3.3 Parallel Lines & Transversals
Postulate 15 - Corresponding Angles Postulate Goal 1: Properties of Parallel Lines Postulate 15 - Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2 1 ≅ 2
Theorem 3.4 - Alternate Interior Angles If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4 3 ≅ 4
Theorem 3.5 - Consecutive Interior Angles If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 5 6 5 + 6 = 180°
Theorem 3.6 - Alternate Exterior Angles If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8 7 ≅ 8
Theorem 3.7 - Perpendicular Transversal If a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other. j h k j k
Example 1: Proving the Alternate Interior Angles Theorem Given: p ║ q Prove: 1 ≅ 2 1 2 3
Proof Statements: p ║ q 1 ≅ 3 3 ≅ 2 1 ≅ 2 Reasons: Given Given: p ║ q Prove: 1 ≅ 2 1 2 3 Statements: p ║ q 1 ≅ 3 3 ≅ 2 1 ≅ 2 Reasons: Given Corresponding Angles Postulate Vertical Angles Theorem Transitive Property of Congruence
Example 2: Using Properties of Parallel Lines Given that m 5 = 65°, find each measure. Tell which postulate or theorem you use. A. m 6 B. m 7 C. m 8 D. m 9 9 6 8 5 7
Solutions: m 6 = m 5 = 65° m 7 = 180° - m 5 =115° 9 m 6 = m 5 = 65° Vertical Angles Theorem m 7 = 180° - m 5 =115° Linear Pair postulate m 8 = m 5 = 65° Corresponding Angles Postulate m 9 = m 7 = 115° Alternate Exterior Angles Theorem 8 6 5 7
Example 3 - Classifying Leaves BOTANY—Some plants are classified by the arrangement of the veins in their leaves. In the diagram below, j ║ k. What is m 1? k j 1 120°
Solution m 1 + 120° = 180° m 1 = 60° Consecutive Interior angles Theorem Subtraction POE k j 120° 1
Example 4: Using Properties of Parallel Lines Goal 2: Properties of Special Pairs of Angles Example 4: Using Properties of Parallel Lines Use the properties of parallel lines to find the value of x. 125° 4 (x + 15)°
Proof Statements: m4 = 125° m4 +(x+15)°=180° 125°+(x+15)°= 180° Given 125° 4 (x + 15)° Statements: m4 = 125° m4 +(x+15)°=180° 125°+(x+15)°= 180° x = 40° Reasons: Corresponding Angles Postulate Linear Pair Postulate Substitution POE Subtraction POE