correctionkey=nl-b;ca-b correctionkey=nl-c;ca-c 19 …€¦ · · 2017-02-2419.3 proving lines...

© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Name Class Date

Explore Writing Converses of Parallel Line TheoremsYou form the converse of and if-then statement "if p, then q" by swapping p and q. The converses of the postulate and theorems you have learned about lines cut by a transversal are true statements. In the Explore, you will write specific cases of each of these converses.

The diagram shows two lines cut by a transversal t. Use the diagram and the given statements in Steps A–D. You will complete the statements based on your work in Steps A–D.

Statements

lines ℓ and m are parallel ∠4 ≅ ∠

∠6 and ∠ are supplementary ∠ ≅ ∠7

A Use two of the given statements together to complete a statement about the diagram using the Same-Side Interior Angles Postulate.

By the postulate: If , then ∠6 and ∠ are supplementary.

B Now write the converse of the Same-Side Interior Angles Postulate using the diagram and your statement in Step A.

By its converse: If ,

then .

C Repeat to illustrate the Alternate Interior Angles Theorem and its converse using the diagram and the given statements.

By the theorem: If , then ∠4 ≅ ∠ .


then .

D Use the diagram and the given statements to illustrate the Corresponding Angles Theorem and its converse.

By the theorem: If , then ∠ ≅ ∠7.

By its converse: .

Resource Locker

ℓ2t

m

134

6578

lines ℓ and m are parallel

∠6 and ∠3 are supplementary



3

6

3 3

6

∠4 ≅ ∠6


3

if ∠3 ≅ ∠7, then lines ℓ and m are parallel


Module 19 955 Lesson 3

19.3 Proving Lines are ParallelEssential Question: How can you prove that two lines are parallel?

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B

IN1_MNLESE389762_U8M19L3 955 9/23/14 8:51 AM

Common Core Math StandardsThe student is expected to:

G-CO.9

Prove theorems about lines and angles. Also G-CO.12

Mathematical Practices

MP.3 Logic

Language ObjectiveExplain to a partner whether the angles formed by two lines cut by a transversal determine parallel lines.

HARDCOVER PAGES 769776

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

Proving Lines are Parallel

ENGAGE Essential Question: How can you prove that two lines are parallel?Possible answer: Look at the angles formed when a transversal crosses the two lines. If a pair of alternate interior angles are congruent, a pair of corresponding angles are congruent, or a pair of same-side interior angles are supplementary, then the lines are parallel.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the photo. Ask students to speculate on advantages a system of parallel east-west and north-south streets at Giza might have had over another system. Then preview the Lesson Performance Task.

955

HARDCOVER

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

© H

ough

ton

Mif

flin

Har

cour

t Pub

lishi

ng C

omp

any

Name

Class Date

Explore Writing Converses of Parallel Line Theorems

You form the converse of and if-then statement "if p, then q" by swapping p and q.

The converses of the postulate and theorems you have learned about lines cut by a transversal

are true statements. In the Explore, you will write specific cases of each of these converses.

The diagram shows two lines cut by a transversal t. Use the diagram and the given statements

in Steps A–D. You will complete the statements based on your work in Steps A–D.

Statements

lines ℓ and m are parallel ∠2 ≅ ∠

∠2 and ∠ are supplementary ∠ ≅ ∠7

Use two of the given statements together to complete a statement about the diagram using

the Same-Side Interior Angles Postulate.

By the postulate: If , then ∠2 and ∠

are supplementary.

Now write the converse of the Same-Side Interior Angles Postulate using the diagram and

your statement in Step A.


then .

Repeat to illustrate the Alternate Interior Angles Theorem and its converse using the

diagram and the given statements.



then .

Use the diagram and the given statements to illustrate the Corresponding Angles Theorem

and its converse.


By its converse: .

Resource

Locker

G-CO.9 Prove theorems about lines and angles. Also G-CO.12

ℓ2t

m

134

6578


∠2 and ∠3 are supplementary



3

6

∠2 ≅ ∠6


5

if ∠5 ≅ ∠7, then lines ℓ and m are parallellines ℓ and m are parallel

Module 19

955

Lesson 3

19 . 3 Proving Lines are Parallel

Essential Question: How can you prove that two lines are parallel?

DO NOT EDIT--Changes must be made through "File info"

CorrectionKey=NL-A;CA-A

IN1_MNLESE389762_U8M19L3 955

4/19/14 12:37 PM

955 Lesson 19 . 3

L E S S O N 19 . 3

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

© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Reflect

1. How do you form the converse of a statement?

2. What kind of angles are ∠4 and ∠6 in Step C? What does the converse you wrote in Step C mean?

Explain 1 Proving that Two Lines are ParallelThe converses from the Explore can be stated formally as a postulate and two theorems. (You will prove the converses of the theorems in the exercises.)

Converse of the Same-Side Interior Angles Postulate

If two lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the lines are parallel.

Converse of the Alternate Interior Angles Theorem

If two lines are cut by a transversal so that any pair of alternate interior angles are congruent, then the lines are parallel.

Converse of the Corresponding Angles Theorem

If two lines are cut by a transversal so that any pair of corresponding angles are congruent, then the lines are parallel.

You can use these converses to decide whether two lines are parallel.

Example 1 A mosaic designer is using quadrilateral-shaped colored tiles to make an ornamental design. Each tile is congruent to the one shown here.

The designer uses the colored tiles to create the pattern shown here.

Use the values of the marked angles to show that the two lines ℓ 1 and ℓ 2 are parallel.

Measure of ∠1: 120° Measure of ∠2: 60°

Relationship between the two angles: They are supplementary.

Conclusion: ℓ 1 ǁ ℓ 2 by the Converse of the Same-Side Interior Angles Postulate.

120°

120° 60°

60°

ℓ1

ℓ2

1

2

ℓ1

ℓ2

Possible answer: Reverse the hypothesis and conclusion; for a statement “if p, then q”, the

converse is “if q, then p.”

Possible answer: alternate interior angles; if the two alternate interior angles ∠4 and ∠6

are congruent, then the lines ℓ and m are parallel.


DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B


IN1_MNLESE389762_U8M19L3 956 9/23/14 8:54 AM

Learning ProgressionsPreviously, students learned the Same-Side Interior Angles Postulate, the Alternate Interior Angles Theorem, and the Corresponding Angles Theorem. They now prove that the converses of the theorems are true given that the Converse of the Same-Side Interior Angles Postulate is also true. Presenting the converses of these statements reinforces the relationship between parallel lines cut by a transversal and the angles these lines form. Parallel lines are important to architecture, construction, and other disciplines. All students should develop fluency with parallel lines and the associated angles as they continue their study of the applications of geometry.

EXPLORE Writing Converses of Parallel Line Theorems

INTEGRATE TECHNOLOGYStudents have the option of exploring the converses of parallel lines theorem activity either in the book or online.

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Before introducing the converses of the Same-Side Interior Angles Postulate, the Alternate Interior Angles Theorem, and the Corresponding Angles Theorem, you may want to spend a few minutes talking about statements and their converses. Have students give the converse for statements based on everyday situations such as, “If you live in Cleveland, then you live in Ohio.” Forming the converses of such statements can help students see that the converse of a true statement may or may not be true.

EXPLAIN 1 Proving that Two Lines are Parallel

QUESTIONING STRATEGIESHow can you determine that a statement is the converse of a theorem? The hypothesis and

conclusion of the theorem are switched.

CONNECT VOCABULARY Have students explore lines of latitude, often called parallels, on a map or globe. They might measure the distance between different parallels at several locations. The actual distance between successive degrees of latitude is about 69 miles.

PROFESSIONAL DEVELOPMENT

Proving Lines are Parallel 956


© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

B Now look at this situation. Use the values of the marked angles to show that the two lines are parallel.

Measure of ∠1: Measure of ∠2:

Relationship between the two

angles:

Conclusion:

Reflect

3. What If? Suppose the designer had been working with this basic shape instead. Do you think the conclusions in Parts A and B would have been different? Why or why not?

Your Turn

Explain why the lines are parallel given the angles shown. Assume that all tile patterns use this basic shape.

4. 5.

ℓ1

ℓ2

1

2

110° 70°

110°70°

ℓ1

ℓ2

1

2

120°

120° 60°

60°

ℓ1

ℓ2

1

2

120° 120°

They are congruent corresponding angles.

ℓ 1 ǁ ℓ 2 by the Converse of the Corresponding Angles Theorem.

No, because the tile pattern formed still has congruent corresponding angle and

supplementary angle pairs that can be used to produce parallel lines.

m∠1 = 120° and m∠2 = 120°

They are congruent alternate interior

angles. The lines are parallel because of

the Converse of the Alternate Interior

Angles Theorem.

m ∠1 = 120° and m∠2 = 60°

The angles are supplementary. The lines

are parallel because of the Converse of

the Same-Side Interior Angles Postulate.


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

IN1_MNLESE389762_U8M19L3 957 4/19/14 12:35 PM

COLLABORATIVE LEARNING

Whole Class ActivityHave groups of students create posters to demonstrate how the Corresponding Angles Theorem and its converse are related and how the Alternate Interior Angles Theorem and its converse are related. Ask them to display their results as a graphic organizer that shows the answers to two questions for each theorem and converse: “How are the theorem and its converse the same?” and “How are the theorem and its converse different?”

AVOID COMMON ERRORSAs you work through the examples, watch for students who use a postulate or theorem for justification when its converse should be used. To help them choose the correct conditional, point out that any given information corresponds to the hypothesis in a conditional statement.

957 Lesson 19 . 3


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Explain 2 Constructing Parallel LinesThe Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ.

The Parallel Postulate

Through a point P not on line ℓ, there is exactly one line parallel to ℓ.

Example 2 Use a compass and straightedge to construct parallel lines.

Construct a line m through a point P not on a line ℓ so that m is parallel to ℓ.

Step 1 Draw a line ℓ and a point P not on ℓ.

Step 2 Choose two points on ℓ and label them Q and R. Use a straightedge to draw

‹ − › PQ .

Step 3 Use a compass to copy ∠PQR at point P, as shown, to construct line m.

line m ǁ line ℓ

In the space provided, follow the steps to construct a line r through a point G not on a line s so that r is parallel to s.

Step 1 Draw a line s and a point G not on s.

Step 2 Choose two points on s and label them E and F. Use a straightedge to draw ‹ −

› GE .

Step 3 Use a compass to copy ∠GEF at point G. Label the side of the angle as line r.line r ǁ line s

P

ℓ

P

RQℓ

P

R

m

Q ℓ

G

F

r

SE


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

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

IN1_MNLESE389762_U8M19L3 958 4/19/14 12:35 PM

DIFFERENTIATE INSTRUCTION

ManipulativesHave students tape two pieces of uncooked spaghetti together to form four angles, then measure one angle. Have them place another piece of spaghetti so that the corresponding angle is congruent to the measured angle. Repeat with different angle pairs. Ask the students to discuss how they can justify whether they created pairs of parallel lines.

EXPLAIN 2 Constructing Parallel Lines

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Make sure students understand that the construction is based on congruent corresponding angles. Have students develop a similar construction for parallel lines that is based on congruent alternate interior angles.

QUESTIONING STRATEGIESHow is the Parallel Postulate used in the parallel lines construction? The Parallel

Postulate guarantees that, for any line, you can always construct a parallel line through a point that is not on the line.

Why must the constructed lines be parallel? Congruent corresponding angles

were constructed, so the Converse of the Corresponding Angles Theorem applies.

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Ask students to compare constructions using a reflective device, tracing paper, compass and straightedge, and geometry software. Discuss the advantages of each. Have students consider not only how easy each is to use, but also how accurate, and how well each one helps them understand the geometric relationships being studied.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Reflect

6. Discussion Explain how you know that the construction in Part A or Part B produces a line passing through the given point that is parallel to the given line.

Your Turn

7. Construct a line m through P parallel to line ℓ.

Explain 3 Using Angle Pair Relationships to Verify Lines are Parallel

When two lines are cut by a transversal, you can use relationships of pairs of angles to decide if the lines are parallel.

Example 3 Use the given angle relationships to decide whether the lines are parallel. Explain your reasoning.

∠3 ≅ ∠5

Step 1 Identify the relationship between the two angles.∠3 and ∠5 are congruent alternate interior angles.

Step 2 Are the lines parallel? Explain. Yes, the lines are parallel by the Converse of the Alternate Interior Angles Theorem.

m∠4 = (x + 20) °, m∠8 = (2x + 5) °, and x = 15.

Step 1 Identify the relationship between the two angles.

m∠4 = (x + 20) ° m∠8 = (2x + 5) °

= ( + 20) °= = (2 ⋅ + 5) °=

So, and are angles.

Step 2 Are the lines parallel? Explain.

m ℓ

P

t

2 1 3 4

6 5 7 8

m

ℓ

In each case, the construction creates two congruent corresponding angles. In Part A,

for example, lines ℓ and m are cut by a transversal and a pair of corresponding angles

are congruent, so the lines are parallel (Converse of the Corresponding Angles Theorem).

1515 35°35°

∠4 ∠8 congruent corresponding

Check students’ constructions.

Yes, the lines are parallel by the Converse of the Corresponding Angles Theorem.



IN1_MNLESE389762_U8M19L3.indd 959 24/09/14 2:06 AM

LANGUAGE SUPPORT

Connect VocabularyHelp students understand how the term converse is used to write the Converse of the Same-Side Interior Angle Postulate, the Converse of the Corresponding Angles Theorem, and the Converse of the Alternate Interior Angles Theorem by first having them explain how to write the converses of simple, non-mathematical statements. Then have them explain how to write the converses of these mathematical statements.

EXPLAIN 3 Using Angle Pair Relationships to Verify Lines are Parallel

CONNECT VOCABULARY Point out to students that the word converse comes from the Latin conversus, which means to turn around. Connect converse to the word conversation, meaning to have a back-and-forth discussion.

QUESTIONING STRATEGIESWhat are some congruent angle pairs that can be used to prove two lines parallel? corresponding

angles or alternate interior angles

What are some angle pairs that must be supplementary to prove two lines

parallel? same-side interior angles

INTEGRATE MATHEMATICAL PRACTICESFocus on TechnologyMP.5 Encourage students to use the geometry software to create congruent pairs of corresponding angles or alternate interior angles. Then have them use the congruent pairs to draw parallel lines. Discuss how congruence of corresponding angles may translate to lines being parallel.

959 Lesson 19 . 3


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

Your Turn

Identify the type of angle pair described in the given condition. How do you know that lines ℓ and m are parallel?

8. m∠3 + m∠6 = 180°

9. ∠2 ≅ ∠6

Elaborate

10. How are the converses in this lesson different from the postulate/theorems in the previous lesson?

11. What If? Suppose two lines are cut by a transversal such that alternate interior angles are both congruent and supplementary. Describe the lines.

12. Essential Question Check-In Name two ways to test if a pair of lines is parallel, using the interior angles formed by a transversal crossing the two lines.

The diagram shows two lines cut by a transversal t. Use the diagram and the given statements in Exercises 1–3 on the facing page.

Statements


m∠ + m∠3 = 180°

∠1 ≅ ∠

∠ ≅ ∠6

t

2 1 3 4

6 5 7 8

m

ℓ

ℓ2t

m

134

6578

same side interior angles; by the Converse of the Same Side

Interior Angles Postulate

corresponding angles; by the Converse of the Corresponding

Angles Theorem

In the previous lesson, we knew lines were parallel and things about angles; here, we

know things about angle pairs, and lines are parallel.

Possible answer: Use given information or measure pairs of angles to decide if alternate

interior angles are congruent or if same-side interior angles are supplementary.

The lines are parallel and all the angles are 90°. The transversal is perpendicular to the

lines.

6

5

4




IN1_MNLESE389762_U8M19L3 960 9/23/14 8:58 AMExercise Depth of Knowledge (D.O.K.) Mathematical Practices

1–3 1 Recall of Information MP.3 Logic

4–7 2 Skills/Concepts MP.3 Logic

8–11 2 Skills/Concepts MP.6 Precision

12 2 Skills/Concepts MP.4 Modeling

13 2 Skills/Concepts MP.5 Using Tools

14 1 Recall of Information MP.3 Logic

15–16 3 Strategic Thinking MP.3 Logic

ELABORATE AVOID COMMON ERRORSA common error when working with converses of the parallel lines postulate or theorems is to assume that the lines are already parallel. Emphasize the need to establish that there are congruent angle pairs (corresponding or alternate interior) or same-side interior angles supplementary before a statement about parallelism can be made.

QUESTIONING STRATEGIESHow are the converses in this lesson different from the postulate/theorems in the previous

lesson? There, we knew the lines were parallel and we were proving things about the angles; here, we know things about the angle pairs, and are proving the lines are parallel.

SUMMARIZE THE LESSONWhat are the angle relationships that would prove two lines parallel?

a pair of congruent corresponding angles; a pair of congruent alternate interior angles; a pair of same-side interior angles that are supplementary

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreWriting Converses of Parallel Line Theorems

Exercises 1–3

Example 1Proving that Two Lines are Parallel

Exercises 4–8

Example 2Constructing Parallel Lines

Exercises 13–14

Example 3Using Angle Pair Relationships to Verify Lines are Parallel

Exercises 9–12



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

1. Use two of the given statements together to complete statements about the diagram to illustrate the Corresponding Angles Theorem. Then write its converse.


By its converse:

2. Use two of the given statements together to complete statements about the diagram to illustrate the Same-Side Interior Angles Postulate. Then write its converse.

By the postulate: If , then m∠ + m∠3 = 180°.

By its converse:

3. Use two of the given statements together to complete statements about the diagram to illustrate the Alternate Interior Angles Theorem. Then write its converse.


By its converse:

4. Matching Match the angle pair relationship on the left with the name of a postulate or theorem that you could use to prove that lines ℓ and m in the diagram are parallel.

A. ∠2 ≅ ∠6

B. ∠3 ≅ ∠5

C. ∠4 and ∠5 are supplementary.

D. ∠4 ≅ ∠8

E. m∠3 + m∠6 = 180°

F. ∠4 ≅ ∠6

Converse of the Corresponding Angles Theorem

Converse of the Same-Side Interior Angles Postulate

Converse of the Alternate Interior Angles Theorem

t

2 1 3 4

6 5 7 8

m

ℓ

5

If ∠1 ≅ ∠5, then lines ℓ and m are parallel.


6

4

If ∠4 ≅ ∠6, then lines ℓ and m are parallel.

If m∠6 + m∠3 = 180°, then lines ℓ and m are parallel.



B, F

C, E

A, D



IN1_MNLESE389762_U8M19L3 961 9/23/14 9:00 AM

AVOID COMMON ERRORSStudents may incorrectly apply the converses presented in this lesson because they assume that lines are parallel and then “prove” that they are. Remind them that the converses of postulates and theorems have different given conditions than those of the original theorems.

961 Lesson 19 . 3


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Use the diagram for Exercises 5–8.

5. What must be true about ∠7 and ∠3 for the lines to be parallel? Name the postulate or theorem.

6. What must be true about ∠6 and ∠3 for the lines to be parallel? Name the postulate or theorem.

7. Suppose m∠4 = (3x + 5) ° and m∠5 = (x + 95) °, where x = 20. Are the lines parallel? Explain.

8. Suppose m∠3 = (4x + 12) ° and m∠7 = (80 - x) °, where x = 15. Are the lines parallel? Explain.

Use a converse to answer each question.

9. What value of x makes the horizontal parts of the letter Z parallel?

10. What value of x makes the vertical parts of the letter N parallel?

t

2 1 3 4

6 5 7 8

m

ℓ

2x°

( 25)°x +5x°

(2 9)°x +

∠7 ≅ ∠3; Converse of the Corresponding Angles Theorem

m∠6 + m∠3 = 180°; Converse of the Same-Side Interior Angles Postulate

m∠4 = 65° and m∠5 = 115°, so m∠4 + m∠5 = 180°. Yes, the lines are parallel by the Converse of the Same-Side Interior Angles Postulate.

m∠3 = 72° and m∠7 = 65°, so m∠3 ≠ m∠7. No, the lines are not parallel because a pair of corresponding angles are not congruent.

When x = 25, x + 25 = 2x = 50; the alternate interior angles are congruent and the horizontal parts of the letter Z are parallel.

When x = 3, 2x + 9 = 5x = 15; the alternate interior angles are congruent and the vertical parts of the letter N are parallel.




IN1_MNLESE389762_U8M19L3 962 4/19/14 12:35 PM

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

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 You may want to have students write a plan for their proof of the theorems in the exercises. Start by assembling the given information that is part of the theorem’s hypothesis, the if part. For example, before students work on the proof of the Converse of the Corresponding Angles Theorem, have them read the theorem and look for key words or phrases. In this case, students should identify corresponding angles as important given information in the theorem. Then ask students to identify the then part, which is to conclude that lines are parallel. For many students, a brief preliminary activity of this type can provide a running start for their work on the proof itself.

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Ask students to discuss in groups how the proof of the Converse of the Alternate Interior Angles Theorem would change if the Converse of Same-Side Interior Angles Postulate were a theorem and the Converse of the Corresponding Angles Theorem were a postulate. Ask them to show a plan for the proof and a diagram illustrating the given information.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

11. Engineering An overpass intersects two lanes of a highway. What must the value of x be to ensure the two lanes are parallel?

12. A trellis consists of overlapping wooden slats. What must the value of x be in order for the two slats to be parallel?

13. Construct a line parallel to ℓ that passes through P.

14. Communicate Mathematical Ideas In Exercise 13, how many parallel lines can you draw through P that are parallel to ℓ? Explain.

H.O.T. Focus on Higher Order Thinking

15. Justify Reasoning Write a two-column proof of the Converse of the Alternate Interior Angles Theorem.

Given: lines ℓ and m are cut by a transversal t; ∠1 ≅ ∠2 Prove: ℓ ∥ m

Statements Reasons

4x°

(2x + 12)°

(3x + 24)° 7x °

ℓ

Pm

t

1

3 2 m

ℓ

When x = 28, 2x + 12 = 68 and 4x = 132; the same-side interior angles are supplementary and the lanes are parallel.

When x = 6, 3x + 24 = 42 and 7x = 42; the corresponding angles are congruent and the slats are parallel.

One; by the Parallel Postulate, there is only one line through a point not on a given line that is parallel to the given line.

Check students’ constructions.

1. Given

7. Converse of Same-Side Interior Angles Postulate

2. Definition of congruence

3. Linear Pair Theorem

4. Definition of supplementary angles

5. Substitution Property of Equality


1. lines ℓ and m are cut by a transversal;

7. ℓ ∥ m

2. m∠1 = m∠2

3. ∠2 and ∠3 are supplementary.

4. m∠2 + m∠3 = 180°

5. m∠1 + m∠3 = 180°


∠1 ≅ ∠2



IN1_MNLESE389762_U8M19L3.indd 963 24/09/14 2:06 AM

PEER-TO-PEER DISCUSSIONAsk students to sketch two parallel lines m and n and a transversal p and then label the angles formed with the numbers 1 through 8. Have them explain three ways of proving lines m and n are parallel.

JOURNALHave students explain how to construct parallel lines using one of the postulates or theorems in the lesson.

963 Lesson 19 . 3


© H

oughton Mifflin H

arcourt Publishing Company

16. Justify Reasoning Write a two-column proof of the Converse of the Corresponding Angles Theorem.

Given: lines ℓ and m are cut by a transversal t; ∠1 ≅ ∠2 Prove: ℓ ∥ m

Statements Reasons

A simplified street map of a section of Harlem in New York City is shown at right. Draw a sketch of the rectangle bounded by West 110th Street and West 121st Street in one direction and Eighth Avenue and Lenox Avenue in the other. Include all the streets and avenues that run between sides of the rectangle. Show St. Nicholas Avenue as a diagonal of the rectangle.

Now imagine that you have been given the job of laying out these streets and avenues on a bare plot of land. Explain in detail how you would do it.

Lesson Performance Task

t

1 3

2 m

ℓ

W 110th StW 111th StW 112th StW 113th St

Eigh

th A

ve

Leno

x A

veW 118th StW 119th St

St Nicholas Ave

W 120th St

W 122nd StW 121st St

W 125th St N

SEW

1. Given

7. Converse of Same-Side Interior Angles Postulate

2. Definition of congruence

3. Linear Pair Theorem


5. Substitution Property of Equality


1. lines ℓ and m are cut by a transversal;

7. ℓ ∥ m

2. m∠1 = m∠2


4. m∠1 + m∠3 = 180°

5. m∠2 + m∠3 = 180°


∠1 ≅ ∠2

Students should use the fact that St. Nicholas Ave. is a transversal of both the parallel streets and the parallel avenues. One approach is to lay out the outside rectangle with right angles at the corners and St. Nicholas Ave. connecting opposite corners. Then measure the angles where St. Nicholas meets the corners and duplicate them as either alternate interior angles or corresponding angles to draw the streets and avenues between the boundaries.


DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-C;CA-C

IN1_MNLESE389762_U8M19L3 964 09/06/15 12:02 PM

116th Street

110th Street

Leno

x A

venu

e

7th A

venu

e

A

B

C

D

E

F I

J

K

L

M

NO

P

Q

R

S

G H

EXTENSION ACTIVITY

The grid on which New York City’s street-and-avenue system is based was designed in 1811. The streets from 11 0 th to 12 1 st are 60 feet wide, except for 11 6 th Street, which is 100 feet wide. The distance between each pair of streets is 200 feet. 8 th Avenue, 7 th Avenue, and Lenox Avenue are each 100 feet wide. The distance between each pair of avenues is 922 feet.

• Find the length and width of the rectangle bounded by 8 th Avenue, 110 th Street, Lenox Avenue, and 121 st Street.

• Find the area of the rectangle in square feet and square miles.

dimensions: 2144 ft by 2960 ft; area: 6,346,240 ft2; 0.23 mi2

CONTEXT CLUESDraw attention to the fact that St. Nicholas Avenue is a transversal of both the parallel streets and the parallel avenues.

QUESTIONING STRATEGIESSketch the map shown below on the board. Ask questions such as the following. (Answers shown are sample answers.)

Name a pair of vertical angles. ∠AMO and ∠MOP

Name an angle congruent to ∠LPQ. ∠OPC

Name an angle that corresponds to ∠JRS. ∠KQR

Name an angle that is supplementary to ∠OPL. ∠LPQ

Name a pair of same-side interior angles. ∠CPQ and ∠DQP

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.



correctionkey=nl-b;ca-b correctionkey=nl-c;ca-c 19 …€¦ · · 2017-02-2419.3 proving lines...

Documents