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LESSON
4.2
Name
Transversals and
Parallel Lines
4.2
Date
Transversals and Parallel Lines
Essential Question: How can you prove and use theorems about angles formed by
transversals that intersect parallel lines?
Resource
Locker
Common Core Math Standards
Explore
The student is expected to:
COMMON
CORE
Class
G-CO.C.9
Exploring Parallel Lines and
Transversals
A transversal is a line that intersects two coplanar lines at two different points. In the figure,
line t is a transversal. The table summarizes the names of angle pairs formed by a transversal.
Prove theorems about lines and angles.
Mathematical Practices
COMMON
CORE
MP.3 Logic
t
1 2
4 3
5 6
8 7
Language Objective
Explain to a partner how to identify the angles formed by two parallel
lines cut by a transversal.
ENGAGE
View the Engage section online. Discuss the photo.
Ask students to describe ways that a real-world
example of parallel lines and a transversal like this
example differ from parallel lines and a transversal
that might appear in a geometry book. Then preview
the Lesson Performance Task.
Corresponding angles lie on the same side of the transversal and on the same
sides of the intersected lines.
∠ 1 and ∠5
Same-side interior angles lie on the same side of the transversal and between
the intersected lines.
∠3 and ∠6
Alternate interior angles are nonadjacent angles that lie on opposite sides of
the transversal between the intersected lines.
∠3 and ∠ 5
Alternate exterior angles lie on opposite sides of the transversal and outside
the intersected lines.
∠1 and ∠ 7
Recall that parallel lines lie in the same plane and never
intersect. In the figure, line ℓ is parallel to line m, written ℓǁm.
The arrows on the lines also indicate that they are parallel.
ℓ
m
ℓ‖ m
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Lesson 2
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HARDCOVER PAGES 153160
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Resource
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Essential
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GE_MNLESE385795_U2M04L2.indd 175
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Credits:
PREVIEW: LESSON
PERFORMANCE TASK
∠3 and ∠6
∠3 and ∠ 5
∠1 and ∠ 7
ℓ
m
Lesson 2
175
Module 4
Lesson 4.2
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Example
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Possible answer: Start by establishing a postulate
about certain pairs of angles, such as same-side
interior angles. The postulate allows you to prove a
theorem about other pairs of angles, such as
alternate interior angles. You can then use the
postulate and the theorem to prove other theorems
about other pairs of angles, such as corresponding
angles.
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©Ruud Morijn Photographer/Shutterstock
Essential Question: How can you
prove and use theorems about angles
formed by transversals that intersect
parallel lines?
Angle Pair
p
175
13/06/15
11:06 PM
6/9/15 1:04 AM
When parallel lines are cut by a transversal, the angle pairs formed are either congruent or
supplementary. The following postulate is the starting point for proving theorems about
parallel lines that are intersected by a transversal.
EXPLORE
Same-Side Interior Angles Postulate
Exploring Parallel Lines and
Transversals
If two parallel lines are cut by a transversal, then the pairs of same-side interior angles
are supplementary.
Follow the steps to illustrate the postulate and use it to find angle measures.
A
t
1 2
4 3
5 6
8 7
B
INTEGRATE TECHNOLOGY
Draw two parallel lines and a transversal, and number the angles formed from 1 to 8.
The properties of parallel lines and transversals can
be explored using geometry software. Students can
display lines and angle measures on screen, rotate a
line so it is parallel to another, and observe the
relationships between angles.
p
q
Identify the pairs of same-side interior angles.
∠4 and ∠5; ∠3 and ∠6
C
QUESTIONING STRATEGIES
What does the postulate tell you about these same-side interior angle pairs?
When two lines are cut by a transversal, how
many pairs of corresponding angles are
formed? How many pairs of same-side angles? 4; 2
Given p ‖ q, then ∠4 and ∠5 are supplementary and ∠3 and ∠6 are supplementary.
D
If m∠4 = 70°, what is m∠5? Explain.
m∠5 = 110°; ∠4 and ∠5 are supplementary, so m∠4 + m∠5 = 180°. Therefore
What does the Same-Side Interior Angles
Postulate tell you about the measure of a pair
of same-side interior angles? The sum of their
measures is 180˚̊.
70° + m∠5 = 180°, so m∠5 = 110°.
Reflect
Explain how you can find m∠3 in the diagram if p ‖ q and m∠6 = 61°.
∠3 and ∠6 are supplementary, so m∠3 + m∠6 = 180°. Therefore m∠3 + 61° = 180°, so
m∠3 = 119°.
2.
What If? If m ‖ n, how many pairs of same-side interior angles are shown in the
figure? What are the pairs?
Two pairs; ∠3 and ∠5, ∠4 and ∠6
Module 4
176
m
n
1 3 5 7
t
2 4
6 8
© Houghton Mifflin Harcourt Publishing Company
1.
Lesson 2
PROFESSIONAL DEVELOPMENT
GE_MNLESE385795_U2M04L2.indd 176
Math Background
6/9/15 1:04 AM
When two parallel lines are cut by a transversal, alternate interior angles have the
same measure, corresponding angles have the same measure, and same-side
interior angles are supplementary. One of these three facts must be taken as a
postulate and then the other two may be proved. In the work text, the statement
about same-side interior angles is taken as the postulate. This is closely related to
one of the postulates stated in Euclid’s Elements: “If a straight line falling on two
straight lines makes the interior angles on the same side less than two right angles,
the two straight lines, if produced indefinitely, meet on that side on which are the
angles less than the two right angles.”
Transversals and Parallel lines
176
Explain 1
EXPLAIN 1
Proving that Alternate Interior Angles
are Congruent
Other pairs of angles formed by parallel lines cut by a transversal are alternate interior angles.
Proving that Alternate Interior
Angles are Congruent
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles
have the same measure.
CONNECT VOCABULARY
To prove something to be true, you use definitions, properties, postulates, and theorems that you
already know.
Have students experience the idea of what the term
alternate means by shading in alternating figures in a
series of figures.
Example 1
Prove the Alternate Interior Angles Theorem.
Given: p ‖ q
Prove: m∠3 = m∠5
t
1 2
4 3
5 6
8 7
QUESTIONING STRATEGIES
p
q
Complete the proof by writing the missing reasons. Choose from the following reasons.
You may use a reason more than once.
Do you have to write a separate proof for
every pair of alternate interior angles in the
figure? Why or why not? No, you can write the proof
for any pair of alternate interior angles. The same
reasoning applies to all the pairs.
• Same-Side Interior Angles Postulate
• Subtraction Property of Equality
• Given
• Substitution Property of Equality
• Definition of supplementary angles
• Linear Pair Theorem
© Houghton Mifflin Harcourt Publishing Company
Statements
Reasons
1. p ‖ q
1. Given
2. ∠3 and ∠6 are supplementary.
2. Same-Side Interior Angles Postulate
3. m∠3 + m∠6 = 180°
3. Definition of supplementary angles
4. ∠5 and ∠6 are a linear pair.
4. Given
5. ∠5 and ∠6 are supplementary.
5. Linear Pair Theorem
6. m∠5 + m∠6 = 180°
6. Definition of supplementary angles
7. m∠3 + m∠6 = m∠5 + m∠6
7. Substitution Property of Equality
8. m∠3 = m∠5
8. Subtraction Property of Equality
Reflect
3.
In the figure, explain why ∠1, ∠3, ∠5, and ∠7 all have the same measure.
m∠1 = m∠3 and m∠5 = m∠7 (Vertical Angles Theorem), m∠3 = m∠5 (Alternate Interior
Angles Theorem), so m∠1 = m∠3 = m∠5 = m∠7 (Transitive Property of Equality).
Module 4
177
Lesson 2
COLLABORATIVE LEARNING
GE_MNLESE385795_U2M04L2 177
Peer-to-Peer Activity
Have students use lined paper or geometry software to draw two parallel lines and
a transversal that is not perpendicular to the lines. Instruct the student’s partner to
shade or mark the acute angles with one color and the obtuse angles with another
color. Let students use a protractor or geometry software to see that all the angles
with the same color are congruent, and that pairs of angles with different colors
are supplementary.
177
Lesson 4.2
10/14/14 7:01 PM
4.
Suppose m∠4 = 57° in the figure shown. Describe two different ways to determine m∠6.
By the Alternate Interior Angles Theorem, m∠6 = 57°. Also ∠4 and ∠5 are supplementary,
EXPLAIN 2
so m∠5 = 123°. Since ∠5 and ∠6 are supplementary, m∠6 = 57°.
Explain 2
Proving that Corresponding Angles
are Congruent
Proving that Corresponding Angles
are Congruent
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Explaining and justifying arguments is at the
Two parallel lines cut by a transversal also form angle pairs called corresponding angles.
Corresponding Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of corresponding angles
have the same measure.
Example 2
heart of this proof-based lesson. You may wish to
have students pair up with a “proof buddy.” Students
can exchange their work with this partner and check
that the partner’s logical arguments make sense.
Complete a proof in paragraph form for the Corresponding Angles Theorem.
Given: p‖q
t
1 2
4 3
Prove: m∠4 = m∠8
5 6
8 7
p
q
QUESTIONING STRATEGIES
By the given statement, p‖q. ∠4 and ∠6 form a pair of alternate interior angles .
m∠4 = m∠6
So, using the Alternate Interior Angles Theorem,
What can you use as reasons in a proof? given
information, properties, postulates, and
previously-proven theorems
.
∠6 and ∠8 form a pair of vertical angles. So, using the Vertical Angles Theorem,
m∠6 = m∠8
. Using the Substitution Property of Equality
in m∠4 = m∠6, substitute m∠4 for m∠6. The result is
m∠4 = m∠8
© Houghton Mifflin Harcourt Publishing Company
Reflect
5.
Use the diagram in Example 2 to explain how you can prove the Corresponding Angles
Theorem using the Same-Side Interior Angles Postulate and a linear pair of angles.
By the Same-Side Interior Angles Theorem, m∠4 + m∠5 = 180°. As a linear pair,
m∠4 + m∠1 = 180°. Therefore m∠4 + m∠1 = m∠4 + m∠5 , so m∠1 = m∠5.
6.
How can you check that the first and last
statements in a two-column proof are
correct? The first statement should match the
“Given” and the last statement should match
the “Prove.”
.
Suppose m∠4 = 36°. Find m∠5. Explain.
m∠5 = 144°; Since ∠1 and ∠4 form a linear pair, they are supplementary. So,
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Draw parallel lines and a transversal on a
m∠1 = 144°. Using the Corresponding Angles Theorem, you know that m∠1 = m∠5.
So, m∠5 = 144°.
Module 4
178
transparency. Trace an acute and an obtuse angle
formed by the lines onto another transparency, and
use them to find congruent angles.
Lesson 2
DIFFERENTIATE INSTRUCTION
GE_MNLESE385795_U2M04L2 178
Kinesthetic Experience
Have students draw a pair of parallel lines and a transversal on translucent paper
squares. Tear the paper between the parallel lines, and overlay the two parts to
show that the angles are congruent.
AVOID COMMON ERRORS
22/05/14 2:20 PM
Some students may have difficulty identifying the
correct angles for two parallel lines cut by a
transversal because they are unfamiliar with the
angles. Have these students use highlighters to
color-code the different angle pairs. For example,
students can use a yellow highlighter to highlight the
term corresponding angles and then use the same
highlighter to mark the corresponding angles.
Transversals and Parallel lines
178
Explain 3
EXPLAIN 3
You can apply the theorems and postulates about parallel lines cut by a transversal to
solve problems.
Using Parallel Lines to Find Angle
Pair Relationships
Example 3

INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Have students look through magazines to find
In the diagram, roads a and b are parallel. Explain how to find the
measure of ∠VTU.

In the diagram, roads a and b are parallel. Explain how to find the
measure of m∠WUV.
It is given that m∠PRS = (9x)° and m∠WUV = (22x + 25)°.
m∠PRS = m∠RUW by the
Students may incorrectly apply the postulates and
theorems presented in this lesson when lines cut by
a transversal are not parallel. Remind them that
the postulates and theorems are only true for
parallel lines.
QUESTIONING STRATEGIES
Postulates may be used to prove theorems;
which postulate was used to prove the two
theorems in this lesson? Same-Side Interior Angles
Postulate
So, m∠RUW + m∠WUV =
and
x= 5
7.
P
S
R
V
(2x - 22)°
a
Q
P
(9x)°
U
R
S
U
T
b
V
(22x + 25)°
W
180° . Solving for x, 31x + 25 = 180,
= 135° .
In the diagram of a gate, the horizontal bars are parallel and the
vertical bars are parallel. Find x and y. Name the postulates and/or
theorems that you used to find the values.
126°
36°
(12x + 2y)°
(3x + 2y)°
x = 10, y = 3; (12x + 2y)° = 126° by the Corresponding Angles Theorem
and (3x + 2y)° = 36° by the Alternate Interior Angles Theorem. Solving the equations
simultaneously results in x = 10 and y = 3.
Module 4
Lesson 2
179
LANGUAGE SUPPORT
GE_MNLESE385795_U2M04L2 179
5/22/14 9:20 PM
Visual Cues
Have students work in pairs to add the following angle definitions and pictures to
an organizer like the one below:
Angle(s)
Alternate interior angles
Same-side interior angles
Lesson 4.2
Q
T
(x + 40)°
Your Turn
Corresponding angles
179
b
.Substitute the value of x to find m∠WUV ;
m∠WUV = (22(5) + 25)°
© Houghton Mifflin Harcourt Publishing Company
AVOID COMMON ERRORS
Corresponding Angles Theorem .
a
∠RUW and ∠WUV are supplementary angles.
QUESTIONING STRATEGIES
ELABORATE
Find each value. Explain how to find the values using postulates,
theorems, and algebraic reasoning.
It is given that m∠PRQ = (x + 40)° and m∠VTU = (2x - 22)°.
m∠PRQ = m∠RTS by the Corresponding Angles Theorem and
m∠RTS = m∠VTU by the Vertical Angles Theorem.
So, m∠PRQ = m∠VTU, and x + 40 = 2x - 22. Solving for x,
x + 62 = 2x, and x = 62. Substitute the value of x to find m∠VTU:
m∠VTU = (2(62) - 22)° = 102°.
pictures with parallel lines and transversals, such as
bridges, fences, furniture, etc. Use markers or colored
tape to mark the lines, and then identify angles that
appear to be congruent and angles that appear to be
supplementary.
How can you check that the postulates and
theorems about parallel lines apply to realworld situations? Sample answer: Measure some
angle pair relationships with lines that appear
parallel, and verify that the postulates and theorems
apply.
Using Parallel Lines to Find Angle Pair
Relationships
Definition
Picture
Elaborate
8.
SUMMARIZE THE LESSON
How is the Same-Side Interior Angles Postulate different from the two theorems in the
lesson (Alternate Interior Angles Theorem and Corresponding Angles Theorem)?
The postulate shows that pairs of angles are supplementary, while the theorems show that
Have students use the diagram to identify the
following:
pairs of angles have the same measure.
9.
Discussion Look at the figure below. If you know that p and q are parallel, and are
given one angle measure, can you find all the other angle measures? Explain.
t
1 2
4 3
p
5 6
8 7
1 2
5 6
q
3 4
7 8
t
Yes; Possible explanation: Consider angles 1–4. If you knew one angle you can use the
p
fact that angles that are linear pairs are supplementary and the Vertical Angles Theorem.
q
You could use either the Alternate Interior Angles Theorem or the Corresponding Angles
Theorem to find one of the angle measures for angles 5–8. Then you can use the Linear
Parallel lines p and q
Pair Theorem and the Vertical Angles Theorem to find all of those angle measures.
Transversal t
Congruent angles: Corresponding ∠1 ≅ ∠3; ∠2 ≅
∠4; ∠5 ≅ ∠7; ∠6 ≅ ∠8
10. Essential Question Check-In Why is it important to establish the Same-Side
Interior Angles Postulate before proving the other theorems?
You need to use the Same-Side Interior Angles Postulate to prove the Alternate Interior
Alternate interior ∠2 ≅ ∠7; ∠3 ≅ ∠6
Angles Theorem, and to prove the Corresponding Angles Theorem you need to use either
Supplementary angles: Same-side interior ∠2 and
∠3; ∠6 and ∠7
the Same-SIde Interior Angles Postulate or the Alternate Interior Angles Theorem.
Evaluate: Homework and Practice
In the figure below, m‖n. Match the angle pairs with the correct
label for the pairs. Indicate a match by writing the letter for the
angle pairs on the line in front of the corresponding labels.
A. ∠4 and ∠6
C
Corresponding Angles
B. ∠5 and ∠8
A
Same-Side Interior Angles
C. ∠2 and ∠6
D
Alternate Interior Angles
D. ∠4 and ∠5
B
m
1 3 5 7
t
2 4
6 8
Depth of Knowledge (D.O.K.)
COMMON
CORE
Mathematical Practices
1
1 Recall of Information
MP.2 Reasoning
2
1 Recall of Information
MP.3 Logic
3
1 Recall of Information
MP.2 Reasoning
2 Skills/Concepts
MP.2 Reasoning
15
3 Strategic Thinking
MP.4 Modeling
16
2 Skills/Concepts
MP.4 Modeling
17
2 Skills/Concepts
MP.3 Logic
4–14
EVALUATE
ASSIGNMENT GUIDE
Lesson 2
180
Exercise
• Online Homework
• Hints and Help
• Extra Practice
Vertical Angles
Module 4
GE_MNLESE385795_U2M04L2.indd 180
n
© Houghton Mifflin Harcourt Publishing Company
1.
20/03/14 9:47 AM
Concepts and Skills
Practice
Explore
Exploring Parallel Lines
and Transversals
Exercises 1–4
Example 1
Proving that Alternate Interior
Angles are Congruent
Exercises 5–14
Example 2
Proving that Corresponding Angles
are Congruent
Exercises 5–14
Example 3
Using Parallel Lines to Find Angle
Pair Relationships
Exercises 15–18
Transversals and Parallel lines
180