4-2 Vocabulary Auxiliary line Corollary Interior Exterior Interior angle Exterior angle Remote interior angle

4.2 Angle Relationships in Triangles Geometry

Thm 4-2-1 Triangle Sum Theorem The sum of angle measures of a triangle is 180°. m∠A + m∠B + m∠C = 180° B A C

Ex. 1 1a.)Find the 1b.) Find the Y 62° 40° W X Z 12°

Corollary – a theorem whose proof follows directly from another theorem. (2 corollaries to the Triangle Sum Theorem.) Corollary 4-2-2 The acute angles of a right triangle are complementary. Corollary 4-2-3 The measure of each angle of an equiangular triangle is 60°.

Ex. 2 One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?

Thm 4-2-4 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. m∠4 = m∠1 + m∠2 1 2 4 3

Ex. 3 Find the 15 (5x-60)° A C D (2x+3)°

4-2-5 Thm 3rd angles Thm If 2 s of one Δ are  to 2 s of another Δ, then the 3rd pr. s are also .

Ex. 4 Find m∠K and m∠J K )) (4y²)° J (6y² - 40 )° )) ) H F ) )) G I

Ex 5: find x ) ) 22o X = 14 )) 87o )) (4x+15)o

If Δ ABC is  to Δ XYZ, which angle is  to C?

Prove Corollary 4-2-3 using 2 column proofs Ex. 6 Given: Δ ABC is equiangular. Prove: mA = m B = m C = 60°

Assignment

Congruent Figures ( )))) ))) (( ( )))) ))) (( B A ___ 2 figures are congruent if they have the exact same size and shape. When 2 figures are congruent the corresponding parts are congruent. (angles and sides) Quad ABDC is congruent to Quad EFHG ___ ___ )))) ))) ___ (( D C F ( E ___ ___ ___ )))) ___ ))) (( H G

Ex 6: ABCD is  to HGFE, find x and y. 9cm A B E 91o F (5y-12)o 86o 113o D C H G 4x-3cm X = 3, y = 25

Thm 4.4 Props. of  Δs A Reflexive prop of Δ  - Every Δ is  to itself (ΔABC  ΔABC). Symmetric prop of Δ  - If ΔABC  ΔPQR, then ΔPQR  ΔABC. Transitive prop of Δ  - If ΔABC  ΔPQR & ΔPQR  ΔXYZ, then ΔABC  ΔXYZ. B C P Q R X Y Z

Given: seg RP  seg MN, seg PQ  seg NQ , seg RQ  seg MQ, mP=92o and mN is 92o. Prove: ΔRQP  ΔMQN N R 92o Q 92o P M

Statements Reasons 1. 1. given 2. mP=mN 2. subst. prop = 3. P  N 3. def of  s 4. RQP  MQN 4. vert s thm 5. R  M 5. 3rd s thm 6. ΔRQP  Δ MQN 6. def of  Δs