12/5/12 Study Guide 5.1, 5.2

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GOAL
Learn how to . . .
' identify pairs of
angles formed by
transversals and
lines
So you can. . .
' find the measures
of these angles
' analyze real-world
examples of
intersecting lines
LL2
Terms to Know
TFansversal (p.219)
Parallel Lines and Transversals
Application
When a violinist plays two strings together with one stroke of the bow, it is
called a double stop. A double stop is an example of a transversal intersecting
two lines.
a line that intersects two or more other lines in the same
plane at different points
Same-side interior angles (p.219)
two angles that lie on the same side of a transversal
between the two lines that it intersects
Alternate interior angles (p.219)
two angles that lie on opposite sides of a transversal
between the two lines that it intersects
Corresponding angles (p. 219)
two angles that lie on the same side of a transversal, in
corresponding positions with respect to the two lines that
it intersects
string (line)
Example / Illustration
Ll and L2 are same-side
interior angles.
L3 and L4 are alternate
interior angles.
L5 and L6 are corresponding
angles.
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Uruo¡nsrANDtNc rrle Mntn loens
Angles formed by a transversal
when a transversal intersects two lines, several types of angles are formed.
Among these angle types are same-side interior angles, alternate interior angles,
and corresponding angles.
Use the diagram at the r¡ght.
1. Which line is the transversal?
2. Name all pairs of same-side interior angles.
3. Name all pairs of alternate interior angles.
4. Name all pairs of corresponding angles.
Use the diagram at the fight. Classify each pair of angles as corre
sponding angles, altetnate interior angles, ot same-side inte¡íor angles.
5. LI4 and Ll5
7. LI3 and LI5
Example 1
Draw a transversal j that intersects lines k and ^/. Number the angles formed. Name all
pairs of same-side interior angles, alternate interior angles, and corresponding angles.
Solution r
Use the road map shown at the r¡ght.
9. Grand Avenue is a transversal of which roads?
10. Maple Street is a transversal of which roads?
Corre sponding angles postulnte
lf two parallel lines are intersected by a transversal,
then corresponding angles are congruent.
Same-side interior angles
L3 and L6
L4 and L5
Alternate interior angles
L3 and L5
L4 and L6
6. LI2 and LI6
8. LI4 and Ll\
Corresponding angles
LI and L5
L2 and L6
L3 and L7
L4 and L8
Oak St. Elm St. Maple St.
rc kll !,,tnen
Ll = L2.
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In the diagram at the right, m ll n'
Find the measure of each angle.
a. Ll
b. L2
c. L3
Solution r
a. By the Conesponding Angles Postulate, mLl = ll0" '
b. Ll and L2 are supplementary because they form a linear
pair. So, mL2 = 180" - mLI = 180' - 110o = 70o'
c. Since Ll and L3 ate vertical angles, mL3 = mLl = II0" '
In the diagram at the right, j ll
of each angle.
Ll-. LI
L3. L3
k and n ll m. Find the measure
L2.
L2
L4. L4
For Exercises 15 and 16, find the values of x, y, and z.
15.
17. Open-ended Draw a transversal f that intersects parallel lines ru and n.
Measure one of the angles formed. On your diagram, write the measures of
all eight angles formed. How many different angle measures are there?
16.
Spiral Review
19. The endpoinrs of AB areÁ(l, 0, 5) and B(7,2, -1). Find the midpoint and
length of AB. (Section 4.6)
Tell whether each statement is True ot False.lf the statement is false, sketch
a counterexamPle. (Section 2'5)
19. If a figure is a parallelogram, then it is a rectangle'
20. A parallelogram is a quadrilateral with both pairs of opposite sides parallel'
LL4
Study Guide, GEOMETRY: EXPLORATIONS AND APPLICATIONS
Copyright @ McDougal Littell lnc. All rights reserved'
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t
GOAt
Learn how to . . .
' find the measures
of alternate interior
angles and sameside
interior angles
' identify trapezoids
So you can. . .
' prove statements
about these angles
' find congruent
angles in real-world
objects
Terms to Know
UnoensrnNDtNc rHE MAIN lo¡ns
Theorems about parallel lines &nd trønsversals
Alternate Interior Angles Theorem
If two parallel lines are intersected by
a transversal, then alternate interior
angles are congruent.
If klll,then L|= L3.
Properties of Parallel Lines
Applicøtion
Here are two geometric patterns that are used for quilts. Each is a regular
hexagon that can be translated and repeated many times to cover a quilt top.
Each hexagon contains six shaded trapezoids.
TFapezoid (p.228)
a quadrilateral with exactly one pair of parallel sides
Bases of a trapezoid (p.228)
the two parallel sides of a trapezoid
Legs of a trapezoid (p.228)
the two non-parallel sides of a trapezoid
Same-Side Interior Angles Theorem
If two parallel lines are intersected by a
transversal, then same-side interior
angles are supplementary.
If j ll k, then mL4 + mL5 = 180".
l
k
Example / lllustration
base
TrapezoidABCD
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L

Use the diagram at the right' Find the measure of each anEle'
t. Ll
3. L3
Find the values of x, Y, and z'
Trapezoids
a. Find the measures of Z1 and L2'
Solution ¡
2. L2
4. L4
b. Find the values of x and Y'
a. By the Alternate Interior Angles Theorem' mLl = 75" '
By the Same-Side Interior Angles Theorem' mL2 = 180' - 75'' or 105"'
b. By the Same-Side Interior Angles Theorem' xo = 180" - 126" ' ot x = 54'
By the Alternate Interior Angles Theorem' 2y" = 126" and therefore i = 63'
A trapezoid is a quadrilateral with exactly one pair of parallel sides called the
bases. The other two (non-parallel) sides are called legs'
116
Use traPezoid ABCD at the right'
a. Name the bases and the legs'
b. Find the measures of LA, LC' and LD'
Study Guide, GEOMETRY: EXPLORATIONS AND APPLICATIONS
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Solution r
a. bases: BC and AD

Example 3 E
Write the key steps of a proof of the Same-Side
Interior Angles Theorem.
Given: j llk
Prove: mL7 + mL2 = I80"
Solution ¡r
Key steps:
l. Ll = L3 (Corresponding Angles Postulate)
2. mL3 + mL2 = 180o (Linear Pair Theorem)
3. mLl + mLZ = 180" (Substitution Property)
15. Use the key steps given in Example 3 to
Same-Side Interior Angles Theorem.
16. Write the key steps of a proof of the
Alternate Interior Angles Theorem.
Givenz kll I
Prove: Ll = L2
17. Write a two-column proof.
Given: Trapezoid ABCD, wrrh[õ 116ð
Prove: mLl + mL2 = I80"
Key step: LI and L2 are supplementary.
(Same-Side Interior Angles Theorem)
write a paragraph proof of the
18. Write a paragraph proof of this theorem: All pairs of consecutive angles of a
p arallelo gram are supplementary'
Spiral Review
Find each unknown angle measwe. (Section 2.2)
19.
118
20.
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