3.3 Parallel Lines & Transversals

Objectives/Assignment

• Prove and use results about parallel lines and transversals • Use properties of parallel lines to solve real-life problems • Assignment: 1-29 all, Quiz page 149

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

.

6 5  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 3 1 2

Proof

Statements: 1. p ║ q 2.

 1 ≅ 3.

4.

 3 ≅  1 ≅  3  2  2 Given: p ║ q Prove:  1 ≅  2 3 1 2 Reasons: 1. Given 2. Corresponding Angles Postulate 3. Vertical Angles Theorem 4. 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 6 7 5 9 8

Solutions:

a. m  6 = m  5 = 65 ° • Vertical Angles Theorem b. m  7 = 180 ° m  5 =115 ° 6 7 • Linear Pair postulate c. m  8 = m  5 = 65 ° • Corresponding Angles Postulate d. m  9 = m  7 = 115 ° • Alternate Exterior Angles Theorem 5 9 8

Example 3 - Classifying Leaves

BOTANY —Some plants are classified by the arrangement of the veins in their leaves. In the diagram below, j is m  1?

║ k. What k j 120 ° 1

Solution

1. m  1 + 120 ° = 180° 2. m  1 = 60 ° k j 120 ° 1 1. Consecutive Interior angles Theorem 2. Subtraction POE

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: 1. m  4 = 125 ° 2. m  4 +(x+15) °=180° 3. 125 ° +(x+15) °= 180° 4. x = 40° Given 125 ° 4 (x + 15) ° Reasons: 1. Corresponding Angles Postulate 2. Linear Pair Postulate 3. Substitution POE 4. Subtraction POE