2.5 Proving angles congruent

1. 2.5 Proving angles congruent

2. Draw two intersecting lines. 1 3 4 2 Number the angles as shown Fold angle 1 onto angle 2. Fold angle 3 onto angle 4. Make a conjecture about vertical angles. Vertical angles are congruent.

3. Theorem: Vertical Angles Theorem Vertical angles are congruent. 1 3 4 2 < 1 < 2, < 3 < 4 ~ ~ = =

4. To prove a theorem, a “Given” list shows you what you know from the hypothesis of the theorem. You will prove the conclusion of the theorem. what you know: Given what you must show: Prove Diagram shows what you know. 40 40

5. Using the Vertical Angles Theorem: Find the value of x. (4x)0 (3x + 35)0 Find the measure of the labeled pair. 4x = 3x + 35 • 3x -3x x = 35 4(35) = 3(35) + 35 = 140 140

6. Using the Vertical Angles Theorem: Find the value of x. (y)0 (4x)0 (3x + 35)0 Find the measure of the other pair. y + 140 = 180 - 140 -140 y = 40

7. Theorem: Congruent Supplements Theorem; If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

8. Proving Theorem: Given: < 1 and < 2 are supplementary. <3 and <2 are supplementary. Prove: < 1 <3 By definition of supplementary angles, m<1 + m<2 = 180 and m< 3 + m< 2 = 180. By substitution, m< 1 + m< 2 = m< 3 + m< 2. Subtract m< 2 from both sides m< 1 = m< 3 3 1 2 ~ =

9. Theorem: If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Theorem: All right angles are congruent. Theorem: If two angles are congruent and supplementary, then each is a right angle.

10. Assignment Geometry Homework (Worksheet) Practice 2.5: Problems #1-9 odd