DO NOT EDIT--Changes must be made through … transversal that is not perpendicular to the lines. Instruct the student’s partner to shade or mark the acute angles with one color

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Explore 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.

Angle Pair Example

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.

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Module 4 175 Lesson 2

4.2 Transversals and Parallel LinesEssential Question: How can you prove and use theorems about angles formed by

transversals that intersect parallel lines?

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Common Core Math StandardsThe student is expected to:

G-CO.C.9

Prove theorems about lines and angles.

Mathematical Practices

MP.3 Logic

Language ObjectiveExplain to a partner how to identify the angles formed by two parallel lines cut by a transversal.

COMMONCORE

COMMONCORE

HARDCOVER PAGES 153160

Turn to these pages to find this lesson in the hardcover student edition.

Transversals and Parallel Lines

ENGAGE Essential Question: How can you prove and use theorems about angles formed by transversals that intersect parallel lines?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.

PREVIEW: LESSONPERFORMANCE TASKView 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.

175

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Explore 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.

Angle Pair

Example

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.

Resource Locker

G-CO.C.9 Prove theorems about lines and angles.

COMMONCORE

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Module 4

175

Lesson 2

4 . 2 Transversals and Parallel Lines

Essential Question: How can you prove and use theorems about angles formed by

transversals that intersect parallel lines?

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175 Lesson 4 . 2

L E S S O N 4 . 2

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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.

Same-Side Interior Angles Postulate

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 Draw two parallel lines and a transversal, and number the angles formed from 1 to 8.

B Identify the pairs of same-side interior angles.

C What does the postulate tell you about these same-side interior angle pairs?

D If m∠4 = 70°, what is m∠5? Explain.

Reflect

1. Explain how you can find m∠3 in the diagram if p ‖ q and m∠6 = 61°.

2. What If? If m ‖ n, how many pairs of same-side interior angles are shown in the figure? What are the pairs?

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∠4 and ∠5; ∠3 and ∠6

Given p ‖ q, then ∠4 and ∠5 are supplementary and ∠3 and ∠6 are supplementary.

m∠5 = 110°; ∠4 and ∠5 are supplementary, so m∠4 + m∠5 = 180°. Therefore

70° + m∠5 = 180°, so m∠5 = 110°.

∠3 and ∠6 are supplementary, so m∠3 + m∠6 = 180°. Therefore m∠3 + 61° = 180°, so

m∠3 = 119°.

Two pairs; ∠3 and ∠5, ∠4 and ∠6





PROFESSIONAL DEVELOPMENT

Math BackgroundWhen 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.”

EXPLORE Exploring Parallel Lines and Transversals

INTEGRATE TECHNOLOGYThe 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.

QUESTIONING STRATEGIESWhen two lines are cut by a transversal, how many pairs of corresponding angles are

formed? How many pairs of same-side interior angles? 4; 2

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˚̊.

PROFESSIONAL DEVELOPMENT

Transversals and Parallel lines 176


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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.

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles have the same measure.

To prove something to be true, you use definitions, properties, postulates, and theorems that you already know.

Example 1 Prove the Alternate Interior Angles Theorem.

Given: p ‖ q

Prove: m∠3 = m∠5

Complete the proof by writing the missing reasons. Choose from the following reasons. You may use a reason more than once.

• Same-Side Interior Angles Postulate

• Given

• Definition of supplementary angles

• Subtraction Property of Equality

• Substitution Property of Equality

• Linear Pair Theorem

Statements Reasons

1. p ‖ q

2. ∠3 and ∠6 are supplementary.

3. m∠3 + m∠6 = 180°

4. ∠5 and ∠6 are a linear pair.

5. ∠5 and ∠6 are supplementary.

6. m∠5 + m∠6 = 180°

7. m∠3 + m∠6 = m∠5 + m∠6

8. m∠3 = m∠5

Reflect

3. In the figure, explain why ∠1, ∠3, ∠5, and ∠7 all have the same measure.

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1. Given

2. Same-Side Interior Angles Postulate

3. Definition of supplementary angles

4. Given

5. Linear Pair Theorem

6. Definition of supplementary angles

7. Substitution Property of Equality

8. Subtraction Property of Equality

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).


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GE_MNLESE385795_U2M04L2 177 10/14/14 7:01 PM

COLLABORATIVE LEARNING

Peer-to-Peer ActivityHave 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.

EXPLAIN 1 Proving that Alternate Interior Angles are Congruent

CONNECT VOCABULARY Have students experience the idea of what the term alternate means by shading in alternating figures in a series of figures.

QUESTIONING STRATEGIESDo 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.

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4. Suppose m∠4 = 57° in the figure shown. Describe two different ways to determine m∠6.

Explain 2 Proving that Corresponding Angles are Congruent

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 Complete a proof in paragraph form for the Corresponding Angles Theorem.

Given: p‖q


By the given statement, p‖q. ∠4 and ∠6 form a pair of .

So, using the Alternate Interior Angles Theorem, .

∠6 and ∠8 form a pair of vertical angles. So, using the Vertical Angles Theorem,

. Using the

in m∠4 = m∠6, substitute for m∠6. The result is .

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.

6. Suppose m∠4 = 36°. Find m∠5. Explain.

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By the Alternate Interior Angles Theorem, m∠6 = 57°. Also ∠4 and ∠5 are supplementary,

so m∠5 = 123°. Since ∠5 and ∠6 are supplementary, m∠6 = 57°.

alternate interior angles

Substitution Property of Equality

m∠4 = m∠6

m∠4 = m∠8

m∠6 = m∠8

m∠4

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.

m∠5 = 144°; Since ∠1 and ∠4 form a linear pair, they are supplementary. So,

m∠1 = 144°. Using the Corresponding Angles Theorem, you know that m∠1 = m∠5.

So, m∠5 = 144°.




GE_MNLESE385795_U2M04L2 178 22/05/14 2:20 PM

DIFFERENTIATE INSTRUCTION

Kinesthetic ExperienceHave 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.

EXPLAIN 2 Proving that Corresponding Angles are Congruent

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Explaining and justifying arguments is at the 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.

QUESTIONING STRATEGIESWhat can you use as reasons in a proof? given

information, properties, postulates, and

previously-proven theorems

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.”

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Draw parallel lines and a transversal on a transparency. Trace an acute and an obtuse angle formed by the lines onto another transparency, and use them to find congruent angles.

AVOID COMMON ERRORSSome 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.



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Explain 3 Using Parallel Lines to Find Angle Pair Relationships

You can apply the theorems and postulates about parallel lines cut by a transversal to solve problems.

Example 3 Find each value. Explain how to find the values using postulates, theorems, and algebraic reasoning.

In the diagram, roads a and b are parallel. Explain how to find the measure of ∠VTU.

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°.

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 .

∠RUW and are supplementary angles.

So, m∠RUW + m∠WUV = . Solving for x, 31x + 25 = 180,

and .Substitute the value of x to find ;

m∠WUV = (22 (5) + 25) ° .

Your Turn

7. 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.

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(3x + 2y)°

(12x + 2y)°36°

126°

Corresponding Angles Theorem

m∠WUV

= 135°

x = 5

∠WUV

180°

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.



GE_MNLESE385795_U2M04L2 179 5/22/14 9:20 PM

LANGUAGE SUPPORT

Visual CuesHave students work in pairs to add the following angle definitions and pictures to an organizer like the one below:

Angle(s) Definition PictureAlternate interior angles

Corresponding angles

Same-side interior angles

EXPLAIN 3 Using Parallel Lines to Find Angle Pair Relationships

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Have students look through magazines to find 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.

QUESTIONING STRATEGIESHow can you check that the postulates and theorems about parallel lines apply to real-

world situations? Sample answer: Measure some

angle pair relationships with lines that appear

parallel, and verify that the postulates and theorems

apply.

ELABORATE AVOID COMMON ERRORSStudents 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 STRATEGIESPostulates may be used to prove theorems; which postulate was used to prove the two

theorems in this lesson? Same-Side Interior Angles

Postulate

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Elaborate

8. How is the Same-Side Interior Angles Postulate different from the two theorems in the lesson (Alternate Interior Angles Theorem and Corresponding Angles Theorem)?

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.

10. Essential Question Check-In Why is it important to establish the Same-Side Interior Angles Postulate before proving the other theorems?

1. 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 Corresponding Angles

B. ∠5 and ∠8 Same-Side Interior Angles

C. ∠2 and ∠6 Alternate Interior Angles

D. ∠4 and ∠5 Vertical Angles

Evaluate: Homework and Practice

• Online Homework• Hints and Help• Extra Practice

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The postulate shows that pairs of angles are supplementary, while the theorems show that

pairs of angles have the same measure.

You need to use the Same-Side Interior Angles Postulate to prove the Alternate Interior

Angles Theorem, and to prove the Corresponding Angles Theorem you need to use either

the Same-SIde Interior Angles Postulate or the Alternate Interior Angles Theorem.

Yes; Possible explanation: Consider angles 1–4. If you knew one angle you can use the

fact that angles that are linear pairs are supplementary and the Vertical Angles Theorem.

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

Pair Theorem and the Vertical Angles Theorem to find all of those angle measures.

C

A

D

B


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GE_MNLESE385795_U2M04L2.indd 180 20/03/14 9:47 AMExercise Depth of Knowledge (D.O.K.)COMMONCORE 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

4–14 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

1 5 6 7 8

2 3 4

p q

t

SUMMARIZE THE LESSONHave students use the diagram to identify the following:

Parallel lines p and q

Transversal t

Congruent angles: Corresponding ∠1 ≅ ∠3; ∠2 ≅

∠4; ∠5 ≅ ∠7; ∠6 ≅ ∠8

Alternate interior ∠2 ≅ ∠7; ∠3 ≅ ∠6

Supplementary angles: Same-side interior ∠2 and

∠3; ∠6 and ∠7

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreExploring Parallel Linesand Transversals

Exercises 1–4

Example 1Proving that Alternate Interior Angles are Congruent

Exercises 5–14

Example 2Proving that Corresponding Angles are Congruent

Exercises 5–14

Example 3Using Parallel Lines to Find Angle Pair Relationships

Exercises 15–18



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2. Complete the definition: A is a line that intersects two coplanar lines at two different points.

Use the figure to find angle measures. In the figure, p ‖ q.

3. Suppose m∠4 = 82°. Find m∠5. 4. Suppose m∠3 = 105°. Find m∠6.



Use the figure to find angle measures. In the figure, m ‖ n and x ‖ y.




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x

transversal

m∠1 = 109°, by the Corresponding Angles Theorem.

m∠6 = 76°, by the Alternate Interior Angles Theorem.


m∠6 = 75°, by the Same-Side Interior Angles Postulate.

m∠5 = 98°, by the Same-Side Interior Angles Postulate.

m∠2 = 74°, by the Corresponding Angles Theorem.




m∠11 = 108°, by the Corresponding Angles Theorem and Linear Pair Theorem.

m∠14 = 66°, by the Corresponding Angles Theorem, Linear Pair Theorem, and Corresponding Angles Theorem.

m∠12 = 86°, by the Alternate Interior Angles Theorem and Corresponding Angles Theorem.


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GE_MNLESE385795_U2M04L2.indd 181 20/03/14 9:46 AMExercise Depth of Knowledge (D.O.K.)COMMONCORE Mathematical Practices

18–21 3 Strategic Thinking MP.3 Logic

22–23 3 Strategic Thinking MP.3 Logic

AVOID COMMON ERRORS 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.

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Some students may have difficulty identifying the correct angles for two parallel lines cut by a transversal because some combinations of angles can be visually distracting. Suggest that these students re-draw or trace the diagram for each exercise, labeling only the angles necessary for the exercise.

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15. Ocean waves move in parallel lines toward the shore. The figure shows the path that a windsurfer takes across several waves. For this exercise, think of the windsurfer’s wake as a line. If m∠1 = (2x + 2y) ° and m∠2 = (2x + y) °, find x and y. Explain your reasoning.

In the diagram of movie theater seats, the incline of the floor, ƒ, is parallel to the seats, s.

16. If m∠1 = 60°, what is x?

17. If m∠1 = 68°, what is y?

18. Complete a proof in paragraph form for the Alternate Interior Angles Theorem.

Given: p ‖ q


It is given that p ‖ q, so using the Same-Side Interior Angles Postulate, ∠3 and ∠6

are . So, the sum of their measures is and m∠3 + m∠6 = 180°.

You can see from the diagram that ∠5 and ∠6 form a line, so they are a ,

which makes them . Then m∠5 + m∠6 = 180°. Using the

Substitution Property of Equality, you can substitute in m∠3 + m∠6 = 180° with

m∠5 + m∠6. This results in m∠3 + m∠6 = m∠5 + m∠6. Using the Subtraction Property

of Equality, you can subtract from both sides. So, .

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70°

1f

s

(5y - 7)°

3x °

x = 40; by the Corr. ∠s Thm. and the Lin. Pair Thm., 3x° + m∠1 = 180°, so 3x + 60 = 180, and x = 40.

y = 15; by the Alt. Int. ∠s Thm., m∠1 = (5y - 7) °, so 68 = 5y - 7, and y = 15.

supplementary 180°

linear pair

supplementary

180°

m∠6 m∠3 = m∠5

x = 15 and y = 40; m∠2 = 70° by the Corresponding Angles Theorem and m∠1 + m∠2 = 180° by the Same-Side Interior Angles Postulate. So, m∠1 = 180° - 70° = 110°. m∠2 = (2x + y) ° and m∠2 = 70°, so (2x + y) = 70° by the Substitution Property of Equality and 2x = 70 - y. m∠1 = (2x + 2y) °, so m∠1 = (70 - y + 2y) ° = (70 + y) ° , so (70 + y) º = 110º and y = 40 by the Substitution Property of Equality.

Since 2x + y = 70 and y = 40, 2x = 30 and x = 15.


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PEERTOPEER DISCUSSIONAsk students to discuss with a partner the parallel line diagrams for various pairs of angles introduced in this lesson. Ask them to write and solve a word problem about parallel lines and their associated angles on index cards, and switch with their partners to solve the problem.

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Remind students about how to use the Same-Side Interior Angles Postulate and the theorems about parallel lines to write equations that will help them find angle measures related to parallel lines.



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19. Write a proof in two-column form for the Corresponding Angles Theorem.

Given: p ‖ q


Statements Reasons

20. Explain the Error Angelina wrote a proof in paragraph form to prove that the measures of corresponding angles are congruent. Identify her error, and describe how to fix the error.Angelina’s proof:I am given that p ‖ q. ∠1 and ∠4 are supplementary angles because they form a linear pair, so m∠1 + m∠4 = 180°. ∠4 and ∠8 are also supplementary because of the Same-Side Interior Angles Postulate, so m∠4 + m∠8 = 180°. You can substitute m∠4 + m∠8 for 180° in the first equation above. The result is m∠1 + m∠4 = m∠4 + m∠8. After subtracting m∠4 from each side, I see that ∠1 and ∠8 are corresponding angles and m∠1 = m∠8.

21. Counterexample Ellen thinks that when two lines that are not parallel are cut by a transversal, the measures of the alternate interior angles are the same. Write a proof to show that she is correct or use a counterexample to show that she is incorrect.

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1. Given1. p‖q

2. Alternate Interior Angles Theorem

3. Vertical Angles Theorem

4. Substitution Property of Equality

2. m∠3 = m∠5

3. m∠1 = m∠3

4. m∠1 = m∠5

∠4 and ∠8 are not same-side interior angles. ∠4 and ∠5 are same-side interior angles. So, in the paragraph proof, replace ∠8 with ∠5 to see that m∠1 = m∠5.

A possible diagram is shown, with two nonparallel lines cut by a transversal. I can measure the angles in my drawing with a protractor as a counterexample. ∠4 and ∠5 are alternate interior angles, but m∠4 = 90° and m∠5 = 130°, so the measures are not the same when the lines are not parallel.


DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A


JOURNALHave students list examples in the real world where they have seen parallel lines cut by a transversal. Ask them to name the pairs of angles that are congruent.

183 Lesson 4 . 2


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H.O.T. Focus on Higher Order Thinking

Analyzing Mathematical Relationships Use the diagram of a staircase railing for Exercises 22 and 23.

_ AG ‖

_ CJ and

_ AD ‖

_ FJ . Choose the best

answer.

22. Which is a true statement about the measure of ∠DCJ?

A. It is 30°, by the Alternate Interior Angles Theorem.

B. It is 30°, by the Corresponding Angles Theorem.

C. It is 50°, by the Alternate Interior Angles Theorem.

D. It is 50°, by the Corresponding Angles Theorem.

23. Which is a true statement about the value of n?

A. It is 25, by the Alternate Interior Angles Theorem.

B. It is 25, by the Same-Side Interior Angles Postulate.

C. It is 35, by Alternate Interior Angles Theorem.

D. It is 35, by the Corresponding Angles Theorem.

Lesson Performance TaskWashington Street is parallel to Lincoln Street. The Apex Company’s headquarters is located between the streets. From headquarters, a straight road leads to Washington Street, intersecting it at a 51° angle. Another straight road leads to Lincoln Street, intersecting it at a 37° angle.

a. Find x. Explain your method.

b. Suppose that another straight road leads from the opposite side of headquarters to Washington Street, intersecting it at a y° angle, and another straight road leads from headquarters to Lincoln Street, intersecting it at a z° angle. Find the measure of the angle w formed by the two roads. Explain how you found w.

A

F

G

H

J

B

C

D

50°

(3n + 7)°

82°

30°

r°

Apex Company

51°

x°

37°Lincoln Street

Washington Street

51°

37°

a. Draw a line parallel to the two streets and passing through the vertex of the angle with measure x°. Because alternate interior angles formed by parallel lines and a transversal are congruent, the angle measuring x° is divided into a 51° angle and a 37° angle, so x = 51 + 37 = 88.

b. Use the method from part a. The top part of the unknown angle measures y° and the bottom part measures z°. So, w = y + z.




51°y°

z°

w° x°

37°

Washington Street

ApexCompany

Lincoln Street

EXTENSION ACTIVITY

The easiest way to solve the problems in the Lesson Performance Task is to draw a line parallel to Washington and Lincoln streets through the angle with its vertex at the Apex Company. But what if there are two such possible lines, or even more?

Have students research and report on “Playfair’s Axiom,” which states that, in a plane, no more than one line can be drawn through a given point that is parallel to a given line. Students should be able to grasp this concept intuitively, as a second line through a point would form a nonzero angle with the parallel line and thus could not also be parallel.

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Students may not have encountered geometry problems that can be solved by drawing a line or some other element on a given figure. Students may object that a problem of this kind isn’t “fair.” Point out that many problems omit information. To solve a problem, you use problem-solving skills, working your way logically from the statement of the problem to its solution. This problem introduces a new approach students can add to their arsenal of problem-solving skills.

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 If students have difficulty drawing the figure for part b of the Lesson Performance Task, show them this figure:

Scoring Rubric2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning.0 points: Student does not demonstrate understanding of the problem.



Documents

DO NOT EDIT--Changes must be made through … transversal that is not perpendicular to the lines. Instruct the student’s partner to shade or mark the acute angles with one color