Congruent Angles Associated with Parallel Lines Section 5.3

The Parallel Postulate Through a point not on a line, there is exactly one line parallel to the given line. a II b a b P

Theorem 37: If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent. (short form: II lines => alt. int. angles ≅ )

Given: m II n Prove: angle 3 ≅ angle 4 Reason: II lines => alt. int. angles ≅ 3 4

Theorem 38 If two parallel lines are cut by a transversal, then any pair of the angles formed are either congruent or supplementary.

(180-x)° x° x°

Theorem 39 If two parallel lines are cut by a transversal, each pair of alternate exterior angles are congruent. (short form: II lines => alt. ext. angles ≅ )

Given: m II n Prove: angle 1 ≅ angle 8 Reason: II lines => alt. ext. angles ≅ 8 1

Theorem 40 If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent. (Short form: II lines => corresponding angles ≅ )

Given: a II b Prove: angle x ≅ angle y Reason: II lines => corresponding angles ≅ x a by

Theorem 41 If two parallel lines are cut by a transversal, each pair of interior angles on the same side of the transversal are supplementary. (II lines => same side int angles ≅ ) 1 2 Angle 1 is supplementary to angle 2. < <

Theorem 42 If two parallel lines are cut by a transversal, each pair of exterior angles on the same side of the transversal are supplementary. (II lines => same side ext. angles ≅ ) 1 2 Angle 1 is supplementary to angle 2. < <

Theorem 43 In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other. c is perpendicular to b. a b c < <

Theorem 44 If two lines are parallel to a third line they are parallel to each other. Also called “Transitive Property of Parallel Lines.” Given: b II a, a II c b is parallel to c a cbcb

Crook Problem Draw a line through the point where the two transversals meet. Using the theorems from 5.3, find each part of x. Add both parts to get the value of x. < < a b 46 ° 137 ° x

Crook Problem Answer Top: II lines ⇒ oppint angles ≅ Bottom: II lines ⇒ same side int angles suppl = 89 x = 89° < < a b 46 ° 137 ° x 46 ° 43 °

Sample Proof Given: AB II DC AB ≅ DC Prove: AD ≅ BC A D C B EF

Sample Proof Statements Reasons A D C B EF 1. AB II DC 1. given 2. AB ≅ DC 2. given 3.<BAC ≅ <DCA 3. II lines ⇒ alt int<s ≅ 4. AC ≅ AC 4. reflexive 5. △ ABC ≅ △ CDA 5. SAS (2, 3, 4) 6. AD ≅ BC 6. CPCTC

Sample Proof StatementsReasons 1. AB II DE1. Given 2. BD ≅ AE 2. Given 3. C is the midpoint of BD and AE 3. Given 4. Angle BCA ≅ angle CED 4. IIlines => alt. int. angles ≅ 5. Angle ABC ≅ angle CDE 5. Same as 4 6. AC ≅ CE, BC ≅ CD 6. Midpoints divide segments into two ≅ segments. 7. ACB ≅ ECD 7. ASA (4,5,6) Given: AB II DE, BD ≅ AE, C is the midpoint of BD and AE. Prove: ACB ≅ ECD B A E C D

Sample Problem Given: a II b, d II e, angle 8 = 80⁰ Find the measures of: Angle 1 Angle 2 Angle 3 Angle 4 Angle 5 Angle 6 Angle 7 a b dede

Answers Angle 1 = 80° Angle 2 = 100° Angle 3 = 100° Angle 4 = 80° Angle 5 = 100° Angle 6 = 100° Angle 7 = 100°

Works Cited Milauskas, George, and Robert Whipple. Geometry for Enjoyment & Challenge. Boston: Houghton Mifflin Company, Print.