B ELL RINGER

2-5 P ROVING ANGLES CONGRUENT

V ERTICAL ANGLES Two angles whose sides are opposite rays

A DJACENT ANGLES Two coplanar angles with a common side, a common vertex, and no common interior points

C OMPLEMENTARY ANGLES Two angles that ADD up to 90˚ Each angle is called the complement of the other 1 2 A B 50° 40°

S UPPLEMENTARY A NGLES Two angles that ADD up to 180° Each angle is called the supplement of the other 105° 75° 3 4

E XAMPLES ! 1. Identify Angle Pairs a. Complementary b. Supplementary c. Vertical d. Adjacent

2. What can you tell from the picture? a. About ∠1 and ∠2? b. What type of angles are ∠2 and ∠3? c. What type of angles are ∠4 and ∠5? d. What type of angles are ∠1 and ∠4?

V ERTICAL A NGLES T HEOREM (2-1) A statement that is proven (true) is called a THEOREM The steps taken to show a theorem is true is called a PROOF Vertical angles are congruent ( ≅) ∠1 ≅ ∠2 and ∠3 ≅ ∠

P ROOFS In a proof, a “GIVEN” list shows what you know from the hypothesis of the theorem. You must prove the conclusion of the theorem.

P ROVING T HEOREM 2-1 Given: ∠1 and ∠2 are vertical angles. Prove: ∠1 ≅ ∠2 Proof: 1. We know by the definition of ___________________________________ that m∠1 + m∠3 = 180° and m∠2 + m∠3 = _________ 2. By substitution, m∠1 + m∠3 ________ m∠2 + m∠3 3. Subtract m∠3 from both sides. 4. You get m∠1 ______ m∠2 or ∠1 ______ ∠

U SING THE VERTICAL ANGLES THEOREM Find the value of x. Find the value of each angle. 4x (3x + 35)

C ONGRUENT S UPPLEMENTS T HEOREM (2-2) If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. ∠1 and ∠2 are supplementary (∠2 is a supplement of ∠1) ∠1 and ∠3 are supplementary (∠3 is a supplement of ∠1) So ∠2 ≅ ∠

P ROVING T HEOREM 2-2 Given: ∠1 and ∠2 are supplementary ∠1 and ∠3 are supplementary Prove: ∠2 ≅ ∠3 Proof: 1. By the definition of ___________________________ m∠1 + m∠2 = 180° and m∠1 + m∠3 = 180°. 2. By substitution, m∠1 + m∠2 _________ m∠1 + m∠3 3. _____________ m∠1 from each side. 4. You get ________________________________ or ________________ 1 2 3

C ONGRUENT C OMPLEMENTS T HEOREM (2-3) If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Theorem 2-4: All right angles are congruent. Theorem 2-5: If two angles are congruent and supplementary, then each is a right angle.

T ICKET O UT THE DOOR

P RACTICE !! Pg , 10, 13, 15, 19, 20-25, 31, and 39-42