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Postulate 3-2: Converse of the Corresponding Angles Postulate

- If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

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

- If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.

Theorem 3-4: Converse of the Sam-Side Interior Angles Theorem

- If two lines and a transversal form same-side interior angles that 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.

Statements:

1 2

2 3

1 3

m ║ n

Reasons:

Given

Vertical Angles

Transitive prop.

Corresponding angles converse

Example 1: Proof of Alternate Interior ConverseProof of the Consecutive Interior Angles Converse

Given: 4 and 5 are supplementary

Prove: g ║ h

g

6

5

4

h

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.

Solution:

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

x + 4x = 180

5x = 180

X = 36

4x = 144

So, if x = 36, then j ║ k.

Find the value of x that makes j ║ k.4x

x

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.

H

E

G

58

61

62

59

C

A

B

D

A. Is EB parallel to HD?

B. Is EA parallel to HC?

Example 5: Identifying parallel lines

Decide which rays are parallel.

H

E

G

58

61

B

D

- Is EB parallel to HD?
- mBEH = 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.

H

E

G

120

120

C

A

- 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?

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.

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