1. 3.4 Proving Lines are Parallel Mrs. Spitz Fall 2005

2. Standard/Objectives: Standard 3: Students will learn and apply geometric concepts Objectives: Prove that two lines are parallel. Use properties of parallel lines to solve real-life problems, such as proving that prehistoric mounds are parallel. Properties of parallel lines help you predict.

3. HW ASSIGNMENT: 3.4--pp. 153-154 #1-28 Quiz after section 3.5

4. Postulate 16: Corresponding Angles Converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

5. 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.

6. 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.

7. 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.

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

9. Example 1: Proof of Alternate Interior Converse Statements: 1.  1   2 2.  2   3 3.  1   3 4.m ║ n Reasons: 1. Given 2. Vertical Angles 3. Transitive prop. 4. Corresponding angles converse

10. Proof of the Consecutive Interior Angles Converse Given:  4 and  5 are supplementary Prove: g ║ h 6 g h 5 4

11. 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.

12. 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. xx 4x 

13. 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

14. Solution: Because corresponding angles are congruent, the boats’ paths are parallel. Parallel lines do not intersect, so the boats’ paths will not cross.

15. Example 5: Identifying parallel lines Decide which rays are parallel. 62  61  59  58  AB E H G D C A. Is EB parallel to HD? B. Is EA parallel to HC?

16. Example 5: Identifying parallel lines Decide which rays are parallel. 61  58  B E H G D A.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.

17. Example 5: Identifying parallel lines Decide which rays are parallel. 120  A E H G 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.

18. 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.