Proofs with Parallel Lines 6.3 for Use with Exploration 6.3

Home , Parallel (geometry), Transversal (geometry)

Name ______Date ______

Proofs with Parallel Lines 6.3 For use with Exploration 6.3

Essential Question For which of the theorems involving parallel lines and transversals is the converse true?

1 EXPLORATION: Exploring Converses

Work with a partner. Write the converse of each conditional statement. Draw a diagram to represent the converse. Determine whether the converse is true. Justify your conclusion.

a. Corresponding Angles Theorem If two parallel lines are cut by a transversal, then the pairs of

corresponding angles are congruent. 6 1 2 5 8 7 Converse 4 3

b. Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 6 1 2 5 8 7 Converse 4 3

c. Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 6 1 2 5 7 4 3 8 Converse

6.3 Proofs with Parallel Lines (continued)

1 EXPLORATION: Exploring Converses (continued)

d. Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 6 1 2 5 8 7 Converse 4 3

Communicate Your Answer

2. For which of the theorems involving parallel lines and transversals is the converse true?

3. In Exploration 1, explain how you would prove any of the theorems that you found to be true.

Copyright © Big Ideas Learning, LLC All rights reserved. 194 Name ______Date ______Name ______Date ______Notetaking with Vocabulary 6.3 NotetakingFor use after Lesson with 6.3 Vocabulary 6.3 For use after Lesson 6.3 In your own words, write the meaning of each vocabulary term. converseIn your own words, write the meaning of each vocabulary term. converse

parallel lines parallel lines

transversal transversal

corresponding angles corresponding angles

congruent congruent

alternate interior angles alternate interior angles

alternate exterior angles alternate exterior angles

Name ______Date ______consecutive interior angles Nameconsecutive ______interior angles Date ______Practice 6.3 For use after Lesson 6.3 Name ______Date ______6.3 Notetaking with Vocabulary (continued) Theorems In your own words, write the meaning of each vocabulary term. Theorems6.3 Notetakin g with Vocabulary (continued) AlternateCorrespondingconverse Interior Angles Angles Converse Converse NameIf two ______lines are cut by a transversal so the alternate interior angles Date ______AlternateIf Corresponding two lines are Interior cut by Anglesa transversalAngles Converse Converseso the corresponding angles 2 are congruent, then the lines are parallel. 5 4 j Ifare two congruent, lines are then cut bythe a linestransversal are parallel. so the alternate interior angles j parallelIf two6.3 lines lines Notetakinare cut by a gtransversal with Vocabular so the correspondingy (continued) angles 6 2 k are congruent, then the lines are parallel. 4 j 5 kj NameNotes: ______6 Date ______k Alternate Interior Angles Converse jk k Notes: jk Iftransversal two6.3 lines Notetakinare cut by a transversalg with Vocabular so the alternatey (continued) interior angles jk Notes: jk are congruent, then the lines are parallel. 4 j 5 Alternate Interior Angles Converse k Notes:corresponding angles If two lines are cut by a transversal so the alternate interior angles jk Alternate Exterior Angles Converse are congruent, then the lines are parallel. 5 4 j AlternateIf two lines are Exterior cut by a transversal Angles soConverse the alternate exterior angles are 1 k congruent Notes:congruent, then the lines are parallel. Copyright © Big Ideasj Learning, LLC If two lines are cut by a transversal so the alternate exterior angles are 1 195 jk All rights reserved. Copyright © Big Ideas Learning, LLC congruent, then the lines are parallel. 8 kj Notes: 195 All rights reserved. Alternate Exterior Angles Converse alternate interior angles j 8 k k Notes:  If two lines are cut by a transversal so the alternate exterior angles are 1 j  k congruent, then the lines are parallel. j alternate exterior angles Alternate Exterior Angles Converse 8 k Notes: If two lines are cut by a transversal so the alternate exterior angles are 1 Consecutive Interior Angles Converse j  k congruent, then the lines are parallel. j consecutive interior angles If two lines are cut by a transversal so the consecutive interior angles are Consecutive Interior Angles Converse 8 k Notes:supplementary, then the lines are parallel. If two lines are cut by a transversal so the consecutive interior angles are 3 j j  k supplementary, then the lines are parallel. 5 Notes:Theorems 3 kj Consecutive Interior Angles Converse 5 Notes: IfCorresponding two lines are cut by Anglesa transversal Converse so the consecutive interior angles are k If ∠∠3 and 5 are supplementary, then the lines are parallel. If two lines are cut by a transversal so the corresponding angles supplementary,3 then jjk . Consecutive Interior Angles Converse If ∠∠3 and2 5 are are congruent, then the lines are parallel. 5 j Notes: supplementary, then jk . If two lines are cut by a transversal so the consecutive interior angles are 6 k k supplementary, Transitive Property then the lines of are Parallel parallel. Lines 3 j IfNotes: two lines are parallel to the same line, then they are parallel to each other.If ∠∠jk3 and 5pare q r Transitive Property of Parallel Lines 5 Notes: supplementary, then jk . If two lines are parallel to the same line, then they are parallel to each other. p q k r

Notes:

If ∠∠3 and 5 are TransitiveNotes: Property of Parallel Lines supplementary,If pq and then qr jk , then . If two lines are parallel to the same line, then they are parallel to each other. p q r pr . If pq and qr , then  Notes: Copyrightpr . © Big Ideas Learning, LLC 195 All rights reserved. If two lines are parallel to the same line, th p q r Copyright © Big Ideas Learning, LLC If pq and qr , then All rights reserved. 196  CopyrightNotes: © Big Ideas Learning, LLC pr . All rights reserved. 196

7. Lines m and n are parallel when the marked consecutive 12. Given ∠3 and ∠5 are supplementary. interior angles are supplementary. Prove j � k 32 j x° + 2x° = 180° 5 3x = 180 k 3x 180 — = — 3 3 STATEMENTS REASONS x = 60 1. ∠ 3 and ∠5 are 1. Given supplementary. 8. Lines m and n are parallel when the marked alternate interior angles are congruent. 2. ∠ 2 and ∠3 are 2. Linear Pair Postulate supplementary. 3x° = (2x + 20)° 3. Defi nition of x = 20 3. m ∠3 + m∠5 = 180°, m∠2 + m∠3 = 180° supplementary angles 9. Let A and B be two points on line m. Draw ⃖ AP��⃗ and 4. m ∠3 + m∠5 = 4. Transitive Property of construct an angle ∠1 on n at P so that ∠PAB and ∠1 are m∠2 + m∠3 Equality corresponding angles. 5. 5. m ∠2 = m∠5 Subtraction Property of Equality P 1 n 6. ∠ 2 ≅ ∠5 6. Defi nition of congruent angles m � 7. Corresponding Angles AB 7. j k Converse

10. LetName A and ______B be two points on line m. Draw ⃖ AP��⃗ and 13. yes; Alternate Interior Date ______Angles Converse construct an angle ∠1 on n at P so that ∠PAB and ∠1 are corresponding angles. 6.3 Practice (continued) n

Worked-OutAlternateP Interior Examples Anglesm Converse If two1 lines are cutA by a transversal so the alternate interior angles Example #1 are congruent, thenB the lines are parallel. Find the value of x that makes m || n. Explain your reasoning.

m 150° = y, and n (3x − 15)°

∠2

Lines m and n are parallel when the marked consecutive interior angles are supplementary. 180° = 150° + (3x − 15)° 180 = 135 + 3x 45 = 3x 45 3x — = — 3 3 Chapter 10 x = 15

7. Lines m and n are parallel when the marked consecutive Example #2 interior angles are supplementary. Find the value of x that makes m || n. x° + 2x° = 180° Explain your reasoning. 3x = 180 3x 180 m n — = — 3 3 (180 − x)° k Integrated Mathematics I 461 x = 60 x° Worked-Out Solutions 8. Lines m and n are parallel when the marked alternate interior 3 and 5 are angles are congruent. jk . Lines m and n are parallel when the marked alternate exterior 3x° = (2x + 20)° angles are congruent. x = 20 x° = (180 − x)° 2x = 180 9. Let A and B be two points on line m. Draw ⃖ AP��⃗ and p q r construct an angle ∠1 on n at P so that ∠PAB and ∠1 are 2x 180 — = — corresponding angles. 2 2 x = 90

P 1 n  . m AB

10. Let A and B be two points on line m. Draw ⃖ AP��⃗ and 13.Copyright yes; Alternate © Big Ideas Interior Learning, Angles LLC Converse construct an angle ∠1 on n at P so that ∠PAB and ∠1 are All rights reserved. 196 corresponding angles. 14. yes; Alternate Exterior Angles Converse n 15. no m P 16. yes; Corresponding Angles Converse 1 A

B 17. no

18. yes; Alternate Exterior Angles Converse 11. Given ∠1 ≅ ∠8 1 19. This diagram shows that vertical angles are always Prove j � k 2 j congruent. Lines a and b are not parallel unless x = y, and 8 k you cannot assume that they are equal.

STATEMENTS REASONS 20. It would be true that a � b if you knew that ∠1 and ∠2 were supplementary, but you cannot assume that they are 1. ∠ 1 ≅ ∠8 1. Given supplementary unless it is stated or the diagram is marked as 2. ∠ 1 ≅ ∠2 2. Vertical Angles Congruence such. You can say that ∠1 and ∠2 are consecutive interior Theorem angles. 3. ∠8 ≅ ∠2 3. Transitive Property of Congruence 21. yes; m∠DEB = 180° − 123° = 57° by the Linear Pair 4. j � k 4. Corresponding Angles Converse Postulate. So, by defi nition, a pair of corresponding angles are congruent, which means that ⃖ AC��⃗ � ⃖DF��⃗ by the Corresponding Angles Converse.

22. yes; m∠BEF = 180° − 37° = 143° by the Linear Pair Postulate. So, by defi nition, a pair of corresponding angles are congruent, which means that ⃖ AC��⃗ � ⃖DF��⃗ by the Corresponding Angles Converse.

23. cannot be determined; The marked angles are vertical angles. You do not know anything about the angles formed by the intersection of ⃖DF ��⃗ and ⃖ BE��⃗ .

6.3 Practice (continued)

PracticeExtra Practice A In Exercises 1 and 2, find the value of x that makes mn . Explain your reasoning.

1. 2. n m

95° m 130° (8x + 55)° (200 n − 2x)°

In Exercises 3–6, decide whether there is enough information to prove that mn . If so, state the theorem you would use. n 3. m 4. r

r m

5. s 6. n r s m m n r

Practice10.3 BPractice B

In Exercises 1 and 2, find the value of x that makes st . Explain your reasoning. r 1. 2. st 2(x + 15)° (7x − 20)° s (3x + 20)° r

(4x + 16)° t

In Exercises 3 and 4, decide whether there is enough information to prove that pq . If so, state the theorem you would use. r 3. p q 4.

x ° p r

(180 − x)° q

5. The map of the United States shows the lines of latitude and a longitude. The lines of latitude run horizontally and the lines of longitude run vertically. b c a. Are the lines of latitude parallel? Explain. d b. Are the lines of longitude parallel? Explain.

6. Use the diagram to answer the following. 7. Given: ∠1 ≅∠ 2 and ∠ 2 ≅∠ 3 p q r Prove: ∠14 ≅∠

a b

(4x − 30)° (6y)° 1 4 3(x − 1)° 6(z + 8)° s c

3 d 2

a. Find the values of x, y, and z that makes p  q and qr . Explain your reasoning.

b. Is p  r ? Explain your reasoning.