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3.4 Proving Lines are Parallel

- After Quiz, Copy Postulate 16 and Theorems 3.8,

3.9 and 3.10 into your notes. - (Page 150)

Objectives

- Prove that two lines are parallel.
- Use properties of parallel lines to solve

real-life problems.

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.

Example 1

- Can you prove that lines p and q are parallel? If

so, describe how.

A.

p

p

p

p

p

p

q

q

q

q

q

q

q

q

p

B.

p

q

q

q

- Can you prove that lines p and q are parallel? If

so, describe how.

C.

p

62

62

q

105

p

D.

105

q

Example 1

- Solutions
- A Yes, Alternate Exterior Angles Converse.
- B No
- C Yes, Alternate Interior Angles Converse.
- D Yes, Corresponding Angles Converse.

Example 2Find the value of x that makes j k.

j

k

x?

4x?

- Solution
- Lines j and k will be parallel if the marked

angles are supplementary. - x? 4x? 180 ?
- 5x 180 ?
- X 36 ?
- So, if x 36, then j k.

Example 3 Using Parallel ConversesUsing

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 4 Identifying parallel lines

- Decide which rays are parallel.

H

E

G

61?

58?

62?

59?

C

A

B

D

A. Is EB parallel to HD? B. Is EA parallel to

HC?

Example 4 Identifying parallel lines

- Decide which rays are parallel.

H

E

G

61?

58?

B

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.

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.

HW ASSIGNMENT

- 3.4--pp. 153-154 1-25
- Quiz after section 3.5

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