 b 130 c 90
d 40 e 180 x
8 a 60 b 45 c 35
d 51 e 67.5
180 n
9 a _ b 90 2n c
n
2 _
2
d 45 n e __
180 m n
2
10 16
11 80
12 48
13 234
14 a 18 b 72 c 29.25
15 a 11, 17, 18 b no sides equal
16 a 12, 16, 20
b Satisfies Pythagorean theorem:
12 2 16 2 20 2
17 a 10, 15, 15 b two sides are equal
18 BE 4, EC 5, CF 5, AF 6, AC 11,
AB 10
19 AB 20, CB 20, RB 10
20 AB 24, DR 14, DB 32
21 OB 26, DB 36, RB 16
22 OB 20, AB 2 √ 91 , CB 2 √ 91
23 OB 22, CB 8 √ 6 , AB 8 √ 6
24 AB 20, CB 20, OC 15, OB 25
25 Tangent segments from the same point are
congruent. Base angles are equal, each measuring
60. Therefore, APB is equilateral.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
26 1. Let E be the point of intersection of −−
AB
and −−−
CD .
2. −−
EB and −−
ED are tangent segments to
circle P.
3. −−
EB −−
ED (Two tangent segments
drawn from an external
point are congruent.)
4. −−
EA and −−
EC are tangent segments to
circle O.
5. −−
EA −−
EC
6. −−
EA −−
EB ( Addition postulate)
−−
EC −−−
CD
or −−
AB −−−
CD
13-4 Tangents and Secants 81

27 1. Common external tangents, −−−
AG and −−−
CM
2. Radii −−−
OA and −−−
OC
3. OAG and OCM are right angles.
4. OAG and OCM are right triangles.
5. −−−
OA −−−
OC
6. Tangents −−−
OM and −−−
OG of circle P
7. −−−
OM −−−
OG (Two tangent segments
drawn from an
external point are
congruent.)
8. OAG OCM (HL HL)
9. −−−
AG −−−
CM (CPCTC)
28 1. Circles A and B are congruent with
tangent FG .
2. −−
AF is the radius of A. −−
BG is the radius of
B.
3. −−
AF −−
BG
4. AFC and BGC are right angles.
5. FCA GCB
6. FCA GCB (AAS AAS)
7. −−
AC −−
BC
29 mA 1 _ (
2 CD BE ). 2(mA)
mCOD mBOE. But mBOE mA.
2(mA) mCOD mA. Therefore,
3(mA) mCOD.
30 1. −−
AB is tangent to O at E; −−−
CD is tangent to
O at F.
2. −−
OE OF
3. −−−
OA −−−
OC ; −−
OB −−−
OD
4. −−
OE −−
AB
5. OEA and OEB are right angles.
6. −−
OF −−−
CD
7. OFC and OFD are right angles.
8. OEA OFC (HL HL)
OEB OFD
9. −−
AE −−
EB (Addition
−−
CF −−
FD postulate)
or −−
AB −−−
CD
13-5 Angles Formed by
Tangents, Chords, and
Secants
(pages 368–371)
1 (2) 96
2 (1) 24
3 (2) 144
82 Chapter 13: Geometry of the Circle
4 (4) 173
5 (1) 38
6 (1) 90
7 a 40 b 62 c 225 d 45
e 75 f 100 g 80 h 30
8 a 79 b 38
c x 50, y 60, z 120
9 a 66 b 80 c 73
d 130 e 33 f 73
10 a 90 b 120 c 90 d 60
e 30 f 60 g 30 h 90
11 a 120 b 40 c 80
d 40 e 140 f 100
12 BC CD because in a regular hexagon,
congruent chords subtend congruent arcs.
Chords and a tangent that intercept congruent
arcs are parallel. Therefore, PC −−
BD
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
13 1. −−
FB −−
EC
2. BEC EBF
3. EHJ BHG
4. EHJ BHG
(Alternate interior angles
are congruent.)
5. BH _
HE
HG _
HJ
(Corresponding sides
of similar triangles are
in proportion.)
6. BH JH (The product of means
HE HG equals the product of
14 1.
extremes.)
−−
BA −−−
DA
2. Diameter −−−
DC
3. BAD and DBC are right angles.
4. −−
AB is tangent to O at B.
5. mDBA 1 _ m
2 BD
6. mDCB 1 _ m
2 BD
7. mDBA mDCB
8. DBA DCB
9. DBA DCB

10. BD _
DA
11. BD _
DC
DC
_
DB
DA _
BD
( Corresponding sides of
similar triangles are in
proportion.)
12. (BD) 2 ( The product of means
DC DA equals the product of
extremes.)
13-6 Measures of Tangent
Segments, Chords and
Secant Segments
(pages 374–376)
1 (4) 20
2 (2) 46
3 (1) 17
4 (3) 24
5 (2) 10
6 (1) 3
7 (3) 25
8 (3) cz _
a
9 (3) 17
10 (3) 33
11 (1) 9
12 24
13 19
14 40
15 44
16 12
17 16
18 NQ 6, QP 16
19 x 4, GH 16
20 −−−
AOB −−−
WZ . Radius 12.5; 10 2 (7.5) 2
(12.5) 2
13-7 Circle Proofs
(pages 381–382)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
1 1. A is the midpoint of CD ;
−−−
AOB is a diameter.
2. CA AD
3. 1 _ m
2 CA 1 _ m
2 AD
4. mABC 1 _ m
2 AD
mABC 1 _ m
2 CA
5. mABC mABD
6. ACB and ADB are right angles.
7. mACB mADB
8. −−
AB −−
AB
9. ACB ADB (AAS AAS)
10. −−
CB −−
DB (CPCTC)
2 1. −−−
AD −−
CB
2. AD CB
3. mABD 1 _ m
2 AD
mCDB 1 _ m
2 CB
4. mABD mCDB
5. −−
DB −−
DB
6. mDAB 1 _ m
2 DB
mDCB 1 _ m
2 DB
7. mDAB mDCB
8. ADB CBD (AAS AAS)
9. −−
AB −−−
CD (CPCTC)
3 1. Circle O with diameters −−−−
MOT and −−−−
AOH
2. MAT and HTA are right angles.
3. −−−−
MOT −−−−
AOH
4. −−
AT −−
AT
5. MAT HTA (HL HL)
6. −−−
MA −−
HT
(CPCTC)
4 1. Circle O, tangents PR , PV
2. PR OR ; PV OV
3. ORP and OVP are right angles.
4. mORP mOVP
5. −−−
OR −−−
OV
6. −−
PR −−
PV
7. RPO VPO (SAS SAS)
8. RPO VPO (CPCTC)
5 1. Circle O with T as the midpoint of CH
2. m CT m TH
3. m CTH 180
4. m CT 90
5. mCOT m CT
6. COT and CTH are right angles.
7. mCOT mCTH
13-7 Circle Proofs 83

8. mHCT 1 _ m
2 TH
mCHT 1 _ CT
2 m
9. mHCT mCHT
10. CTO HTC
11. CT _
CH
TO
_
TH (Corresponding
sides of similar
triangles are in
proportion.)
6 1. Circle O with diameter −−−−
DOG ; DR GI
2. mTEG 1 _ m (
2 DR TG )
3. mTRI 1 _ m (
2 GI TG )
4. TG GI TGI
5. mTEG 1 _ m
2 TGI
6. mTRI 1 _ m
2 TGI
7. mTEG mTRI
8. mRTI mRTI
9. TEX TRI
10.
(AA)
TX _
EX
TI _
RI
11. TX RI EX TI (The product of
means equals
the product of
extremes.)
7 1. CT CH
2. −−−
HA −−
TD
3. CTD CHA; TDH HAT
4. TDH is supplementary to TDC;
HAT is supplementary to HAC.
5. TDC HAC
6. TDC HAC
7.
(AAS AAS)
−−
CT −−−
CH (CPCTC)
8. HA TD
8 1.
(Contradiction)
−−
BC −−−
QR
2. −−
BA −−
QP
3. BC RQ
4. BA QP
(Congruent chords
subtend congruent
arcs.)
5. A D
6. B Q
(Inscribed angles
intercepting congruent
arcs are
congruent.)
7. RQP CBA
8. AC PR
(SAS SAS)
84 Chapter 13: Geometry of the Circle
9 a 1. BEA CED (Vertical angles are
congruent.)
2. BDC CAB (Congruent
inscribed angles)
3. BEA CED
b (i) 4 : 5 (ii)
(AA)
16 _
25
10 1. −−
BA −−−
PGD
2. DBA BDP
3. mPBD 1 _ m
2 BGD
4. mBAD 1 _ m
2 BGD
5. mPBD mDAB
6. PBD DAB
7. PBD DAB
8. PD _
BD
BD _
BA (Corresponding
sides of similar
triangles are in
11 1.
proportion.)
−−
AB is the diameter.
2. Line l is tangent to circle O.
3. ABC and AEB are right angles.
4. ABC AEB
5. CAB BAE
6. ABC AEB (AA)
7. AC _
AB
AB _
AE (Corresponding
sides of similar
triangles are in
proportion.)
12 1. −−
AC is the diameter.
2. −−
BD intersects the diameter at point E.
3. −−
AB −−
BE
4. BAE BEA
5. BEA CED
6. BAE CED
7. ABE and DCE are right angles.
8. ABE DCE
9. ABE ECD
10.
(AA)
AC _
ED _
AB
EC
11. AC EC (The product of
ED AB means equals
the product of
extremes.)
13 1. DCG CBD
2. mE 1 _ m (
2 ABC DF )
1 _ m (
2 CB )

3. mCDB 1 _ m (
2 CB )
4. mE mCDB
5. CBD FCE (AA)
6. EF _ EC
CD _
DB (Corresponding
sides of similar
triangles are in
proportion.)
14 1. AZT MZH
2. HMT TAH
3. MZH AZT (AA)
4. MZ _
AZ
ZH _
ZT (Corresponding
sides of similar
triangles are in
proportion.)
15 a 1. BAC BDC
2. ABD ACD
(Congruent inscribed
angles)
3. ACD ABE
b BE 6
(AA)
13-8 Circles in the
Coordinate Plane
(pages 386–387)
1 (2) (x 4) 2 (y 3) 2 36
2 (4) (13, 1)
3 a (x 1) 2 (y 4) 2 25
b (x 3) 2 (y 2) 2 49
c (x 4) 2 (y 1) 2 121
d (x 2) 2 (y 5) 2 64
e (x 2) 2 y 2 121
f x 2 y 2 144
4 a x 2 y 2 49
b (x 4) 2 y 2 25
c x 2 y 2 41
d (x 2) 2 y 2 36
e (x 3) 2 (y 2) 2 9
f (x 1) 2 (y 3) 2 25
5 For parts a–d, check students’ graphs. The
center and radius of each circle are listed
below.
a C(2, 4), r 3
b C(3, 4), r 5
c C(0, 2), r 4
d C(6, 0), r 2
6 a C(0, 0), r 9
b C(0, 0), r 11
c C(0, 0), r 20
d C(0, 0), r 1
e C(0, 0), r √ 2
7 Note: Students’ answers may vary for the
two other points on the circle.
a C(5, 3), r 8; (5, 11), (5, 5)
b C(4, 0), r 10; (6, 0), (14, 0)
c C(2, 5), r 4; (2, 5), (6, 5)
d C(3, 7), r 2 √ 3 ; (3, 7 2 √ 3 ),
(3, 7 2 √ 3 )
8 a x 2 y 2 25
b x 2 y 2 4
c x 2 y 2 34
d x 2 y 2 37
e x 2 y 2 68
9 a (x 2) 2 (y 3) 2 2
b (x 3) 2 (y 7) 2 80
10 a x 2 (y 0.5) 2 12.25
b (x 3) 2 (y 3) 2 5
11 a Circumference 8, area 16
b Circumference 12, area 36
c Circumference 2 √ 13 , area 13
12 a (3, 0)
b (x 3) 2 (y 1) 2 1
c
13 a C(1, 3), r 2 √ 2
b Area 8, circumference 4 √ 2
13-9 Tangents, Secants,
and the Circle in the
Coordinate Plane
(pages 392–393)
1 (1) 0
2 a (3, 11), (3, 1)
b (3) x 1
3 y 0
4 y 0.5x 2.5
5 y x 4
6 y 3
7 y 4
8 y 2x 10
9 a (4, 3), (3, 4) b secant
10 a (0, 4), (4, 0) b secant
11 a (2, 3), (2, 5) b secant
12 a (0, 3) b tangent
13 a (0, 2), (2, 0) b secant
14 a (5, 5) b tangent
15 a (1, 4), (1, 4) b secant
13-9 Tangents, Secants, and the Circle in the Coordinate Plane 85

16 a (2 √ 2 , 2 √ 2 ), (2 √ 2 , 2 √ 2 ) b secant
17 a (10, 0) and (10, 0) b secant
18 a (7, 7) and (7, 7) b secant
19 a (6, 0) b tangent
20 a (8, 6) and (6, 4) b secant
21 a (0, 5) and (4, 3) b secant
22 a Sub-in the given points in (x 3) 2
(y 2) 2 25.
b Midpoint of −−−
ME is (3.5, 2.5);
distance √ 0.5 .
23 a y x 8
b y x 8
c (0, 8)
24 a y 3x 10 b x 10 c P(10, 20)
d PA √ 160 4 √ 10 , PB √ 1000
10 √ 10 , PM 20
PA PB PM 2
4 √ 10 10 √ 10 20 2
400 400
Chapter Review (page 393–397)
1 (1) All chords in a circle are congruent.
2 (2) (x 4) 2 (y 2) 2 9
3 (2) 1
4 (3) 8
5 (3) 6 and 15
6 (2) 12
7 (1) 12
8 (2) 6
9 (2) 68
10 (3) 122
11 (1) 43
12 (3) 86
13 (2) 2 : 1
14 (1) 41
15 (3) 10 inches
16 (3) 125
17 (1) 40
18 (4) 100
19 (2) 230
20 16
21 90
22 104
23 20
24 48
25 12
26 a 40 b 40 c 40 d 40
e 110 f 110 g 35
86 Chapter 13: Geometry of the Circle
27 a 40 b 70 c 20
d 45 e 25 f 65
28 a 30 b 45 c 15
d 135 e 90
29 a C(0, 0), r 20
b Answers may vary. (0, 20), (0, 20)
30 a C(7, 11), r 9
b Answers may vary. (7, 20), (7, 2)
31 a C(3, 13), r 6
b Answers may vary. (3, 19), (3, 7)
32 a C(3, 5), r 2 √ 5
b Answers may vary. (3, 5 2 √ 5 ),
(3, 5 2 √ 5 )
33 (x 3) 2 (y 2) 2 16, C(3, 2), r 4
34 a (x 4) 2 (y 2) 2 49
b (x 5) 2 y 2 2
c x 2 y 2 169
d C(3, 0), r 5, (x 3) 2 y 2 25
e C(4, 4) r √ 10 , (x 4) 2 (y 4) 2 10
2 ) 2
35 6 2 6 2 (6 √
36 y 1 _
x 10
3
37 a (4, 3) and (4, 3) b secant
38 a (0, 3) and (3, 0) b secant
39 a (8, 2) and (2, 8) b secant
40 a (0, 5) and (3, 4) b secant
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
41 1. AOB COB
2. −−
AB −−
CB
3. AB CB
42 1. −−−
DA is the diameter of circle O.
2. Radii −−−
OA , −−−
OC , and −−−
OD
3. −−−
OA −−−
OC −−−
OD (Radii are equal.)
4. −−−
AD −−−
CD
5. AOD COD (SSS SSS)
6. AD CD
7. ABD CBD (Congruent arcs
are intercepted
by congruent inscribed
angles.)

43 1. −−−
MA −−−
OK , −−−
MD −−
OL
2. OAM and ODM are right angles.
3. −−−
AM −−−
DM
4. −−−
OM −−−
OM
5. OAM ODM (HL HL)
6. AOM LOM (CPCTC)
7. MK ML (Congruent angles
have congruent
arcs.)
Locus and
Constructions
14-1 Basic Constructions
(page 404)
Note: For exercises 1–20, check students’ constructions;
procedures may vary.
1 ab Use construction of congruent angles
procedure.
2 Use construction of a perpendicular to a line
through a given point on the line procedure.
3 ac Use construction of an angle bisector
procedure.
4 Use construction of a line perpendicular to
a line through a given point not on the line
procedure.
5 Use construction of a perpendicular bisector
of a line segment procedure.
6 Use construction of a parallel line through a
given point not on the line procedure.
7 Use construction of the median of a triangle
procedure.
8 Use construction of line perpendicular to a
line through a given point not on the line
procedure.
44 1. −−
OE −−
AB , OD −−
AC
2. AEO and ADO are right angles.
3. −−
AB −−
AC
4. −−
OE bisects −−
AB .
5. −−−
OD bisects −−
AC .
6. −−
AE −−−
AD
7. −−−
OA −−−
OA
8. AEO ADO (HL HL)
45 Use equal arcs are subtended by equal
chords.
CHAPTER
14
9 Use construction of a perpendicular bisector
procedure.
10 Use construction of an angle bisector
procedure.
11 a Use construction of a perpendicular line
procedure.
b Use construction of a perpendicular line
procedure and then construct an angle
bisector.
12 a Use construction of an equilateral triangle
procedure.
b Using angle constructed in a, construct an
angle bisector.
c Construct angle bisector of b.
13 Using the vertex of the angle as the center
of a circle and one of the legs as its radius,
construct a circle. Construct the diameter by
extending the radius.
14 Using the vertex of the angle as the center
of a circle and one of the legs as its radius,
construct a circle. Construct the diameter by
extending the radius. Construct a perpendicular
through the center of the circle.
14-1 Basic Constructions 87

15 a Use constructing a congruent angle
procedure.
b Use constructing a congruent angle
procedure.
c Construct a perpendicular bisector of a
segment to form a 90 angle. Then use
constructing a congruent angle procedure
to construct the sum.
d Use constructing an angle bisector
procedure.
e Use constructing an angle bisector
procedure for A. Then use constructing a
congruent angle procedure to construct
the sum.
16 Use construction of an equilateral triangle
procedure.
17 Use the construction of an equilateral
triangle procedure using the length of the
longer segment to construct the leg of the
isosceles triangle.
18 Using the compass, measure the radius and
construct a circle passing through the point
on the tangent line.
19 Use construction of a line tangent to a given
circle through a given point outside the circle
procedure.
20 Use procedures for constructing parallel lines
and perpendicular lines.
14-2 Concurrent Lines and
Points of Concurrency
(pages 407–408)
1 (4) obtuse
2 (3) at one of the vertices of the triangle
3 Check students’ constructions.
4 All at the same point
5 10
6 6 2 _
3
7 36
8 16.5
9 4.5
10 SP 4, SC 6
11 PB 15, PR 30
12 PA 15, QA 45
13 4 1 _
3
14 9
88 Chapter 14: Locus and Constructions
15 3 √ 3
16 QE 4, DE 6
17 a, b, c: all interior
18 a, b, c: all interior
19 a interior b on a side c exterior
20 a interior b at a vertex c exterior
21 Incenter: √ 3 , circumcenter: 2 √ 3
22 Incenter: 2 √ 3 , circumcenter: 4 √ 3
23 Incenter: 3 √ 3 , circumcenter: 6 √ 3
24 Incenter: 6, circumcenter: 12
25 Since AG GC, the base is the same and the
altitude is the same. Therefore, the area is the
same.
14-4 Six Fundamental Loci
and the Coordinate Plane
(pages 413–414)
1 (3) One concentric circle of radius 11 inches
2 (2) one line
3 (2) a point
4 (1) a circle of radius 4 with center at P
5 (4) a circle
Note: For exercises 6–17 check students’ sketches.
6 Circle with given point as center and
radius 3
7 Two parallel lines, one on each side of the
given line
8 One line parallel to and midway between the
given parallel lines
9 The perpendicular bisector of the segment
joining R and S
10 The perpendicular bisector of the segment
AB
11 A circle with radius 2.5 inches and the given
point as the center
12 The line that is the bisector of ABC
13 Two lines each the bisector of the vertical
angles
14 Two concentric circles with radii 5 and 9
15 One concentric circle midway between the
given circles
16 All the points in the interior of a circle with
the given point as the center and with a
radius 2 inches
17 A circle with the given point as the center
and a radius of 3 inches and all the points in
the exterior of that circle

18 Two lines, one on each side, parallel to the
given line and at a distance r from the given
line
19 One line parallel to the two parallel lines and
midway between them
20 The diameter of the circle that is the perpendicular
bisector of the given chord
21 The diameter of the circle that is the perpendicular
bisector of the given chord
22 A ray that is the angle bisector
23 A line perpendicular to the point on the
given line
24 The perpendicular bisector of the chord that
joins the two given points
Locus in the Coordinate Plane
(Pages 418–419)
1 (4) y 2 or y 2
2 (1) x 2
3 (2) x 5
4 (2) y 1
5 (4) (x 1) 2 (y 3) 2 25
6 (3) y 3 and x 0
7 (4) y x 1 and y x 5
8 y 1
9 y 2, y 10
10 y 4
11 y x 3 and y x 3
12 y x 3
13 x 1 and y 0
14 a (x 1) 2 (y 4) 2 16
b x 2 y 2 36
c x 2 (y 1) 2 9
d (x 1) 2 y 2 1
e x 2 (y 2) 2 6.25
f (x 1) 2 (y 3) 2 30.25
15 The showers could be placed anywhere on
the perpendicular bisector of the line segment
joining the diving boards, which is
35 ft from each of the diving boards.
14-5 Compound Locus and
the Coordinate Plane
(pages 421–422)
1 (3) a pair of points
2 (4) the empty set
3 (1) 0
4 (3) a pair of points
5 (3) 2
6 (2) 1
7 a 4 b 2 c 0
8 Find the point of concurrency (circumcenter)
of the perpendicular bisectors of the sides of
the triangle formed.
9 3
10 2
11 Parallel lines 2 inches from the given line,
one on each side
a Circle with radius 3, center R. 1 point
b Circle with radius 6, center R. 2 points
c Circle with radius 7, center R. 3 points
d Circle with radius 9, center R. 3 points
12 For all parts, check students’ sketches.
a Two intersecting circles, but not tangent.
2 points
b Two tangent circles. 1 point
c Two disjoint circles. 0 points
13 Sketch circle and two lines; 4 points.
14 Sketch a line and a circle; 2 points.
15 Sketch two lines bisecting vertical angles,
and two parallel lines; 4 points.
16 Sketch two concentric circles and two lines
bisecting vertical angles; 8 points.
17 The intersection of the line parallel to m and
k and midway between them, and a circle
with A as the center and d as the radius.
Case I d 1 _ f (0 points)
2
Case II d 1 _ f (1 point)
2
Case III d 1 _ f (2 points)
2
18 The intersection of the perpendicular
bisector of segment AB and a circle about
center A with radius x.
Case I Radius x 1 _ d (0 points)
2
Case II Radius x 1 _ d (1 point)
2
Case III Radius x 1 _ d (2 points)
2
Compound Loci and the Coordinate Plane
(pages 423–424)
1 (4) 4
2 (3) 2
3 (4) 4
4 a 2 b 1 c 0
14-5 Compound Locus and the Coorinate Plane 89

5 Two horizontal lines y 11 and y 1, and
vertical line x 4. Locus: 2 points.
6 Locus 2 points: (0, 8) and (8, 0)
7 Locus 2 points: (2, 7) and (2, 1)
8 Locus 2 points: (3, 1) and (2, 2)
9 a y 2 b x 4 c (4, 2)
d (x 2) 2 (y 2) 2 16 e one
10 a Circle with radius, d
b Two lines: x 1 and x 1
c (i) 1 point (ii) 3 points (iii) 4 points
14-6 Locus of Points
Equidistant From a Point
and a Line
(page 427)
Note: For exercises 1–10, check students’
sketches.
1 b (0, 2), (2, 3), (2, 3)
x 2
c y _ 2
4
2 b (2, 1), (2, 1), (6, 1)
2
c y _ x
x
8 _
2 1 _
2
3 b (2, 0), (2, 2), (6, 2)
2
c y
x
_
1
8 _ x
2 1 _
2
4 b (3, 0.5), (0, 1), (6, 1)
x 2
c y _ x 1
6
5 b (1, 0), (3, 4), (3, 4)
y 2
c x _ 1
8
6 b (3, 1.5), (6, 0), (0, 0)
x 2
c y
_ x
6
7 b (2, 2), (6, 4), (2, 4)
x 2
c y
_
1
8 _ x
2 5 _
2
8 b (1, 4), (4, 10), (4, 2)
2
y
c x _
2y
12 _
7
3 _
3
9 b (0, 0), (2, 4), (2, 4)
y 2
c x _
8
10 b (3, 1.5), (1, 2), (6, 2)
x 2
c y _ x 2
6
90 Chapter 14: Locus and Constructions
14-7 Solving Other Linear-
Quadratic and Quadratic-
Quadratic Systems
(page 430)
Note: For exercises 1–20, check students’
sketches.
1 (2, 0), (2, 0)
2 (2 √ 2 , 0), (2 √ 2 , 0)
3 (0, 5), (4, 3)
4 (3, 2)
5 (2, 3), (2, 5)
6 (2, 1), (1, 2)
7 (1, 1), (1, 1)
8 (5, 27), (1, 5)
9 ( 5 _ , 6) , (2, 5)
3
10 (4, 0), (4, 0)
11 ( √ 2 , 0), ( √ 2 , 0)
12 (3, 2), (6, 1)
13 (2 √ 2 , √ 2 ), (2 √ 2 , √ 2 ), ( √ 2 , 2 √ 2 ),
( √ 2 , 2 √ 2 )
14 (0.5, 1.25), (4, 3)
15 (1, 0)
16 (1, 3), (1, 3)
17 (2 √ 2 , √ 5 ), (2 √ 2 , √ 5 ), (2 √ 2 , √ 5 ),
(2 √ 2 , √ 5 )
18 (2, 1), (2, 1)
19 (3, 0)
20 (2, 4), (2, 0)
Chapter Review (pages 430–433)
1 (2) acute triangles
2 (4) median to side −−
AC
3 (2) AAS
4 (2) two circles
5 (2) X lies on the locus of points equidistant
from R and S.
6 (2) y 2x 3
7 (3) (x 3) 2 (y 4) 2 36
8 (1) x 3
9 (3) x 5
10 (1) (0, 0) and (0, 8)
11 (4) (0, 2) and (4, 2)
12 (3) perpendicular bisectors of the sides of the
triangle
13 (3) 4
14 (3) 3
15 (3) 2

16 (4) 4
17 (1) 90 x
18 8
19 x 3, PD 6, RP 12, RD 18
20 x 2, y 4, BE 9, AD 12
Note: For exercises 21–24, check students’
sketches.
21 Two concentric circles. Locus: radius 1 and
radius 7.
22 Circle, radius 3, x 2 y 2 9. Two lines,
x 4, x 4. Locus: 0 points
23 Circle: x 2 y 2 4. Circle: x 2 (y 4) 2 9.
Locus: 1 point, (0, 2)
24 Two lines parallel to line m on opposite sides
and 4 centimeters from m. Circle with center
H and radius 2. Locus: 1 point
25 (x 4) 2 (y 2) 2 1
26 y x 4
27 Two lines: y x 2 and y x 2
Note: For exercises 28–37, check students’
sketches.
28 b (2, 0.5), (5, 1), (1, 1)
2
c y _ x
2x
6 _
1
3 _
6
29 b (5, 7), (1, 3), (3, 1)
2
y
c x _
3y
8 _
1
4 _
8
30 b (2, 0), (2, 2), (6, 2)
x 2
c y
_
x
8 _
2 1 _
2
31 b (1, 0), (1, 4), (1, 4)
y 2
c x _ 1
8
32 (0, 5), (3, 4), (3, 4)
33 (0, 5), (3, 8)
34 (0, 3), (3, 9)
35 (2 √ 7 , 1 √ 7 ), (2
36 (3, 10), (1, 6)
√ 7 , 1 √ 7 )
37 (0, 4), ( √ 7 , 3), ( √ 7 , 3)
38 The intersection of the perpendicular bisector
of AB, and circle with A as center
Case I x 1 _ f (0 points)
2
Case II x 1 _ f (1 point)
2
Case III x 1 _ f (2 points)
2
39 The intersection of one circle with radius of
2.5, concentric with the given circles, and
two lines parallel to the given line, one on
each side at a distance x from the given line.
Case I x 2.5 (4 points)
Case II x 2.5 (2 points)
Case III x 2.5 (0 points)
40 The intersection of circle with center P
and radius d and circle with center Q and
radius d.
Case I x 2d (2 points)
Case II x 2d (1 point)
Case III x 2d (0 points)
41 The intersection of two lines parallel to line
m, one on each side at a distance d from line
m and the circle with center A and radius r.
Case I r d (0 points)
Case II r d (2 points)
Case III r d (4 points)
42 Locus: A line parallel to the north side and
south side of the courtyard and midway
between them. Since the width of the garden
is 60 ft, every point on this line will be equidistant
from the north and south walls and
at least 30 feet from the North Entrance.
43 Check students’ constructions of the orthocenter
of an acute triangle.
44 Check students’ constructions of the perpendicular
bisector of the three sides of a right
triangle.
45 Check students’ constructions of a circle that
passes through the three vertices of an obtuse
triangle.
Chapter Review 91

Each review has a total of 58 possible points. Use the following table, adapted from the Regents
Examinations, to convert the student’s raw score to a scaled score.
Raw Score Scaled Score Raw Score Scaled Score Raw Score Scaled Score
58 100 38 75 19 49
57 99 37 74 18 47
56 98 36 72 17 46
55 97 35 70 16 45
54 96 34 68 15 44
53 93 33 67 14 43
52 92 32 66 13 42
51 91 31 65 12 41
50 90 30 64 11 39
49 89 29 63 10 37
48 87 28 61 9 36
47 86 27 60 8 34
46 85 26 59 7 32
45 84 25 58 6 29
44 82 24 57 5 26
43 80 23 56 4 21
42 79 22 54 3 16
41 78 21 53 2 10
40 77 20 51 1 5
39 76
Chapters 1–2
(pages 434–437)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (4) a → r 2 (2) −−
AB 3 (3) p q
92
Cumulative Reviews

4 (4) 2BC AC
5 (1) If I win a scholarship, then I will play soccer.
6 (3) 2 _
3
7 (3) The triangle may be acute.
8 (4) If two angles are not congruent, then they are not right angles.
9 (2) right
10 (1) isosceles
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 AB CD EF, and an appropriate explanation is given.
1 AB CD EF, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
12 Score Explanation
2 3, and an appropriate explanation is given.
1 3, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 55, and an appropriate explanation is given.
1 55, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 9, and an appropriate explanation is given.
1 9, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–2 93

Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
4 a (i) scalene (ii) right triangle
94 Cumulative Reviews
b (i) isosceles (ii) acute
c (i) equilateral (ii) equiangular
d (i) isosceles (ii) obtuse
e (i) isosceles (ii) right triangle
f (i) scalene (ii) obtuse
3 Answered all but two parts correctly.
2 Answered all but three parts correctly.
1 Answered two parts correctly.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
16 Score Explanation
4 3 : 1, and an appropriate explanation is given.
3 An appropriate method is shown, but AB _ is given instead.
BD
2 An appropriate method is shown, but an incorrect answer is given.
1 3 : 1, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
17 Score Explanation
4 x 77, and an appropriate method is used.
3 An appropriate method is shown, but y is given instead.
2 An appropriate method is shown, but an incorrect answer is given.
1 x 77, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.

Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
6 a distributive property
b associative property of addition
c additive identity
d commutative property of multiplication
5 Student answers all but one part correctly.
3 Student answers only two parts correctly.
1 Student answers only one part correctly.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
19 Score Explanation
6 34, and an appropriate method is given.
5 An appropriate method is used, but a single arithmetical error is made.
4 An appropriate method is used, but two arithmetical errors are made.
3 An appropriate method is used, but student found m1 or m2.
2 An appropriate method is shown, but multiple computational mistakes are made.
1 34, but no appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
20 Score Explanation
6 a {6, 7, 8, 9, 10}
b {0, 2, 4, 6, 8, 10}
c {1, 3, 5}
d {0, 1, 2, 3, 4, 5, 7, 9}
5 One part is missing elements from the solution set.
4 Elements are missing from two solution sets.
3 Answered two parts correctly.
2 Answered all parts, but solution sets are missing elements.
1 Only one part is answered correctly.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–2 95

Chapters 1–3
(pages 438–441)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) 80
2 (2) 54
3 (1) Subtraction postulate
4 (4) If Cecilia is not a senior, then she does not take AP Calculus.
5 (4) 22
6 (3) If a triangle is not equilateral, then it is not isosceles.
7 (1) 20
8 (3) 1 _
2
9 (4) 3 _
2
10 (2) b c
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 70, and an appropriate explanation is given.
1 70, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
12 Score Explanation
2 Affirming the conclusion does not affirm the premise.
a The quadrilateral could be rhombus or any other figure.
b x could be 7 or any other positive number less than 7, so that x is not greater than 8.
1 Student answers only one part correctly.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 mr 20, and an appropriate explanation is given.
1 mr 20, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 RB 2, and an appropriate explanation is given.
1 RB 2, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
96 Cumulative Reviews

Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
4 mA 82, mB 82, and mC 16, and an appropriate explanation is given.
3 An appropriate method is used, but a single arithmetical error is made.
2 An appropriate method is used to find x, but the measures of the angles are not given.
1 mA 82, mB 82, and mC 16, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
16 Score Explanation
4 a ABD and DBE
b ABE and EBC
c DBC
d ABC
3 Student answers all but one part correctly.
2 Student answers only two parts correctly.
1 Student answers only one part correctly.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
17 Score Explanation
4 m4 50, m5 20, mABD 160, and an appropriate method is used.
3 An appropriate method is used, but a single arithmetical error is made.
2 m4 50, m5 20, but student did not give the measure of ABD.
1 m4 50, m5 20, mABD 160, but no appropriate method is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–3 97

Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
6 a If the sum of the measures of two acute angles is 90, the two angles are
complementary.
b If two acute angles are not complementary, then their sum is not 90.
c If the sum of the measures of two acute angles is not 90, then the two acute angles
are not complementary.
d Two acute angles are complementary if and only if the sum of their measures is 90.
5 Student answers all but one part correctly.
3 Student answers only two parts correctly.
1 Student answers only one part correctly.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
19 Score Explanation
1 (2) Reflexive postulate
2 (3) Addition postulate
1 (4) Partition postulate
2 (5) Substitution postulate
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
20 Score Explanation
1 (2) Definition of bisector
1 (3) Definition of midpoint
2 (4) Doubles of equals are equal.
2 (5) Substitution postulate
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–4
(pages 442–445) Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) 220 5 (2) 120
2 (2) a median 6 (1) 3 and 2 are nonadjacent complementary angles.
3 (4) 120 7 (2) b
4 (3) If the diagonals of a quadrilateral 8 (3) BAR and RAI
are not congruent, then the 9 (1) mx my
quadrilateral is not a rectangle. 10 (3) a b 4
98 Cumulative Reviews

Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 a True b False c False d True
1 Student answers only two parts correctly.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
12 Score Explanation
2 11, and an appropriate explanation is given.
1 11, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 68 and 112, and an appropriate explanation is given.
1 68 and 112, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 57.5, and an appropriate explanation is given.
1 57.5, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
4 AC 12, AB 16, BC 16, and an appropriate explanation is given.
3 An appropriate method is used, but a single arithmetical error is made.
2 An appropriate method is used, but multiple arithmetical errors are made.
1 AC 12, AB 16, BC 16, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–4 99

16 Score Explanation
4 a 91, b 35, c 54, d 91, and an appropriate explanation is given.
3 An appropriate method is used, but a single arithmetical error is made.
2 An appropriate method is used, but multiple arithmetical errors are made.
1 a 91, b 35, c 54, d 91, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
17 Score Explanation
4 mPTC 50, and an appropriate explanation is given.
3 An appropriate method is used, but a single arithmetical error is made.
2 An appropriate method is used, but multiple arithmetical errors are made.
1 mPTC 50, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
3 a Given A F and −−
AB −−
EF . −−−
AD −−
CF and by the reflexive property of
congruence −−−
DC −−−
DC . By the addition postulate, −−−−
ADC −−−
FCD . Therefore, by SAS,
ABC DEF.
2 a An appropriate proof with correct conclusions is shown, but one reason is faulty
and/or one statement is missing.
1 a A correct conclusion is reached and a reason is given, but no appropriate method
is shown.
3 b Given D A. C is the midpoint of −−−
AD , then −−
AC −−−
DC .
ACB DCE because vertical angles are congruent. Therefore,
by ASA, ABC DEC.
2 b An appropriate proof with correct conclusions is shown, but one reason is faulty
and/or one statement is missing.
1 b A correct conclusion is reached and a reason is given, but no appropriate method
is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
100 Cumulative Reviews

19 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. B D 1. Given.
2. −−
BC −−−
DC 2. Given.
3. BCA DCE 3. Vertical angles are congruent.
4. ACB ECD 4. ASA ASA.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1 ACB ECD and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
20 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. −−
BD is the median to −−
AC . 1. Given.
2. D is the midpoint of −−
AC . 2. Definition of a median of an isosceles triangle.
3. −−−
AD −−−
DC 3. Definition of a midpoint.
4. −−
AB −−
BC 4. Given.
5. −−
BD −−
BD 5. Reflexive Property of Congruence.
6. I II 6. SSS SSS.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or
three steps are missing.
2 An appropriate proof with correct conclusion is used, but more than three reasons are
faulty or more than three steps are missing or have errors.
1 I II and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–4 101

Chapters 1–5
(pages 446–449)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (4) −−
QB , −−−
QA , ____
QM 6 (1) isosceles
2 (1) q → p 7 (3) −−−
CD bisects −−
AB .
3 (3) 70 8 (2) 3 4
4 (2) 9 (3) 28
5 (4) ADB WDB 10 (3) 78.5
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 6, and an appropriate explanation is given.
1 6, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
12 Score Explanation
2 x 8, A 81, B 99, and an appropriate explanation is given.
1 x 8, A 81, B 99, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 256, and an appropriate explanation is given.
1 256, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
1 a 45, and an appropriate explanation is given.
1 b 90, and an appropriate explanation is given.
1 (a) 45 and (b) 90, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
102 Cumulative Reviews

Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
2 a BCA ACD
2 b −−−
AD −−
AB
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
16 Score Explanation
4 The following proof or another appropriate proof is given.
Statements Reasons
1. −−
DB −−
AC , −−−
AD −−−
DC 1. Given.
2. DBA and ADC are right
angles.
3. 1 is complementary to x. 3. Given.
2. Definition of perpendicular lines.
4. m1 mx 90 4. Definition of complementary angles.
5. mx m2 mADC 5. Addition postulate.
6. mx m2 90 6. Substitution postulate.
7. m1 mx mx m2 7. Substitution postulate.
8. m1 m2 8. Subtraction postulate.
9. 1 2 9. Angles with equal measures are congruent.
3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1 1 2 and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
17 Score Explanation
2 a Since −−−
CD bisects −−
AB at E, −−
AE −−
BE . Given 1 2, and because vertical angles are
congruent, AEC BED. Hence, by ASA, ACE BDE.
1 a A correct conclusion is reached and reason is given, but no appropriate method
is shown.
2 b Given 1 2 and −−
LN −−−
MO . LNM and LNO are right angles. Since right
angles are congruent, LNM LNO. −−
LN −−
LN by the reflexive property of
congruence. Therefore, by ASA, LMN LON.
1 b A correct conclusion is reached and reason is given, but no appropriate method
is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–5 103

Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
6 The following proof or another appropriate proof is given.
104 Cumulative Reviews
Statements Reasons
1. PET 1. Given.
2. −−−
PCB −−
ET ; −−−
TCA −−
EP 2. Given.
3. CBT and CAP are right
angles.
3. Definition of perpendicular lines.
4. CBT CAP 4. Right angles are congruent.
5. ACP BCP 5. Vertical angles are congruent.
6. −−
PA −−
TB 6. Given.
7. ACP BCT 7. AAS AAS.
8. −−
CP −−
CT 8. CPCTC.
9. PCT is isosceles. 9. Definition of isosceles triangle.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or
three steps are missing.
2 An appropriate proof with correct conclusion is used, but more than three reasons are
faulty or more than three steps are missing or have errors.
1 PCT is isosceles and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.

19 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. −−
AC −−
BD 1. Given.
2. −−
AC −−
BC −−
BD −−
BC 2. Subtraction postulate.
3. −−
AB −−−
CD 3. Partition postulate.
4. −−
GB −−−
AD ; −−
EC −−−
AD 4. Given.
5. GBA and ECD are right angles. 5. Definition of perpendicular lines.
6. GBA ECD 6. Right angles are congruent.
7. AGB DEC 7. Given.
8. AGB DEC 8. AAS AAS.
9. A D 9. CPCTC.
10. −−
AF −−
DF 10. Converse of isosceles triangle theorem.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or
three steps are missing.
2 An appropriate proof with correct conclusion is used, but more than three reasons are
faulty or more than three steps are missing or have errors.
1 −−
AF −−
DF and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–5 105

20 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. −−
BD is the median of ABC. 1. Given.
2. −−
BA −−
BC 2. Given.
3. ABC is isosceles. 3. Definition of isosceles triangle.
4. −−
BD −−
AC 4. The median from the vertex angle of an
5.
isosceles triangle is perpendicular to the base.
−−
BD is an altitude to −−
AC . 5. Definition of an altitude.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty
or multiple steps are missing or have errors.
1 −−
BD is an altitude to −−
AC and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–6
(pages 450–453)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) a ab b 2 6 (3) 4z
2 (1) (x 9, y 5)
3 (4)
7 (2)
−−
AE −−
BE 8 (2) −−
XY −−−
MA
4 (2) (x, y) 9 (1) (4, 5)
5 (4) ~r → ~p 10 (2) dilation
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 25, and an appropriate explanation is given.
1 25, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
106 Cumulative Reviews

12 Score Explanation
2 100, and an appropriate explanation is given.
1 100, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 a (3, 4) b (3, 4) c (4, 3) d (3, 2)
1 Student answers only two parts correctly.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 39 and 51, and an appropriate explanation is given.
1 39 and 51, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
4 45-45-90 isosceles triangle and an appropriate explanation is given.
3 An appropriate method is used, but student does not identify the type of triangle.
2 An appropriate method is used, but multiple computational errors are made.
1 45-45-90 isosceles triangle, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
16 Score Explanation
4 36, and an appropriate explanation is given.
3 An appropriate method is used, but one computational error is made.
2 An appropriate method is used, but multiple computational errors are made.
1 36, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–6 107

17 Score Explanation
4 The following proof or another appropriate proof is given.
Statements Reasons
1. −−
AB −−
EB 1. Given.
2. −−
BC −−
BD 2. Given.
3. B B 3. Reflexive property of congruence.
4. ABC EBD 4. SAS SAS.
5. 1 2 5. CPCTC.
3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1 1 2 and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
1 a K(1, 1), A(5, 2), T(3, 5)
1 b K (1, 1) A(5, 2), T (3, 5)
2 c
y
T'
5
4
3
T
2
A'
1
K' K
5 4 3 2 1 1
1
K"
2
3
4
2 3 4
A
x
5
A"
5
T"
2 d r x-axis
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
108 Cumulative Reviews

19 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. Isosceles ABC, with −−
AB −−
BC 1. Given.
2. −−
BD bisects ABC. 2. Given.
3. ABD CBD 3. Definition of angle bisector.
4. −−
BD −−
BD 4. Reflexive property of congruence.
5. ABD CBD 5. SAS SAS.
6. −−−
AD −−−
CD 6. CPCTC.
7. −−
BD bisects −−
AC . 7. Definition of bisector.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1 −−
BD bisects −−
AC and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
20 Score Explanation
1 a C
1 b D
2 c B
2 d C
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
Chapters 1–7
(pages 454–456)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) supplementary 6 (4) −−
AB −−−
RM
2 (4) (0, 6) 7 (2) 3(b a)
3 (1) 6 8 (1) B is the largest angle.
4 (3) I and III only 9 (1) 15
5 (2) 2 10 (1) x _
2
Chapters 1–7 109

Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 6, and an appropriate explanation is given.
1 6, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
12 Score Explanation
2 (7, 5), and an appropriate explanation is given.
1 (7, 5), but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 144, and an appropriate explanation is given.
1 144, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 (3, 3), and an appropriate explanation is given.
1 (3, 3), but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
1 a (4, 0)
b (4, 2)
1 c (6, 3)
d (3, 6)
2 e (8, 3)
f (6, 3)
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
110 Cumulative Reviews

16 Score Explanation
4 The following proof or another appropriate proof is given.
Right angles B and A are congruent because −−
AB −−
DB and
−−
AC −−−
DC . −−−
BW −−−
CW . BWA CWD because vertical angles are congruent.
Therefore, by ASA ABW DCW. −−−
AW −−−
DW because of CPCTC. Hence, AWD is
isosceles by definition of an isosceles triangle.
3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1 AWD is isosceles and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
17 Score Explanation
4 The following proof or another appropriate proof is given.
Statements Reasons
1. BC AB 1. Given.
2. mBAC mBCA 2. The measures of the angles opposite unequal
sides are unequal.
3. 1 _ mBAC
2 1 _ mBCA 3. Division postulate.
2
4. −−−
DA bisects BAC. 4. Given.
5. mDAC 1 _ mBAC 5. Definition of a bisector.
2
6. −−−
DC bisects BCA. 6. Given.
7. mDCA 1 _ mBCA 7. Definition of a bisector.
2
8. mDAC mDCA 8. Substitution postulate.
9. DC DA 9. The lengths of the sides opposite two unequal
angles of a triangle are unequal.
3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1 DC DA and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–7 111

Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
1 a (4, 1)
1 b √
10
2 c y 3x 13
2
d Slope −−−
MD
112 Cumulative Reviews
1 _ , slope of the median is 3. The slopes are negative
3
reciprocals; therefore, the median is perpendicular to −−−
MD .
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
19 Score Explanation
4 a Altitude from D to −−
AC is x 0. Altitude from C to −−−
AD is y 1 _ x 3.
2
Altitude from A to −−−
CD is y x 3.
3 a An appropriate method is used, but one computational error is made.
2 a An appropriate method is used, but multiple computational errors are made.
1 a The altitudes are identified, but no explanation is given.
2 b (0, 3), and appropriate work is shown.
1 b (0, 3), but no work is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
20 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. ABC, −−
AB −−
BC 1. Given.
2. ABC is isosceles. 2. Definition of isosceles triangle.
3. A C 3. Definition of isosceles triangle.
4. mC mT 4. Definition of an exterior angle.
5. mA mT 5. Substitution postulate.
6. PT AP 6. Greater side opposite greater angle.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty
or multiple steps are missing or have errors.

Chapters 1–8
(pages 457–460)
1 PT AP and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) rotation of 180 6 (1) y
2 (1) 5 7 (3) (5, 3)
3 (1) x 2y 5 8 (2) negative and less than the x-intercept of m
4 (1) 23 9 (2) 2
5 (1) {1, 2, 3} 10 (4) (9, 7)
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 m1 36, m2 72, m3 36, mAOE 144, and an appropriate explanation is
given.
1 m1 36, m2 72, m3 36, mAOE 144, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
12 Score Explanation
2 (0, 1), and an appropriate explanation is given.
1 (0, 1), but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 (5, 1), and an appropriate explanation is given.
1 (5, 1), but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 105, and an appropriate explanation is given.
1 105, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–8 113

Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
4 2 : 3, and an appropriate explanation is given.
3 An appropriate method is used, but a single arithmetical error is made.
2 An appropriate method is used, but multiple arithmetical errors are made.
1 2 : 3, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
16 Score Explanation
4 x 7, mP 62, mQ 28, and an appropriate explanation is given.
3 An appropriate method is used, but a single arithmetical error is made.
2 An appropriate method is used, but multiple arithmetical errors are made.
1 x 7, mP 62, mQ 28, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
17 Score Explanation
4 The following proof or another appropriate proof is given.
Statements Reasons
1. −−−
AD −−
BC , −−
DE −−
CE , −−
EA −−
EB 1. Given.
2. ADE BCE 2. SSS SSS.
3. 1 2 3. CPCTC.
4. 3 4 4. Base angles of an isosceles triangle are
congruent.
5. 1 3 2 4 5. Addition postulate.
6. DAB CBA 6. Partition postulate.
3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1 DAB CBA and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
114 Cumulative Reviews

Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
6 2. If two sides are congruent, the triangle is isosceles.
3. Base angles of isosceles triangles are congruent.
4. Exterior angle of a triangle is greater than either nonadjacent interior angle.
5. Substitution postulate.
6. Longest side of a triangle is opposite the angle with the largest measure.
5 One reason is faulty or missing.
4 Two reasons are faulty or missing.
3 Three reasons are faulty or missing.
2 More than three reasons are faulty or missing.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Check students’ graphs. The coordinates are listed below.
19 Score Explanation
1 a A(6, 6), B(12, 6), C(6, 15)
2 b A(6, 6), B(12, 6), C(6, 15)
2 c A(6, 4), B(6, 10), C(15, 4)
1 d D 3
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
20 Score Explanation
1 a Student graphs points A and B correctly.
3 b x 4 and an appropriate explanation is given. Student graphs point C correctly.
2 c 0, and an appropriate explanation is given.
1 Correct answers for b and c, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–8 115

Chapters 1–9
(pages 461–463)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) right
2 (4) −−
6 (1) 34
AE bisects BAC.
7 (3) SSS
3 (4) 4, 5, 6
8 (1) 290
4 (1) (5, 3)
9 (3) 3,600
5 (2) congruent
10 (1) 30
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 x is divisible by 2, and an appropriate explanation is given.
1 x is divisible by 2, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
12 Score Explanation
2 155, and an appropriate explanation is given.
1 155, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 (9, 2), and an appropriate explanation is given.
1 (9, 2), but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 (4, 2), and an appropriate explanation is given.
1 (4, 2), but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
116 Cumulative Reviews

Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
4 a True b True c False
d True e False f True
3 Student answers all but one part correctly.
2 Student answers two or three parts incorrectly.
1 Student answers four or five parts incorrectly.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
16 Score Explanation
4 18, and an appropriate explanation is given.
3 An appropriate method is used, but a single arithmetical error is made.
2 An appropriate method is used, but multiple arithmetical errors are made.
1 18, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
17 Score Explanation
4 a Yes b Yes c Yes
d No e No f Yes
3 Student answers all but one part correctly.
2 Student answers two or three parts incorrectly.
1 Student answers four or five parts incorrectly.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
6 AB √ 85 , BC √ 85 , AC √ 68 , and an appropriate explanation is given.
5 An appropriate method is shown, but one computational mistake is made.
4 An appropriate method is shown, but two computational mistakes are made.
3 An appropriate method is shown, but wrong sides of the triangle are shown congruent.
2 An appropriate method is shown, but more than two computational mistakes are
made.
1 An appropriate method is shown, but no conclusion is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–9 117

19 Score Explanation
6 The following proof or another appropriate proof is given.
118 Cumulative Reviews
Statements Reasons
1. ABC is equilateral. 1. Given.
2. mBAC 60 2. Interior angles of an equilateral
triangle are 60.
3. BCA and BCE are a linear pair. 3. Definition of linear pair.
4. BCE is supplementary to BCA. 4. If two angles form a linear pair, then
they are supplementary.
5. mBCA mBCE 180 5. Definition of supplementary angles.
6. mBCE 120 6. Subtraction postulate.
7. −−−
CD bisects BCE. 7. Given.
8. mDCE 1 _
mBCE 8. Definition of angle bisector.
2
9. mDCE 60 9. Substitution.
10. mDCE mBAC 10. Substitution.
11. DCE BAC 11. Angles with equal measures are
congruent.
12. CD −−
AB 12. Lines with congruent corresponding
angles are parallel.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1 CD −−
AB and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.

20 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. −−
BC −−−
AD 1. Given.
2. −−
BC −−−
AD 2. Given.
3. EAD FCB 3. Alternate interior angles are congruent.
4. −−
BF −−
ED 4. Given.
5. BFC DEA 5. Alternate interior angles are congruent.
6. BFC DEA 6. AAS AAS.
7. −−
BF −−
ED 7. CPCTC.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1 −−
BF −−
ED and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–10
(pages 464–466)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (2) 360
2 (4) 140
3 (1) 22
4 (3) y x
5 (3) 8
6 (3) a rhombus
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
7 (1) y 1 _ x
3 10 _
3
8 (4) The diagonals bisect the angles of
the parallelogram.
9 (1) adjacent
10 (3) m4 mB
2 120, and an appropriate explanation is given.
1 120, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–10 119

12 Score Explanation
2 31, and an appropriate explanation is given.
1 31, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 7 √ 2 , and an appropriate explanation is given.
1 7 √ 2 , but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 9, and an appropriate explanation is given.
1 9, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
4 6, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but multiple computational errors are made.
1 6, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
16 Score Explanation
4 A(3, 6), B(2, 1), C(9, 1), and the transformations are performed correctly.
3 Student arrives at the correct answer, but performs one incorrect transformation.
2 Student performs transformations in the wrong order.
1 A(3, 6), B(2, 1), C(9, 1), but no appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
120 Cumulative Reviews

17 Score Explanation
4 51, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but multiple computational errors are made.
1 51, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
3 a Slope −−−
DA
1 _ , slope
3 −−
2
AR _ , slope
3 −−
1
RT _ ,
3 −−
TD is a vertical line. Since
two sides have the same slope, −−−
DA −−
RT . DART is a trapezoid.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
3 b AK TA, and an appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
19 Score Explanation
3 a The following proof or another appropriate proof is given.
Statements Reasons
1. −−−
AD −−
BE 1. Given.
2. −−
AE −−
BD
3.
2. Given.
___
DE ___
DE 3. Reflexive property of congruence.
4. ADE BED 4. SSS SSS.
5. BDE AED 5. CPCTC.
1 BDE AED and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
3 b The following proof or another appropriate proof is given.
Statements Reasons
1. −−−
AD −−
BE 1. Given.
2. −−
AE −−
BD 2. Given.
3. −−
AB −−
AB 3. Reflexive property of congruence.
4. ADB BEA 4. SSS SSS.
5. FAB FBA 5. CPCTC.
1 FAB FBA and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–10 121

20 Score Explanation
6 The following proof or another appropriate proof is given.
122 Cumulative Reviews
Statements Reasons
1. Quadrilateral ABCD and
diagonal −−−−
AFEC
1. Given.
2. −−
BF −−
AC , −−
DE −−
AC 2. Given.
3. BFA and DEC are right
angles.
3. Definition of perpendicular lines.
4. BFA DEC 4. Right angles are congruent.
5. −−
BF −−
DE , −−
AE −−
CF 5. Given.
6. ADE CBF 6. SAS SAS.
7. −−−
AD −−
BC 7. CPCTC.
8. DAE BCF 8. CPCTC.
9. −−−
AD −−
BC 9. Alternate interior angles of two parallel lines
are congruent.
10. ABCD is a parallelogram. 10. Definition of a parallelogram.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1 ABCD is a parallelogram and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–11
(pages 467–469)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (1) b _
a
6 (3) 160 cm
2 (3) equilateral and equiangular
2
7 (1) 2 √ 6
3 (3) 120
4 (1) −−
AC
5 (2) 9
8 (3) 3 _
5
9 (3) The sphere is inscribed in the cube.
10 (1) parallelogram

Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 x 116, y 32, and an appropriate explanation is given.
1 x 116, y 32, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
12 Score Explanation
2 n 1, and an appropriate explanation is given.
1 n 1, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 116.5, and an appropriate explanation is given.
1 116.5, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 50, and an appropriate explanation is given.
1 50, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
4 3 √ 10 , and an appropriate explanation is given.
3 An appropriate method is used, but a single arithmetical error is made.
2 An appropriate method is used, but multiple arithmetical errors are made.
1 3 √ 10 , but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–11 123

16 Score Explanation
4 DN √ 106 , NA √ 106 , making DNA isosceles.
Slope −−−
9
DN _ and slope
5 −−−
5
NA _ . Since the slopes are negative reciprocals of each
9
other, −−−
DN −−−
NA . Therefore, DNA is an isosceles right triangle.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but DNA is not proven to be an isosceles right
triangle.
1 An appropriate method is used, but multiple computational errors are made.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
17 Score Explanation
4 6 √ , and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but multiple computational errors are made.
1 6 √ , but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
124 Cumulative Reviews

Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. −−−
DA −−
AB , −−−
DC −−
CB 1. Given.
2. BAD and BCD are right angles. 2. Definition of perpendicular lines.
3. BAD BCD 3. Right angles are congruent.
4. 1 2 4. Given.
5. BAE BCE 5. Subtraction postulate.
6. −−
BD −−
AC at E 6. Given.
7. BEA and BEC are right angles. 7. Definition of perpendicular lines.
8. BEA BEC 8. Right angles are congruent.
9. −−
BE −−
BE 9. Reflexive property of congruence.
10. ABE CBE 10. AAS AAS.
11. 3 4 11. CPCTC.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/
or one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1 3 4 and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
19 Score Explanation
2 a Student proves that MIKE is a rhombus by showing that
MI IK KE EM √ 40 or by some other appropriate method.
1 a The correct conclusion is reached and a reason is given, but no appropriate method
is shown.
2 b Student shows that MIKE is not a square by showing that the
diagonals are not perpendicular. Slope −−
1
MI _ and the slope
3 −− 3.
IK
1 b The correct conclusion is reached and a reason is given, but no appropriate method
is shown.
2 c 32, and an appropriate explanation is given.
1 c 32, but no appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–11 125

20 Score Explanation
2 a 3, and an appropriate explanation is given.
1 3, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
2 b 36, and an appropriate explanation is given.
1 36, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
2 c 1 _ , and an appropriate explanation is given.
4
1 1 _ , but no explanation is given.
4
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–12
(pages 470–472)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) It has a slope of 0.
2 (2) are congruent
3 (3) 50
4 (3) cylinder
5 (4) 130
6 (3) 16 x 2
7 (2) 10 √ 2
8 (1) 1 : 3
9 (1) 4 _
25
10 (3) 4x y 2
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 225, and an appropriate explanation is given.
1 225, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
126 Cumulative Reviews

12 Score Explanation
2 (0, 4) and (6, 0), and an appropriate explanation is given.
1 (0, 4) and (6, 0), but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
1 a 9 _ , and an appropriate explanation is given.
16
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
1 b 27 _ , and an appropriate explanation is given.
64
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 102, and an appropriate explanation is given.
1 102, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
2 a 22, and an appropriate explanation is given.
1 a 22, but no appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
2 b 78, and an appropriate explanation is given.
1 b 78, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
16 Score Explanation
4 2 √ 3 , and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but multiple computational errors are made.
1 2 √ 3 , but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–12 127

17 Score Explanation
4 6 √
5 , and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but multiple computational errors are made.
1 6 √ 5 , but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
6 Student uses an appropriate method, such as showing that both pairs of opposite sides
are parallel, by showing that slope −−− 0, slope −−− 0; slope −−−
b
AD ME MA _
a
, slope −−
b
ED _
a
.
5 An appropriate method is used, but one computational error is made.
4 An appropriate method is used, but two computational errors are made.
3 An appropriate method is used, but three computational errors are made.
2 More than three computational errors are made.
1 An appropriate method is used, but does not prove that MADE is a parallelogram.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
128 Cumulative Reviews

19 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. ABC is isosceles. 1. Given.
2. −−
AB −−
AC 2. Definition of isosceles triangle.
3. AMC is isosceles. 3. Given.
4. −−−
AM −−−
CM 4. Definition of isosceles triangle.
5. −−−
PM −−−
QM 5. Given.
6. −−−
AM −−−
QM −−−
CM −−−
PM 6. Addition postulate.
7. −−−
AQ −−
CP 7. Partition postulate.
8. BAC BCA 8. Isosceles triangle theorem.
9. MAC MCA 9. Isosceles triangle theorem.
10. BAC MAC
BCA MCA
10. Subtraction postulate.
11. BAM BCP 11. Partition postulate.
12. ABQ CBP 12. SAS SAS.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1 ABQ CBP and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–12 129

20 Score Explanation
6 The following proof or another appropriate proof is given.
130 Cumulative Reviews
Statements Reasons
1. Parallelogram ABCD 1. Given.
2. A C 2. Opposite angles of a parallelogram are
congruent.
3. m1 m2 3. Given.
4. 1 2 4. Angles with equal measures are
congruent.
5. 1 and 4 are a linear pair.
2 and 3 are a linear pair.
6. 1 and 4 are supplementary. 2 and
3 are supplementary.
5. Definition of linear pair.
6. Two angles that form a linear pair are
supplementary.
7. 3 4 7. Supplements of congruent angles are
congruent.
8. AFE CHG 8. AA.
9. FE _
FA
GH
_
CH
9. Corresponding parts of similar
triangles are in proportion.
10. FE CH GH FA 10. The product of means equals the
product of extremes.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1 FE CH GH FA and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–13
(pages 473–475)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (4) ABD CDB
2 (2) a median
3 (1) 12
4 (1) 1 : 2
5 (1) 2
6 (3) R 270
7 (1) 30
8 (1) 12
9 (3) III and IV
10 (4) −−
TA −−
TO

Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 (16, 14), and an appropriate explanation is given.
1 (16, 14), but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
12 Score Explanation
2 55, and an appropriate explanation is given.
1 55, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
13 Score Explanation
2 38, and an appropriate explanation is given.
1 38, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 30.5, and an appropriate explanation is given.
1 30.5, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
4 y x 3, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but the equation of a parallel line is given.
1 y x 3, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–13 131

16 Score Explanation
4 mADE 93, mBGA 87, mABC 87, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but one of the angle measures found is incorrect.
1 mADE 93, mBGA 87, mABC 87, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
17 Score Explanation
4 h r _
, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but h is found in terms of r and s.
1 h r _
, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
6 h 12, and an appropriate explanation is given.
5 An appropriate method is used, but one computational error is made.
4 An appropriate method is used, but two computational errors are made.
3 An appropriate method is used, but three computational errors are made.
2 More than three computational errors are made.
1 h 12, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
132 Cumulative Reviews

19 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. Parallelogram ABCD;
−−−
EAB ; −−−
DCF
1. Given.
2. −−
AB −−−
DC 2. Definition of a parallelogram.
3. −−
EB −−
DF 3. Collinearity.
4. AEH CFG 4. Alternate interior angles are congruent.
5. BAH BCD 5. Opposite angles of a parallelogram are
congruent.
6. GCF and BCD are a linear
pair. HAE and BAH are a
linear pair.
7. GCF and BCD are supplementary.
HAE and BAH are
supplementary.
6. Definition of linear pair.
7. Two angles that form a linear pair are
supplementary.
8. GCF HAE 8. Supplements of congruent angles are
congruent.
9. a EAH FCG 9. AA.
10. EA _
AH
FC
_
CG
10. Corresponding parts of similar triangles are in
proportion.
11. b EA CG AH CG 11. The product of means equals the product of
extremes.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 An appropriate proof with correct conclusion is shown for part a only.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1 EAH FCG and EA CG AH CG, and a reason is given, but no appropriate
method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–13 133

20 Score Explanation
Chapters 1–14
(pages 476–479)
1 a m
BC 72, and an appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
1 b mEDC 108, and an appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
1 c mBCR 36, and an appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
1 d mAKB 72, and an appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
2 e mRF 72, and an appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (1) x 3
2 (3) 3 : 5
3 (3) 125
4 (2) 45
5 (3) 2 √ 2
134 Cumulative Reviews
6 (2) 8.8
7 (2) 2x 3
8 (1) {C, D, G, H}
9 (4) y 3x 5
10 (4) similar triangles
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11 Score Explanation
2 720, and an appropriate explanation is given.
1 720, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
12 Score Explanation
2 144, and an appropriate explanation is given.
1 144, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.

13 Score Explanation
2 13, and an appropriate explanation is given.
1 13, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
14 Score Explanation
2 80, and an appropriate explanation is given.
1 80, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15 Score Explanation
4 Each side is 2b, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but h is found in terms of r and s.
1 Each side is 2b, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
16 Score Explanation
4 3 _ , and an appropriate explanation is given.
10
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but h is found in terms of r and s.
1 3 _ , but no explanation is given.
10
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
17 Score Explanation
4 30,776.3 in. 3 , and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but h is found in terms of r and s.
1 30,776.3 in. 3 , but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–14 135

Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score Explanation
3 a
x y 7
136 Cumulative Reviews
y
10
9
8
7
6
5
4
3
2
1
3y 2x 6
7 6 5 4 3 2 1
1
1 2 3 4 5 6 7 8 9 10
y 2
2
3
4
5
6
7
x
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
3 b 45, and an appropriate explanation is given.
2 An appropriate method is used, but a single computational error is made.
1 45, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
19 Score Explanation
4 a Slope −−−
AX
1 and slope −−− 1. Since the slopes are negative reciprocals of each
ME
other, the lines are perpendicular. The midpoint of −−−
AX is
(4, 3), which is point E. Therefore, −−−
ME is the perpendicular bisector.
3 An appropriate method is used, but a single arithmetical error is made.
2 An appropriate method is used to show the lines are perpendicular, but no midpoint is
found.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
2 b Shows that MAX is isosceles by MX MA √ 68 , and an appropriate explanation
is given.
1 An appropriate method is used, but a single computational error is made.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.

20 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. AE CE 1. Given.
2. ___
AE ___
CE 2. Congruent arcs have congruent chords.
3. ABE EBD 3. Congruent central angles have congruent
chords.
4. −−
FE is tangent to O at E. 4. Given.
5. FEB is a right angle. 5. Definition of a tangent.
6. FEB and BED are a linear
pair.
7. FEB and BED are
supplementary.
6. Definition of linear pair.
7. Two angles that form a linear pair are
supplementary.
8. BED is a right angle. 8. A supplement to a right angle is a right
angle.
9. Diameter −−−
BOE 9. Given.
10.
BCE is a semicircle. 10. Definition of a semicircle.
11. BAE is a right angle. 11. An angle inscribed in a semicircle is always a
right angle.
12. BAE BED 12. Right angles are congruent.
13. ABE EBD 13. AA.
14. BE _
BA
BD _
BE
14. Corresponding parts of similar triangles are in
proportion.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2 An appropriate proof with correct conclusion is shown, but multiple reasons are faulty
or multiple steps are missing or have errors.
1 BE _
BD
BA _ , and a reason is given, but no appropriate method is shown.
BE
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–14 137

Practice Regents Examination One
(pages 483–489)
Part I
Allow a total of 56 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) 1 _
2
2 (2) 8 √ 2
3 (1) 1 _ b
2 2
4 (1) 48
5 (2) 8 √ 3
6 (2) 12
7 (4) r : 2
8 (4) 106
9 (1) 8
10 (3) AEC is a
right angle.
138
11 (3) (9, 3)
12 (1) 6
13 (4) 16 : 81
14 (3) 18
15 (4) 1
16 (2) 26
17 (4) 16
18 (2) y
19 (2) 6 √ 2
20 (1) 86
21 (2) 18
22 (2) (0, 1)
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
29 Score Explanation
Practice Geometry
Regents Examinations
23 (4) 4
24 (3) right
25 (2) 9
26 (3) x 3
27 (4) I and II only
28 (3) 1 _ V
8
2 (x 2) 2 (y 4) 2 2.25, and an appropriate explanation is given.
1 (x 2) 2 (y 4) 2 2.25, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.

30 Score Explanation
2 100, and an appropriate explanation is given.
1 100, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
31 Score Explanation
2 30, and an appropriate explanation is given.
1 30, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
32 Score Explanation
2 3 √ 2 , and an appropriate explanation is given.
1 3 √ 2 , but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
33 Score Explanation
2 16, and an appropriate explanation is given.
1 16, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
34 Score Explanation
2 (1, 0), and an appropriate explanation is given.
1 (1, 0), but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
35 Score Explanation
4 y x 1, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but multiple computational errors are made.
1 y x 1, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Regents Examination One 139

36 Score Explanation
4 36, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but multiple computational errors are made.
1 36, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
37 Score Explanation
4 27, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but multiple computational errors are made.
1 27, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
38 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. −−
LA −−
EG 1. Given.
2. LAR GER 2. Alternate interior angles are congruent.
3. ALR EGR 3. Alternate interior angles are congruent.
4. ALR EGR 4. AA.
5. AR _
RE
RL _
RG
5. Corresponding parts of similar triangles are in proportion.
6. Rhombus PARL 6. Given.
7. AR PA; RL PL 7. Definition of a rhombus.
8. AR RG RL RE 8. The product of means equals the product of extremes.
9. PA RG RE PL 9. Substitution postulate.
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or
three steps are missing.
2 An appropriate proof with correct conclusion is used, but more than three reasons are
faulty or more than three steps are missing or have errors.
1 PA RG RE PL and a reason is given, but no appropriate method is shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
140 Practice Geometry Regents Examinations

Practice Regents Examination Two
(pages 490–496)
Part I
Allow a total of 56 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (1) −−
EK −−
EY
2 (4) 150
3 (1) 4 √ 3
4 (3) 1 _
2
5 (3) 36
6 (4) R 90
7 (4) Use the slope formula
to prove diagonals are
perpendicular.
8 (2) 105
9 (4) {2, 5, √ 29 }
10 (2) 120
11 (2) 20 √ 3
12 (3) 24
13 (1) 28
14 (3) 160
15 (3) 2
16 (1) 4 _ 3
units
3
17 (4) 6 s 2x
18 (4) 11
19 (3) 8
20 (3) 15
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
29 Score Explanation
21 (2) 5
22 (1) 64 cm 3
23 (4) 4
24 (3) perpendicular
bisectors of the sides
of the triangle
25 (2) 5 √ 2
26 (4) 36
27 (1) BC AB
28 (3) equal in area
2 69, and an appropriate explanation is given.
1 69, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
30 Score Explanation
2 (x 4) 2 (y 5) 2 9, and an appropriate explanation is given.
1 (x 4) 2 (y 5) 2 9, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
31 Score Explanation
2 (2x, 3y), and an appropriate explanation is given.
1 (2x, 3y), but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Regents Examination Two 141

32 Score Explanation
2 108, and an appropriate explanation is given.
1 108, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
33 Score Explanation
2 6.8, and an appropriate explanation is given.
1 6.8, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
34 Score Explanation
2 117, and an appropriate explanation is given.
1 117, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
35 Score Explanation
4 11.55, and an appropriate explanation is given.
3 An appropriate method is used, but answer is not rounded to the nearest hundredth.
2 An appropriate method is used, but multiple computational errors are made.
1 11.55, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
36 Score Explanation
4 OC 21, mCOD 69.5, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but only one of the two answers is found.
1 OC 21, mCOD 69.5, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
142 Practice Geometry Regents Examinations

37 Score Explanation
4 Secant points of intersection are (0, 5) and (2, 4), and an appropriate explanation is
given.
3 Secant is given, but graph is incorrect.
2 Graph is shown, but secant is not given.
1 Secant is given, but graph is not shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
38 Score Explanation
6 Any appropriate method is used to prove that ABCD is a parallelogram, but not a rectangle.
For example: showing that the diagonals bisect each other but are not congruent,
or showing that no consecutive sides have negative reciprocal slopes.
5 An appropriate method is used, but a single computational error is made.
4 An appropriate method is used, but two computational errors are made.
3 An appropriate method is used to prove that ABCD is a parallelogram but does not
show that it is not a rectangle.
2 An appropriate method is used to prove that ABCD is a parallelogram, but more than
two computational errors are made and student does not show that it is not a rectangle.
1 An appropriate method is used, but ABCD is not shown to be a parallelogram, and
multiple computational errors are made.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Regents Examination Three
(pages 497–504)
Part 1
Allow a total of 56 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (2) 50
2 (4) 16 : 25
3 (4) (7, 7 √ 3 )
4 (2) 18 cm 2
5 (3) {2, 7, 10}
6 (3) (3, 6.5)
7 (3) 80 √ 3
8 (3) ABCD is a parallelogram
whose diagonals
bisect each other.
9 (2) 2 _
3
10 (2) 156
11 (2) (x 3) 2 (y 7) 2 2
12 (3) y 1 _ x 4
2
13 (3) d 5
14 (4) 11
15 (1) 60
16 (2) 32 √ 3
17 (2) 16 16 √ 2
18 (4) x 1
19 (1) 5
20 (2) y 5x 2
21 (3) 4 √ 3
22 (1) perpendicular bisectors
of the sides
23 (3) 120
24 (4) obtuse
25 (3) 60
26 (4) 210
27 (1) The perpendicular
bisector of the chord
of a circle bisects the
intercepted arc.
28 (1) (3, 5)
Practice Regents Examination Three 143

Part II
For each question, use the specific criteria to award a maximum of 2 credits.
29 Score Explanation
2 216 cubic inches, and an appropriate explanation is given.
1 216 cubic inches, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
30 Score Explanation
1 a PA 9, and an appropriate explanation is given.
1 b m CB 53 and an appropriate explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
31 Score Explanation
2 16 inches, and an appropriate explanation is given.
1 16 inches, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
32 Score Explanation
2 60 _ , and an appropriate explanation is given.
13
1 60 _ , but no explanation is given.
13
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
33 Score Explanation
2 32, and an appropriate explanation is given.
1 32, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
34 Score Explanation
2 4, and an appropriate explanation is given.
1 4, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
144 Practice Geometry Regents Examinations

Part III
For each question, use the specific criteria to award a maximum of 4 credits.
35 Score Explanation
4 16, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but incorrect radius is found.
1 16, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
36 Score Explanation
4 40, and an appropriate explanation is given.
3 An appropriate method is used, but a single computational error is made.
2 An appropriate method is used, but multiple computational errors are made.
1 40, but no explanation is given.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
37 Score Explanation
4 2. Reflexive property of congruence
3. Perpendicular lines meet to form right angles.
4. All right angles are congruent.
5. Two right triangles are similar if an acute angle of one triangle is congruent to an
acute angle of the other.
6. Corresponding sides of similar triangles are in proportion.
7. The product of means equals the product of extremes.
3 One reason is incorrect.
2 Two reasons are incorrect.
1 More than two reasons are incorrect.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Regents Examination Three 145

Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
38 Score Explanation
6 The following proof or another appropriate proof is given.
Statements Reasons
1. Parallelogram ABCD,
with −−−
ABG
146 Practice Geometry Regents Examinations
1. Given.
2. −−
AB −−−
CD , −−−
AG −−−
CD 2. Definition of a parallelogram.
3. MGB MDC 3. Alternate interior angles are congruent.
4. BMG CMD 4. Vertical angles are congruent.
5. M is the midpoint of −−
BC . 5. Given.
6. −−−
BM −−−
MC 6. Definition of midpoint.
7. a BGM CDM 7. AAS AAS.
8. −−
BG −−−
CD 8. CPCTC.
9. −−
AB −−−
CD 9. Opposite sides of a parallelogram are congruent.
10. b −−
AB −−
BG 10. Transitive postulate of congruence
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4 A faulty conclusion or no conclusion is drawn from correct statements and reasoning.
3 An appropriate proof with correct conclusion is shown, but no conclusion is made
for −−
AB −−
BG .
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1 BGM CDM and −−
AB −−
BG , and a reason is given, but no appropriate proof is
shown.
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.