1. 3.3 Parallel Lines & Transversals

2. 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

3. 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

4. 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

5. 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°

6. 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

7. 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

8. Example 1: Proving the Alternate Interior Angles Theorem
Given: p ║ q
Prove: 1 ≅ 2 1 2 3

9. 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

10. 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

11. 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

12. 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°

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

14. 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)°

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