3.4 Proving Lines are Parallel

Goal 1: Proving Lines are Parallel Postulate 16: Corresponding Angles Converse (pg 143 for normal postulate 15) If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. j k j ll k

Theorem 3.8: Alternate Interior Angles Converse (pg 143 Theorem 3.4) If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. j 3 1 k

Theorem 3.9: Consecutive Interior Angles Converse (pg 143 Theorem 3.5) If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. j 2 1 k

Theorem 3.10: Alternate Exterior Angles Converse (pg 143 Theorem 3.6) If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. 4 j k 5

Prove the Alternate Interior Angles Converse Given: 1  2 Prove: m ║ n 3 m 2 1 n

Example 1: Proof of Alternate Interior Converse Statements: 1  2 2  3 1  3 m ║ n Reasons: Given Vertical Angles Transitive prop. Corresponding angles converse Given: 1  2 Prove: m ║ n 3 m 2 1 n

Example 2: Proof 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. g 6 5 Given: 4 and 5 are supplementary Prove: g ║ h 4 h

Find the value of x that makes j ║ k. Example 3: Applying the Consecutive Interior Angles Converse Find the value of x that makes j ║ k. 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 4x x

Goal 2: Using Parallel Converses Example 4: 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 (cont.) Decide which rays are parallel. H E G 58 61 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 (cont.) 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.