Section 5.2 Proving That Lines are Parallel Steven Shields and Will Swisher Period 1

The Exterior Angle Inequality Theorem An exterior angle is formed when one side of a triangle is extended. An exterior angle is formed when one side of a triangle is extended. Exterior Angle

Theorem 30 The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. Exterior Angle Remote Interior Angles

Theorem 30-Sample Problem Write a valid inequality and find the restrictions on x. Write a valid inequality and find the restrictions on x. 50 2x < 2x-20 < < 2x < < 2x < /2 < x < 200/2 35 < x < 100

Identifying Parallel Lines When two lines are cut by a transversal, eight angles are formed. By proving certain angles congruent, you can prove lines II. When two lines are cut by a transversal, eight angles are formed. By proving certain angles congruent, you can prove lines II

Theorem 31 If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are II. (Alt. int. II lines) If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are II. (Alt. int. II lines) 4 3 If <3 congruent <4, then a II b ab

Theorem 31-Sample Problem 5x 2x+15 Is a II b? If these lines are II, the alt. int. angles would be congruent. 5x=25 5(5)=25 x=5 2(5)+15=25 Yes, they are II because the alt. int. <s both equal 25. a b 25 ab

Theorem 32 If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are II. (alt. ext. II lines) If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are II. (alt. ext. II lines) 1 2 ab If <1 congruent <2, then a II b

Theorem 32-Sample Problem x+20 4x 52 x y Is x II y? x x = 180 (These angles are suppl.) 5x + 20 = 180 x + 20 = 52 5x = 160 (32) + 20 = 52 x = = 52 Therefore, the lines are parallel because alt. ext. II lines y

Theorem 33 If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are II. ( corr. II lines) If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are II. ( corr. II lines) 1 2 If <1 congruent <2, then m II n m n

Theorem 33-Sample Problem 34 P Q R S T If <3 congruent <4, then which lines are II? Write the theorem to prove your answer. QT II RS with transversal PS because Corr. II lines.

Theorem 34 If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, the lines are II. If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, the lines are II. 1 2 If <1 suppl. <2, then c II d. c d

Theorem 34-Sample Problem 8x 12x-20 If x=10, is w II z? Explain. w zYes they are parallel because one angle would be 80 and the other 100, so they would be suppl. Therefore the lines are II by theorem 34.

Theorem 35 If two lines are cut by a transversal such that two exterior angles on the same side of the transversal are supplementary, the lines are II. If two lines are cut by a transversal such that two exterior angles on the same side of the transversal are supplementary, the lines are II. 1 2 If <1 suppl. <2, then a II b.

Theorem 35-Sample Problem 2x6x+60 5x a b Is a II b ? 6x x = 180 6(15) (15) = 180 8x = = 180 X = 15 Therefore a is not II to b because the same side ext <s do not add up to 180.

Theorem 36 If two coplanar lines are perpendicular to a third line, they are parallel. If two coplanar lines are perpendicular to a third line, they are parallel. a b c a II b

Practice Problems Name the theorem that proves a II b a b a b a b

Practice Problems Cont. A B C D E Given: <1 congruent <2 Prove: BD II CE

Practice Problems Cont. 125 x 5. Find the restrictions on x. ___< x < ___

Answers 1. Corr. II lines. 2. Alt. ext. II lines. 3. Same side int. II lines. 4. StatementsReasons 1. <1 congruent <2 1. Given 2. BD II CE 2. Corr. II lines 5. 0 < x < 125

Work Cited Rhoad, Richard, George Milauskas, Robert Whipple. Geometry for Enjoyment and Challenge. Boston: McDougal Littell, Rhoad, Richard, George Milauskas, Robert Whipple. Geometry for Enjoyment and Challenge. Boston: McDougal Littell, 1997.