3.4 Proving Lines are Parallel Mrs. Spitz Fall 2005
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.
HW ASSIGNMENT: 3.4--pp #1-28 Quiz after section 3.5
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.
Prove the Alternate Interior Angles Converse Given: 1 2 Prove: m ║ n m n
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
Proof of the Consecutive Interior Angles Converse Given: 4 and 5 are supplementary Prove: g ║ h 6 g h 5 4
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.
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. xx 4x
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
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. 62 61 59 58 AB E H G D C A. Is EB parallel to HD? B. Is EA parallel to HC?
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.
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.
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.