Section 5.2Section 5.2Proving That Lines are Proving That Lines are

ParallelParallel

Steven Shields and Will Steven Shields and Will SwisherSwisher

Period 1Period 1

The Exterior Angle Inequality TheoremThe Exterior Angle Inequality Theorem

• An exterior angle is formed when one An exterior angle is formed when one side of a triangle is extended. side of a triangle is extended.

Exterior Angle

Theorem 30Theorem 30

• The measure of an exterior angle of a The measure of an exterior angle of a triangle is greater than the measure triangle is greater than the measure of either remote interior angle.of either remote interior angle.

Exterior Angle

Remote Interior Angles

Theorem 30-Sample Theorem 30-Sample ProblemProblem• Write a valid inequality and find the Write a valid inequality and find the

restrictions on x.restrictions on x.

50 2x-20

50 < 2x-20 < 180

50+20 < 2x < 180+20

70 < 2x < 200

70/2 < x < 200/2

35 < x < 100

Identifying Parallel LinesIdentifying Parallel Lines

• When two lines are cut by a When two lines are cut by a transversal, eight angles are formed. transversal, eight angles are formed. By proving certain angles congruent, By proving certain angles congruent, you can prove lines you can prove lines IIII..

1 2

3 45

67

8

3

Theorem 31Theorem 31

• If two lines are cut by a transversal If two lines are cut by a transversal such that two alternate interior such that two alternate interior angles are congruent, the lines are angles are congruent, the lines are IIII. . (Alt. int. <s congruent => (Alt. int. <s congruent => IIII lines) lines)

43

If <3 congruent <4, then a II b

a b

Theorem 31-Sample Theorem 31-Sample ProblemProblem

5x

2x+15

Is a II b?

If these lines are II, the alt. int. angles would be congruent.

5x=25 5(5)=25

x=5 2(5)+15=25

Yes, they are II because the alt. int. <s both equal 25.

a b

25

a b

Theorem 32Theorem 32

• If two lines are cut by a transversal If two lines are cut by a transversal such that two alternate exterior such that two alternate exterior angles are congruent, the lines are angles are congruent, the lines are IIII. . (alt. ext. <s congruent => (alt. ext. <s congruent => IIII lines) lines)

1

2

a b

If <1 congruent <2, then a II b

Theorem 32-Sample Theorem 32-Sample ProblemProblem

x+204x

52

xyIs x II y?

x + 20 + 4x = 180 (These angles are suppl.)

5x + 20 = 180 x + 20 = 52

5x = 160 (32) + 20 = 52

x = 32 52 = 52

Therefore, the lines are parallel because alt. ext. <s congruent => II lines

y

Theorem 33Theorem 33• If two lines are cut by a transversal If two lines are cut by a transversal

such that two corresponding angles such that two corresponding angles are congruent, the lines are are congruent, the lines are IIII. ( corr. . ( corr. <s congruent => <s congruent => IIII lines) lines)

1

2

If <1 congruent <2, then m II n

m n

Theorem 33-Sample Theorem 33-Sample ProblemProblem

3 4P

Q R

ST

If <3 congruent <4, then which lines are II? Write the theorem to prove your answer.

QT II RS with transversal PS because Corr. <s congruent => II lines.

Theorem 34Theorem 34

• If two lines are cut by a transversal If two lines are cut by a transversal such that two interior angles on the such that two interior angles on the same side of the transversal are same side of the transversal are supplementary, the lines are supplementary, the lines are IIII..

1

2

If <1 suppl. <2, then c II d.

c

d

Theorem 34-Sample Theorem 34-Sample ProblemProblem

8x12x-20

If x=10, is w II z? Explain.

w

zYes they are parallel because one angle would be 80 and the other 100, so they would be suppl. Therefore the lines are II by theorem 34.

Theorem 35Theorem 35

• If two lines are cut by a transversal If two lines are cut by a transversal such that two exterior angles on the such that two exterior angles on the same side of the transversal are same side of the transversal are supplementary, the lines are supplementary, the lines are IIII..

1

2

If <1 suppl. <2, then a II b.

Theorem 35-Sample Theorem 35-Sample ProblemProblem

2x

6x+60

5x

a

b

Is a II b ?

6x + 60 + 2x = 180 6(15) + 60 + 5(15) = 180

8x = 120 225 = 180

X = 15

Therefore a is not II to b because the same side ext <s do not add up to 180.

Theorem 36Theorem 36

• If two coplanar lines are perpendicular to a If two coplanar lines are perpendicular to a third line, they are parallel.third line, they are parallel.

a b

c

a II b

Practice ProblemsPractice Problems

Name the theorem that proves a II b.

1.2.

3.

80100

a

b

a

b

a

b

Practice Problems Cont.Practice Problems Cont.A

B

C

D

E

Given: <1 congruent <2

Prove: BD II CE

1

2

4.

Practice Problems Cont.Practice Problems Cont.

125x

5.

Find the restrictions on x.

___< x < ___

AnswersAnswers

1. Corr. <s congruent => 1. Corr. <s congruent => IIII lines.lines.

2. Alt. ext. <s congruent => 2. Alt. ext. <s congruent => IIII lines.lines.

3. Same side int. <s suppl. 3. Same side int. <s suppl. => => IIII lines. lines.

4.4.StatementsStatements ReasonsReasons

1. <1 congruent <21. <1 congruent <2 1. Given1. Given

2. BD 2. BD IIII CE CE 2. Corr. <s congruent => 2. Corr. <s congruent => IIII lineslines

5. 0 < x < 125

Work CitedWork Cited

• Rhoad, Richard, George Milauskas, Robert Whipple. Rhoad, Richard, George Milauskas, Robert Whipple. Geometry for Geometry for Enjoyment and Challenge. Enjoyment and Challenge. Boston: McDougal Boston: McDougal Littell, 1997.Littell, 1997.