33 Proving Lines ParallelMathematics Florida Standards

Extends MAFS.912.G-C0.3.9 Prove theorems about

lines and angles. Theorems include:... when a transversalcrosses parallel lines, alternate interior angles are

congruent and corresponding angles are congruent...

MP 1, 7

Objective To determine whether two lines are parallel

How cart you usetheorems youalready know tosolve this maze

problem?

MATHEMATICAL

PRACTICES

Lesson

Vocabularyflow proof

Getting Ready! X

The maze below has two intersecting sets of poralle! paths. A mouse makesfive turns in the maze to get to a piece of cheese. Follow the mouse's paththrough the maze. What are the number of degrees at each turn? Explainhow you know.

In the Solve It, you used parallel lines to find congruent and supplementary

relationships of special angle pairs. In this lesson, you will do the converse. You will

use the congruent and supplementary relationships of the special angle pairs to prove

lines parallel.

Essential Understanding You can use certain angle pairs to decide whether twolines are parallel.

Theorem 3-4 Converse of the Corresponding Angles Theorem

Theorem

If two lines and a transversal

form corresponding angles

that are congruent, then the

lines are parallel.

Then

€llm

You will prove Theorem 3-4 In Lesson 5-5.

156 Chapter 3 Parallel and Perpendicular Lines

A

Identifying Parallel Lines

Which line is the

transversal for z.1

and Z.2?

Line m Is the transversal

because it forms one side

of both angles.

Which lines are parallel if LI ̂ Z.2? Justify your answer.

LI and Z.2 are corresponding angles. If Z.1 = Z.2, then a

by the Converse of the Corresponding Angles Theorem.

Got It? 1. Which lines are parallelif Z.6 = Z.7? Justifyyour answer.

You can use the Converse of the Corresponding Angles Theorem to prove the following

converses of the theorems and postulate you learned in Lesson 3-2.

Theorem 3-5 Converse of the Alternate Interior Angles Theorem

Theorem

If two lines and a

transversal form alternate

interior angles that are

congruent, then the two

lines are parallel.

Then ...

e\\m

Theorem 3-6 Converse of the Same-Side Interior Angles Postulate

Theorem

If two lines and a

transversal form same-

side interior angles that

are supplementary, then

the two lines are parallel.

If...

mZ3 + mL& = 180

Then ...

€llm

Theorem 3-7 Converse of the Alternate Exterior Angles Theorem

Theorem

If two lines and a

transversal form alternate

exterior angles that are

congruent, then the two

lines are parallel.

Then ...

€ II m

C PowerGeometry.com Lesson 3-3 Proving Lines Parallel 157

The proof of the Converse of the Alternate Interior Angles Theorem below looks

different than any proof you have seen so far in this course. You know two forms of

proof—paragraph and two-column. In a third form, called flow proof, arrows show the

logical connections between the statements. Reasons are written below the statements.

Proof Proof of Theorem 3-5: Converse of the Alternate Interior Angles Theorem

Given: z.4 = z.6

Prove: i II m

^6

Given

Vertical A are

Transitive

Property of =

€ II m

If corresp. A are sthen the lines are

Proof

Given: A1 = A7

Prove: f || m

Writing a Flow Proof of Theorem 3-7

•Xir^ow

• A^ = A7

From the diagram you know• Z.1 and Z.3 are vertical

• Z.5and Z.7 are vertical

• Z.1 and Z.5 are corresponding• Z.3 and Z.7 are corresponding

One pair ofcorresponding anglescongruent to provedim

Psicin

Use a pair of congruentvertical angles to relateeither Z. 1 or Z. 7 to its

corresponding angle.

z:l s z.7

Given

^3 = ̂1

Vertical A are

A3 = A7

Transitive

Property of =

€||m

If corresp. A are s

then the lines are

Got It? 2, Use the same diagram from Problem 2 to Prove Theorem 3-6,Given: mZ.3 -I- mZ.6 = 180

Prove: II m

158 Chapters Parallel and Perpendicular Lines

T' •How do Z.1 and Z.2

relate to each other

in the diagram?Z.1 and ̂ 2 are both

exterior angles and theylie on opposite sides ofthe transversal.

The four theorems you have just learned provide you with four ways to determine if two

lines are parallel.

Determining Whether Lines are Parallel

The fence gate at the right is made up of pieces of

wood arranged in various directions. Suppose

A1 s z.2. Are lines r and s parallel?

; Explain.

Yes, r II s. Z.1 and Z.2 are alternate exteriorangles. If two lines and a transversal form

congruent alternate exterior angles, then the

lines are parallel (Converse of the Alternate

Exterior Angles Theorem).

^ Go. It? 3. In Problem 3, what is another way toexplain why r || s? Justify your answer.

in

You can use algebra along with the postulates and theorems from Lesson 3-2 and

Lesson 3-3 to help you solve problems involving parallel lines.

Work backward.

Think about what must

be true of the givenangles for a and b to beparallel.

Problem 4 Using Algebra

Algebra What is the value of x for which a jj h?

The two angles are same-side interior angles. By the Converse

of the Same-Side Interior Angles Postulate, a || if the anglesare supplementary.

Def. of supplementary angles

Simplify.

Subtract 120 from each side.

Divide each side by 2.

(2a:+ 9)+ 111 = 180

2x + 120 = 180

2x = 60

x = 30

Got It? 4. What is the value of w for which c || rf?

3w - 2

2x + 9

C PowerGeometry.com I Lesson 3-3 Proving Lines Parallel 159

Lesson Check

Do you know HOW?

State the theorem or postulate that proves a || b.

1. a

2.

Do you UNDERSTAND? ̂ ^PRACIICE^S4. Explain how you know when to use the Alternate

Interior Angles Theorem and when to use the

Converse of the Alternate Interior Angles Theorem.

5. Compare and Contrast How are flow proofs andtwo-column proofs alike? How are they different?

6. Error Analysis A classmate ^

says that AB || DC based on thediagram at the right. Explain your

classmate's error. 183'= 97=

3. What is the value ofy for which a || b in Exercise 2?J

Practice and Problem-Solving Exercises

Practice Which lines or segments are parallel? Justify your answer.

MATHEMATICAL

PRACTICES

^ See Problem 1.

7.

9. C

M A R

11. Developing Proof Complete the flow proof below.

Given: /LI and /.S are supplementary.

Prove: a II h

^ See Problem 2.

Z.1 and AS are

supplementary.

a.

d._?_

Supplements of the

same z are s.

a II fa

e. ?

b._L

Def. of linear pair

Z.1 and Z.2 are

supplementary.

c. ?

160 Chapter 3 Parallel and Perpendicular Lines

12. Parking Two workers paint lines for angled

parking spaces. One worker paints a line so

that mZ.1 = 65. The other worker paints a

line so that ml.2 = 65. Are their lines parallel?

Explain.

Algebra Find the value of j; for which C || m.

13. /^ /55°

y /(X + 25)° ^

3x - 33

{2x + 26)

Apply Developing Proof Use the given information to determine which lines,

if any, are parallel. Justify each conclusion with a theorem or postulate.

17. Z.2 is supplementary to Z.3.

19. Z.6 is supplementary to Z.7.

21. m.i7 = 65, mZ.9=115

23. ̂ 1 = Z.8

25. Zll = ̂7

Algebra Find the value of x for which C || m.

27. ^ if im

18. Z.1 = Z.3

20. = Z.12

22. Z.2 = ̂10

24. Z.8 = ̂6

26. Z.5 = Z.10

(5x + 40}

\

^ See Problem 3.

^ See Problem 4.

9/1110/12

29. Write a paragraph proof.rtSof

Given: Z.2 is supplementary to Z7.

Prove: € II m

C PowerGeometry.com I Lesson 3-3 Proving Lines Parallel 161

30. Think About a Plan If the rowing crew at the rightstrokes in unison, the oars sweep out angles of equal

measure. Explain why the oars on each side of the

shell stay parallel.

• What type of information do you need to prove

lines parallel?

• How do the positions of the angles of equal

measure relate?

Algebra Determine the value ofa; for which r || s.Then find mZ.1 and mz.2.

31. m/.l = 80 - jc, = 90 - 2a:

32. mZ.1 = 60 - 2a:, m^2 = 70 - 4a:

33. mZ.1 = 40 - 4a:, mZ.2 = 50 - Bx

34. mZl = 20 - 8x, mA2 = 30 - 16x

Use the diagram at the right below for Exercises 35-41.

?ib

Open-Ended Use the given information. State another fact about one of the

given angles that will guarantee two lines are parallel. Tell which lines will be

parallel and why.

35. Z1 = Z3

37. s /Lll

36. mZ.8 = 110, mZ.9 — 70

38. Z.I1 and Z.12 are supplementary.

39. Reasoning Ifz.1 = Z.7, what theorem or postulate can you use to

show that € II n?

Write a flow proof.

40. Given: i || n, Z.12 = Z.8Prove: j\\k

Given: j || k, mZL8 + mZ.9 = 180Prove: e 1! n

Challenge Which sides of quadrilateral PLAN must be parallel? Explain.

42. m^P = 72, mZL = 108, m/.A = 72, mAN = 108

43. mAP = 59, m^L = 37, m^A = 143, m/LN= 121

44. mAP = 67, ml.L = 120, mZJ\. = 73, mAN = 100

45. mAP = 56, mAL = 124, mAA = 124, mAN = 56

46. Write a two-column proof to prove the following: If a transversal intersects twoProof parallel lines, then the bisectors of two corresponding angles are parallel. {Hint:

Start by drawing and marking a diagram.)

162 Chapter 3 Parallel and Perpendicular Lines

SAT/Aa

Standardized Test Prep

Use the diagram for Exercises 47 and 48.

47. For what value of x is c || rf?

21 CO 43

CO 23 CO 53

48. If c II d, what is m/Lll

CO 24 CO 136

CO 44 CO 146

49. Which of the following is always a valid conclusion for the hypothesis?

If two angles are congruent, then ? .

CO they are right angles CO they have the same measure

CO they share a vertex CO they are acute angles

A

x + 21

Short

. Response

50. What is the value of x in the diagram at tlie right?

CO 1-6 CO 17

CO 10 CO 19 "*~"

51. Draw a pentagon. Is your pentagon convex or concave? Explain.

{8x-10r

Mixed Review

Find mLl and mLZ. Justify each answer.

52.

^ See Lesson 3-2.

D

Get Ready! To prepare for Lesson 3-4, do Exercises 54-57.

Determine whether each statement is always, sometimes, or never true.

54. Perpendicular lines meet at right angles.

55. Two lines in intersecting planes are perpendicular.

56. Two lines in the same plane are parallel.

57. Two lines in parallel planes are perpendicular.

♦ See Lessons 1-6 and 3-1.

I Lesson 3-3 Proving Lines Parallel 163