1. 3-2 Angles & Parallel Lines

2. ***Accepted to be true without proof***
What is a Postulate?
Describes a fundamental relationship between the basic terms of geometry
***Accepted to be true without proof***
Postulate

3. What is a Theorem?
A statement or conjecture that can be proven true by using logical reasoning in conjunction with definitions and postulates.
Theorem

6. Properties of Parallel Lines
Postulate: Corresponding Angles Postulate
If a transversal intersects two parallel lines, then corresponding angles are congruent t
line l || line m 1 l 2 m
Properties of Parallel Lines

7. Properties of Parallel Lines
Theorem: Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are congruent. t
line l || line m l 3 1 2 m
Properties of Parallel Lines

8. Properties of Parallel Lines
Theorem: Alternate Exterior Angles Theorem
If a transversal intersects two parallel lines, then alternate exterior angles are congruent. t
line l || line m 3 l m 2
Properties of Parallel Lines

9. Properties of Parallel Lines
Theorem: Consecutive Interior Angles Theorem
If a transversal intersects two parallel lines, then same-side interior angles are supplementary. t
line l || line m l 3 1 2 m
Properties of Parallel Lines

10. Concept

11. If a transversal is perpendicular to two parallel lines, all eight angles are congruent.

12. Finding Angle Measures
a || b
c || d c
<1
<2
<3
<4
<5
<6
<7
<8 d 8 7 6 a
50° 2 5 4 b 1 3
(1, 2, 4, 3, 8, 7, 5, 6)
Finding Angle Measures

13. Using Algebra to Find Angle Measures
Find the value of x and y.
x =
y = ▲ ▲
50° y
70° x ▲ 2x y
(y – 50) ▲

14. Given: 4 5, m4 = 2x – 10 & m5 = x + 15 Prove: x = 25
Algebraic Proof:
Find Values of Variables
Given: 4 5, m4 = 2x – 10 & m5 = x + 15
Prove: x = 25
4  5 Given
m4 = m5 Definition of congruent angles
2x – 10 = x Given
x – 10 = 15 Subtraction
x = 25 Addition
Answer: x = 25
Example 3

15. * This proves why alternate interior angles are congruent *t a 4 3 1 b 1. 1. 2. 2. 3. 3. 4. 4.
* This proves why alternate interior angles are congruent *
Two-Column Proof

16. Prove: <1 and <2 are Supplementary
Given: a || b
Prove: <1 and <2 are
Supplementary
Statements Reasons t 3 a 2 1 b 1. 1. 2. 2. 3. 3. 4. 4. 5. 5.
Two-Column Proof

17. HOMEWORK Pg. 183-185 #’s 1-6, 8-19,24-28 even, 38, 39, 42