•Prove that 2 lines are parallel.•Use properties of parallel lines to solve problems.

3-4 Proving Lines are Parallel

Corresponding Angles Converse Postulate

•If 2 lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

1

2

Theorem 3.8 (AIA Converse): If 2 lines are cut by a transversal so that AIA are congruent then the lines are parallel.

Proving AIA Converse

1

2

3Given: 1 2Prove: p q

1. 1 2 1. Given2. 1 33. 2 34. p q

2. Vert. ’s Theorem3. Trans. POC4. Corres. ’s Converse

p

q

Statements Reasons

Theorem 3.9 (CIA Converse): If 2 lines are cut by a transversal so that CIA are supplementary then the lines are parallel.

Proving CIA ConverseGiven: Angles 4 and 5 are supplementary.

Prove: p and q are parallel

64

5

p

q

2. Linear Pair Postulate

1. 4 and 5 are supplementary. 1. Given

2. 5 and 6 are supplementary.

3. 4 6

4. p q

Statements Reasons

4. AIA Converse

3. Supplements Theorem

Identify the Parallel Rays

6258

5961

A B C D

E F

3-5 Using Properties of Parallel Lines

•Use properties of parallel lines in problem solving

•Construct parallel lines

Theorem 3.11: If 2 lines are parallel to the same line, they are parallel to each other

Given:Prove:

p q and q r p r

pq

r1

23

1. p q, q r2. 1 2

1. Given

3. 2 34. 1 35. p r

2. Corres. ’s Post. 3. Corres. ’s Post. 4. Trans. POC5. Corres. ’s Converse

Theorem 3.12: If 2 lines in the same plane are perpendicular to the same line, they are parallel to each other

Given:

Prove:

,m p n p m n

m n

p1 2

1. m p, n p2. 1 & 2 are right angles.

1. Given

3. m1 = m22. Def. of lines3. Right Theorem 4. 1 2 4. Def. of ’s

5. m n 5. Corres. ’s Converse