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Section 5.2 Proving That Lines are Parallel

Section 5.2 Proving That Lines are Parallel. By Jeremy Cummings, Tarek Khalil , and Jai Redkar. The Exterior Angle Inequality Theorem. The measure of the exterior angle of a triangle is greater than the measure of either remote interior angle. The Logic Behind This Theorem:

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Section 5.2 Proving That Lines are Parallel

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  1. Section 5.2Proving That Lines are Parallel By Jeremy Cummings, TarekKhalil, and Jai Redkar

  2. The Exterior Angle Inequality Theorem • The measure of the exterior angle of a triangle is greater than the measure of either remote interior angle. The Logic Behind This Theorem: ∠1+∠2=180 ∠2+∠3+∠4=180 ∠1+ ∠2 =∠2+∠3+∠4 ∠1=∠3+∠4 ∠1>∠3 ∠1>∠4

  3. The Exterior Angle Inequality Theorem Sample Problem Find the retrictions on x. 62° x How this would be done: 1. x <62 because of the exteriorangle inequality theorem- 62° is the exterior angle and x is the remote interior 2. x > 0 because every angle in a triangle is greater than 0 3. So, the answer is 0<x<62

  4. The Exterior Angle Inequality Theorem Practice Problem Write an inequality that states the restrictions on x: Do the problem and then continue to see work and answer. Work and Answer 25< 5x-5 <125 25<5x-5 5x-5<125 30<5x 5x<130 6<x x<26 6<x<26 5x-5 125° 25°

  5. Identifying Parallel Lines Theorem 31 • If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are parallel. • Short Form: Alt. int. ∠' s ≅ => ∥ lines Given: ∠ 1≅ ∠2 Prove: y∥z

  6. Identifying Parallel Lines Theorem 32 • If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are parallel. • Short Form: Alt. ext. ∠' s ≅ => ∥ lines Given: ∠ 1≅ ∠2 Prove: y∥z

  7. Identifying Parallel Lines Theorem 33 • If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel. • Short Form: Corr.∠' s ≅ => ∥ lines Given: ∠ 1≅ ∠2 Prove: y∥z

  8. Identifying Parallel Lines Theorem 34 • If two lines are cut by a transversal such that two interior angles on the same sideof the transversal are supplementary, the lines are parallel. • Short Form: Same side int. ∠' s suppl. =>∥lines Given: ∠1 suppl ∠2 Prove: y∥z

  9. Identifying Parallel Lines Theorem 35 • If two lines are cut by a transversal such that two exterior angles on the same sideof the transversal are supplementary, the lines are parallel. • Short Form: Same side ext. ∠' s suppl. =>∥lines Given: ∠1 suppl ∠2 Prove: y∥z

  10. Theorem 36 • If two coplanar lines are perpendicular to a third line, they are parallel. Given: x⊥z and y⊥z Prove: x∥y

  11. Sample Problem Dealing With Theorems • Transversal t cuts lines k and n. m ∠ 1 = (148 - 3x)° and m ∠ 2 = (5x + 10)°. Find the value of x that makes k ∥ n. k n 2 1 t • How to do this: • In order for k ∥ n, ∠ 1 has to be suppl. to ∠ 2 because of the theorem “Same side int. ∠'s suppl. =>∥lines.” • So, m ∠ 1 = (148 - 3x)° + m ∠ 2 = (5x + 10)°=180 because suppl. angles =180°. • Through algebra, • 148-3x+5x+10=180 • 2x+158=180 • 2x=22 • x=11

  12. Practice Problem Write a 2- column proof and then continue to see the correct steps. Given: BD bisects∠ABC BC≅CD Prove: CD∥BA Statements Reasons 1. Given 2. Given 3. If a ray bisects an ∠, then it divides the ∠ into 2 ≅ ∠’s. 3. ∠ABD ≅ ∠ CBD 4. ∠ CDB≅ ∠ CBD 5. ∠ CDB ≅ ∠ABD 5. Transitive 6. Alt. int. ∠’s ≅ =>∥ lines

  13. Works Cited “9-1: Proving Lines Parallel.” Ekcsk12.org. Edwards-Knox Central School, n.d. Web. 18 Jan. 2011. “Perpindicular and Parallel Lines.” edHelper.com. edHelper.com, n.d. Web. 18 Jan. 2011. Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challange. New Edition. Evanston, Illinois: McDougal, Littell and Company, 2004. 216-18. Print.

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