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3.4 Proving Lines are Parallel

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.

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3.4 Proving Lines are Parallel

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  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 3 m 2 1 n

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

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

  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. 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. Find the value of x that makes j ║ k. 4x x

  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. H E G 58 61 62 59 C A B D A. Is EB parallel to HD? B. Is EA parallel to HC?

  16. Example 5: Identifying parallel lines 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.

  17. Example 5: Identifying parallel lines 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.

  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.

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