3.4 Proving Lines are Parallel

Standard/Objectives:

Standard 3: Students will learn and apply geometric concepts

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Objectives:

• Prove that two lines are parallel.
• Use properties of parallel lines to solve real-life problems, such as proving that prehistoric mounds are parallel.
• Properties of parallel lines help you predict.

HW ASSIGNMENT:

• 3.4--pp. 153-154 #1-28

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• Quiz after section 3.5

Postulate 16: Corresponding Angles Converse

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

Theorem 3.8: Alternate Interior Angles Converse

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

Theorem 3.9: Consecutive Interior Angles Converse

• If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

Theorem 3.10: Alternate Exterior Angles Converse

• If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.

Prove the Alternate Interior Angles Converse

Given: 1  2

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Prove: m ║ n

1

2

3

m

n

Example 1: Proof of Alternate Interior Converse

Statements:

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• 1  2
• 2  3
• 1  3
• m ║ n

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Reasons:

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• Given
• Vertical Angles
• Transitive prop.
• Corresponding angles converse

Proof of the Consecutive Interior Angles Converse

Given: 4 and 5 are supplementary

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Prove: g ║ h

6

g

h

5

4

Paragraph Proof

You are given that 4 and 5 are supplementary. By the Linear Pair Postulate, 5 and 6 are also supplementary because they form a linear pair. By the Congruent Supplements Theorem, it follows that 4  6. Therefore, by the Alternate Interior Angles Converse, g and h are parallel.

Find the value of x that makes j ║ k.

Solution:

Lines j and k will be parallel if the marked angles are supplementary.

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x + 4x = 180 

5x = 180 

X = 36 

4x = 144 

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So, if x = 36, then j ║ k.

x

4x

Using Parallel Converses:�Using Corresponding Angles Converse

SAILING. If two boats sail at a 45 angle to the wind as shown, and the wind is constant, will their paths ever cross? Explain

Solution:

Because corresponding angles are congruent, the boats’ paths are parallel. Parallel lines do not intersect, so the boats’ paths will not cross.

Example 5: Identifying parallel lines

Decide which rays are parallel.

62

61

59

58

A

B

E

H

G

D

C

A. Is EB parallel to HD?

B. Is EA parallel to HC?

Example 5: Identifying parallel lines

Decide which rays are parallel.

61

58

B

E

H

G

D

• Is EB parallel to HD?

mBEH = 58

m DHG = 61 The angles are corresponding, but not congruent, so EB and HD are not parallel.

Example 5: Identifying parallel lines

Decide which rays are parallel.

120

120

A

E

H

G

C

• B. Is EA parallel to HC?

m AEH = 62 + 58

m CHG = 59 + 61

AEH and CHG are congruent corresponding angles, so EA ║HC.

Conclusion:

Two lines are cut by a transversal. How can you prove the lines are parallel?

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Show that either a pair of alternate interior angles, or a pair of corresponding angles, or a pair of alternate exterior angles is congruent, or show that a pair of consecutive interior angles is supplementary.