3-4 Proving Lines are Parallel

1. 3-4 Proving Lines are Parallel Prove that 2 lines are parallel. Use properties of parallel lines to solve problems.

2. 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

3. 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 Given: 1  2 Prove: p q 3 p 1 2 q Statements Reasons 1. 1  2 1. Given 2. Vert. ’s Theorem 2. 1  3 3. Trans. POC 3. 2  3 4. Corres. ’s Converse 4. p q

4. 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 Converse p Given: Angles 4 and 5 are supplementary. Prove: p and q are parallel 6 5 4 q Reasons Statements 1. 4 and 5 are supplementary. 1. Given 2. 5 and 6 are supplementary. • Linear Pair Postulate 3.  Supplements Theorem 3. 4 6 4. p q 4. AIA Converse

5. Identify the Parallel Rays E F A B D C

6. 3-5 Using Properties of Parallel Lines • Use properties of parallel lines in problem solving • Construct parallel lines

7. Theorem 3.11: If 2 lines are parallel to the same line, they are parallel to each other p Given: Prove: 1 q r 2 3 1. pq, qr 1. Given 2. 1  2 2. Corres. ’s Post. 3. 2  3 3. Corres. ’s Post. 4. 1  3 4. Trans. POC 5. pr 5. Corres. ’s Converse

8. Theorem 3.12: If 2 lines in the same plane are perpendicular to the same line, they are parallel to each other Given: Prove: 1 2 p m n 1. m  p, n  p 1. Given 2. 1 & 2 are right angles. 2. Def. of  lines 3. m1 = m2 3. Right   Theorem 4. 12 4. Def. of  ’s 5. m n 5. Corres. ’s Converse