3-2 Angles & Parallel Lines
***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
What is a Theorem? A statement or conjecture that can be proven true by using logical reasoning in conjunction with definitions and postulates. Theorem
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
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
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
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
Concept
If a transversal is perpendicular to two parallel lines, all eight angles are congruent.
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
Using Algebra to Find Angle Measures Find the value of x and y. x = y = ▲ ▲ 50° y 70° x ▲ 2x y (y – 50) ▲
Given: 4 5, m4 = 2x – 10 & m5 = x + 15 Prove: x = 25 Algebraic Proof: Find Values of Variables Given: 4 5, m4 = 2x – 10 & m5 = x + 15 Prove: x = 25 4 5 Given m4 = m5 Definition of congruent angles 2x – 10 = x + 15 Given x – 10 = 15 Subtraction x = 25 Addition Answer: x = 25 Example 3
* 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
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
HOMEWORK Pg. 183-185 #’s 1-6, 8-19,24-28 even, 38, 39, 42