Geo 3.2 notes_parallel_converse

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Review of Parallel Lines Theorems and their Converses.

Review of Parallel Lines Theorems and their Converses.

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  • 1. 3.1 Review: Angle Pairs Corresponding Angles 1 2 ∠1 & ∠7 ∠4 & ∠10 4 3 ∠3 & ∠5 ∠13 & ∠15 78 9 16 10 15 6 5 Alternate Int. Angles 11 14 ∠4 & ∠8 ∠8 & ∠12 12 13 ∠3 & ∠7 ∠3 & ∠9 Consecutive Int. Angles ∠2 & ∠9 ∠8 & ∠11 ∠3 & ∠8 ∠14 & ∠15 Wednesday, October 13, 2010 1
  • 2. 3.1 Review: Angle Pairs Alternate Ext. Angles 1 2 ∠1 & ∠5 ∠16 & ∠12 4 3 ∠2 & ∠6 ∠1 & ∠15 78 9 16 10 15 Same-Side Ext. Angles 6 5 11 14 ∠6 & ∠13 ∠5 & ∠2 12 13 ∠6 & ∠1 ∠9 & ∠12 Vertical Angles ∠2 & ∠4 ∠11 & ∠13 ∠3 & ∠1 ∠9 & ∠15 Wednesday, October 13, 2010 2
  • 3. Solve for the missing angles. BM 14 and 15 ∠1= 127° (linear with 53°) ∠2=53° (vertical with 53°) ∠3=127° (vertical with ∠1) ∠5=53° (corr. with 53°) ∠4=37° (comp. with ∠5) ∠6=90° (vertical with rt. ∠) ∠7=37° (corr. with ∠4) ∠8=143° (linear with ∠7) ∠9=37° (vertical with ∠7) ∠10=143° (vertical with ∠8) Wednesday, October 13, 2010 3
  • 4. Solve for x and y. 83 + x = 180 (Linear Pair) x = 97° 83 = y - 13 (alt. ext. ∠s) y = 96° Wednesday, October 13, 2010 4
  • 5. BM 16!! Prove the Alternate Exterior Angles Theorem Given: m || l Prove: ∠1 ≅ ∠2 Statements Reasons 1 1. m || l 1. Given 2 3 l 2. ∠1 ≅ ∠3 2. Corr. ∠s are ≅ m 3. ∠3 ≅ ∠2 3. Vertical ∠s are ≅ 4. ∠1 ≅ ∠2 4. Transitive Wednesday, October 13, 2010 5
  • 6. 3.4 Proving Lines are Parallel Benchmark 17 Wednesday, October 13, 2010 6
  • 7. What are Proofs? A “step-by-step” justification of what you are doing. Start with the given. Define what you know. (Statement, then Reason) End with your desired conclusion. Wednesday, October 13, 2010 7
  • 8. Write the converse: If t wo lines are parallel, then their corresponding angles are congruent. Corresponding Angles Converse Postulate If t wo lines have corresponding angles congruent, then the lines are parallel. Wednesday, October 13, 2010 8
  • 9. Write the converse: If t wo lines are parallel, then their alternate interior angles are congruent. Alternate Interior Angles Converse Theorem If t wo lines have alternate interior angles congruent, then the lines are parallel. Wednesday, October 13, 2010 9
  • 10. Write the converse: If t wo lines are parallel, then their consecutive interior angles are supplementary. Consecutive Interior Angles Converse Theorem If t wo lines have consecutive interior angles supplementary, then the lines are parallel. Wednesday, October 13, 2010 10
  • 11. Is it possible to show lines m and n are parallel? Why? Ex. 1: 50° Yes, Alternate 50° Interior Angles n Converse Ex. 2: m 80° Yes, Consecutive 100° Interior Angles n Converse Wednesday, October 13, 2010 11
  • 12. What value makes these lines parallel? •These angles are alternate exterior angles 4x + 4° •If alternate exterior angles are congruent (equal), then the lines are parallel 92° 4x + 4 = 92 4x = 88 x = 22 Wednesday, October 13, 2010 12
  • 13. BM #13: Proving Parallel Lines c Given the angle d relationship, which lines 1 2 4 3 are parallel and why? 78 9 16 10 6 5 15 Ex. 3: ∠1≅∠5 11 14 b a || b by alt. ext. converse 12 13 a Ex. 4: ∠8≅∠12 Ex. 5: ∠2 and ∠9 are c || d by alt. int. converse supplementary c || d by con. int. converse Wednesday, October 13, 2010 13
  • 14. Summary: The Angle Theorems state “If lines are parallel, then angles are ≅ or supplementary.” The Converse Theorems state “If angles are ≅ or supplementary, then the lines are parallel.” Wednesday, October 13, 2010 14
  • 15. Assignment: #18 3-2 WS p. 295 (Due at the end of class) #19 p. 137 ##1-9 (odd), 10-20 (all), 26, 27, 30, 51, 52, 56-62 (even) Wednesday, October 13, 2010 15