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Pre-AP Bellwork

Pre-AP Bellwork. 1) Solve for p. (3p – 5)°. 3-2 Proving Lines Parallel. Postulate 3-2: Converse of the Corresponding Angles Postulate. If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

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Pre-AP Bellwork

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  1. Pre-AP Bellwork 1) Solve for p. (3p – 5)°

  2. 3-2 Proving Lines Parallel

  3. Postulate 3-2: Converse of the Corresponding Angles Postulate • If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

  4. Theorem 3-3: Converse of the Alternate Interior Angles Theorem • If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.

  5. Theorem 3-4: Converse of the Sam-Side Interior Angles Theorem • If two lines and a transversal form same-side interior angles that are supplementary, then the lines are parallel.

  6. 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.

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

  8. 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

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

  10. 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.

  11. 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

  12. 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

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

  14. 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?

  15. 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.

  16. 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.

  17. 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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