8.5 Use Properties of Trapezoids and Kites

Your Notes
8.5 Use Properties of Trapezoids
and Kites
Goal p Use properties of trapezoids and kites.
VOCABULARY
Trapezoid
Bases of a trapezoid
Base angles of a trapezoid
Legs of a trapezoid
Isosceles trapezoid
Midsegment of a trapezoid
Kite
Copyright © Holt McDougal. All rights reserved. Lesson 8.5 Geometry Notetaking Guide 221

Your Notes
8.5 Use Properties of Trapezoids
and Kites
Goal p Use properties of trapezoids and kites.
VOCABULARY
Trapezoid A trapezoid is a quadrilateral with exactly
one pair of parallel sides.
Bases of a trapezoid The parallel sides of a
trapezoid are the bases.
Base angles of a trapezoid A trapezoid has two pairs
of base angles. Each pair shares a base as a side.
Legs of a trapezoid The nonparallel sides of a
trapezoid are the legs.
Isosceles trapezoid An isosceles trapezoid is a
trapezoid in which the legs are congruent.
Midsegment of a trapezoid The midsegment of
a trapezoid is the segment that connects the
midpoints of its legs.
Kite A kite is a quadrilateral that has two pairs of
consecutive congruent sides, but opposite sides
are not congruent.
Copyright © Holt McDougal. All rights reserved. Lesson 8.5 Geometry Notetaking Guide 221

Your Notes
Example 1 Use a coordinate plane
Show that CDEF is a trapezoid.
Solution
Compare the slopes of opposite sides.
Slope of }
DE 5 5
Slope of }
CF 5 5 5
1
y
D(1, 3) E(4, 4)
C(0, 0)
The slopes of }
DE and }
CF are the same, so }
DE }
CF .
Slope of }
EF 5 5 5
Slope of }
CD 5 5 5
The slopes of }
EF and }
CD are not the same, so }
EF is
to }
CD .
Because quadrilateral CDEF has exactly one pair of
, it is a trapezoid.
THEOREM 8.14
If a trapezoid is isosceles, then each
pair of base angles is .
If trapezoid ABCD is isosceles, then
∠A > ∠ and ∠ > ∠C.
THEOREM 8.15
If a trapezoid has a pair of congruent
, then it is an isosceles
trapezoid.
If ∠A > ∠D (or if ∠B > ∠C), then trapezoid
ABCD is isosceles.
4
B C
F(6, 2)
A D
B C
A D
THEOREM 8.16
A trapezoid is isosceles if and only
B C
if its diagonals are .
Trapezoid ABCD is isosceles if and only if > .
A D
222 Lesson 8.5 Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.
x

Your Notes
Example 1 Use a coordinate plane
Show that CDEF is a trapezoid.
y
D(1, 3) E(4, 4)
Solution
Compare the slopes of opposite sides. 1
Slope of C(0, 0) 4
} 4 2 3 1
DE 5 } 5 }
4 2 1 3
Slope of } 2 2 0 2
CF 5 } 5 } 5
6 2 0 6 1
}
3
The slopes of }
DE and }
CF are the same, so }
DE i }
CF .
Slope of } 2 2 4 22
EF 5 } 5 } 5 21
6 2 4 2
Slope of }
CD 5
3 2 0
}
1 2 0
5 3
}
1 5 3
The slopes of }
EF and }
CD are not the same, so }
EF is
not parallel to }
CD .
Because quadrilateral CDEF has exactly one pair of
parallel sides , it is a trapezoid.
THEOREM 8.14
If a trapezoid is isosceles, then each
pair of base angles is congruent .
If trapezoid ABCD is isosceles, then
∠A > ∠ D and ∠ B > ∠C.
THEOREM 8.15
If a trapezoid has a pair of congruent
base angles , then it is an isosceles
trapezoid.
If ∠A > ∠D (or if ∠B > ∠C), then trapezoid
ABCD is isosceles.
B C
F(6, 2)
A D
B C
A D
THEOREM 8.16
A trapezoid is isosceles if and only
B C
if its diagonals are congruent . A D
Trapezoid ABCD is isosceles if and only if }
AC > }
BD .
222 Lesson 8.5 Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.
x

Your Notes
Example 2 Use properties of isosceles trapezoids
Kitchen A shelf fitting into a
cupboard in the corner of a
L
kitchen is an isosceles trapezoid.
Find m∠N, m∠L, and m∠M.
508
M
K N
Solution
Step 1 Find m∠N. KLMN is an , so
∠N and ∠ are congruent base angles, and
m∠N 5 m∠ 5 .
Step 2 Find m∠L. Because ∠K and ∠L are consecutive
interior angles formed by @##$ KL intersecting two
parallel lines, they are . So,
m∠L 5 2 5 .
Step 3 Find m∠M. Because ∠M and ∠ are a pair
of base angles, they are congruent, and
m∠M 5 m∠ 5 .
So, m∠N 5 , m∠L 5 , and m∠M 5 .
Checkpoint Complete the following exercises.
1. In Example 1, suppose the coordinates of point E are
(7, 5). What type of quadrilateral is CDEF? Explain.
2. Find m∠C, m∠A, and m∠D A
in the trapezoid shown.
1358
B C
Copyright © Holt McDougal. All rights reserved. Lesson 8.5 Geometry Notetaking Guide 223
D

Your Notes
Example 2 Use properties of isosceles trapezoids
Kitchen A shelf fitting into a
cupboard in the corner of a
L
kitchen is an isosceles trapezoid.
Find m∠N, m∠L, and m∠M.
508
M
K N
Solution
Step 1 Find m∠N. KLMN is an isosceles trapezoid , so
∠N and ∠ K are congruent base angles, and
m∠N 5 m∠ K 5 508 .
Step 2 Find m∠L. Because ∠K and ∠L are consecutive
interior angles formed by @##$ KL intersecting two
parallel lines, they are supplementary . So,
m∠L 5 1808 2 508 5 1308 .
Step 3 Find m∠M. Because ∠M and ∠ L are a pair
of base angles, they are congruent, and
m∠M 5 m∠ L 5 1308 .
So, m∠N 5 508 , m∠L 5 1308 , and m∠M 5 1308 .
Checkpoint Complete the following exercises.
1. In Example 1, suppose the coordinates of point E are
(7, 5). What type of quadrilateral is CDEF? Explain.
Parallelogram; opposite pairs of sides are
parallel.
2. Find m∠C, m∠A, and m∠D A
in the trapezoid shown.
m∠C 5 1358, m∠A 5 458,
m∠D 5 458
1358
B C
Copyright © Holt McDougal. All rights reserved. Lesson 8.5 Geometry Notetaking Guide 223
D

Your Notes
THEOREM 8.17: MIDSEGMENT THEOREM FOR
TRAPEZOIDS
The midsegment of a trapezoid is A B
parallel to each base and its length
is one half the sum of the lengths
M N
of the bases.
C
If }
MN is the midsegment of trapezoid ABCD, then
}
MN i , }
MN i } , and MN 5 ( 1 ).
In the diagram, }
Example 3 Use the midsegment of a trapezoid
of trapezoid PQRS. Find MN.
Solution
Use Theorem 8.17 to find MN.
MN is the midsegment 16 in.
P Q
MN 5 ( 1 ) Apply Theorem 8.17.
Checkpoint Complete the following exercise.
3. Find MN in the trapezoid at P
the right.
30 ft
S
M
N
M
Q
12 ft
R
D
S R
9 in.
5 ( 1 ) Substitute for PQ and
for SR.
5 Simplify.
The length MN is inches.
224 Lesson 8.5 Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.
N

Your Notes
THEOREM 8.17: MIDSEGMENT THEOREM FOR
TRAPEZOIDS
The midsegment of a trapezoid is A B
parallel to each base and its length
is one half the sum of the lengths
M N
of the bases.
C
If }
MN is the midsegment of trapezoid ABCD, then
}
MN i }
AB , }
MN i }
DC , and MN 5 1
} ( AB 1 CD ).
2
In the diagram, }
Example 3 Use the midsegment of a trapezoid
of trapezoid PQRS. Find MN.
Solution
Use Theorem 8.17 to find MN.
Checkpoint Complete the following exercise.
3. Find MN in the trapezoid at P
the right.
MN 5 21 ft
MN is the midsegment 16 in.
P Q
MN 5 1
}
2 ( PQ 1 SR ) Apply Theorem 8.17.
30 ft
S
M
N
M
Q
12 ft
R
D
S R
9 in.
5 1
}
2 ( 16 1 9 ) Substitute 16 for PQ and
9 for SR.
5 12.5 Simplify.
The length MN is 12.5 inches.
224 Lesson 8.5 Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.
N

Your Notes
Homework
THEOREM 8.18
If a quadrilateral is a kite, then its
diagonals are . B
If quadrilateral ABCD is a kite,
then ⊥ .
THEOREM 8.19
If a quadrilateral is a kite, then
exactly one pair of opposite angles
are congruent.
B
If quadrilateral ABCD is a kite and }
BC > }
BA ,
then ∠A ∠C and ∠B ∠D.
Example 4 Apply Theorem 8.19
Find m∠T in the kite shown at the right.
Solution
Copyright © Holt McDougal. All rights reserved. Lesson 8.5 Geometry Notetaking Guide 225
C
A
C
A
Q 708
By Theorem 8.19, QRST has exactly one
pair of opposite angles.
T
888
S
Because ∠Q À ∠S, ∠ and ∠T must be
congruent. So, m∠ 5 m∠T. Write and solve
an equation to find m∠T.
m∠T 1 m∠R 1 1 5 Corollary to
Theorem 8.1
m∠T 1 m∠T 1 1 5 Substitute m∠T
for m∠R.
(m∠T) 1 5 Combine like
terms.
m∠T 5 Solve for m∠T.
Checkpoint Complete the following exercise.
4. Find m∠G in the kite shown G H
858
at the right.
J
758
I
D
D
R

Your Notes
Homework
THEOREM 8.18
If a quadrilateral is a kite, then its
diagonals are perpendicular . B
If quadrilateral ABCD is a kite,
then }
AC ⊥ }
BD .
THEOREM 8.19
If a quadrilateral is a kite, then
B
exactly one pair of opposite angles
are congruent.
If quadrilateral ABCD is a kite and }
BC > }
BA ,
then ∠A > ∠C and ∠B À ∠D.
Example 4 Apply Theorem 8.19
Find m∠T in the kite shown at the right.
Solution
Copyright © Holt McDougal. All rights reserved. Lesson 8.5 Geometry Notetaking Guide 225
C
A
C
A
Q 708
By Theorem 8.19, QRST has exactly one
pair of congruent opposite angles.
Because ∠Q À ∠S, ∠ R and ∠T must be
T
888
S
congruent. So, m∠ R 5 m∠T. Write and solve
an equation to find m∠T.
m∠T 1 m∠R 1 708 1 888 5 3608 Corollary to
Theorem 8.1
m∠T 1 m∠T 1 708 1 888 5 3608 Substitute m∠T
for m∠R.
2 (m∠T) 1 1588 5 3608 Combine like
terms.
m∠T 5 1018 Solve for m∠T.
Checkpoint Complete the following exercise.
4. Find m∠G in the kite shown
at the right.
m∠G 5 1008
G
858
H
J
758
I
D
D
R