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Warm-Up Exercises Tell whether the pair of triangles is congruent or not and why. 2. ABE and CBD 1. ANSWER Yes; HL Thm. ANSWER SAS Post. Lesson 4.6 ASA and AAS The side of a triangle that falls between two given angles is called the included side of the angles. It is the one side common to both angles. ___________ C AC is the included side of A and C CB is the included side of C and B A B AB is the included side of A and B You can show that two triangles are congruent by using two _________ angles and the included ____________ side of the triangles. Angle-Side-Angle angles and the ______________ If two ___________ included side of one triangle are congruent to the corresponding angles and included side of another triangle, then the triangles are congruent. S B Postulate 5-3 ASA Postulate A C R If A R and AC RT and C T then ΔABC ΔRST T Warm-Up Exercises Lesson 4.6: ASA & AAS • ASA Postulate – If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. ABC DEF ASA QRS POS ASA Angle-Angle-Side CA and CB are the nonincluded sides of A and B C A B You can show that two triangles are congruent by using two _________ angles and a ________________. nonincluded side AAS If two _________ of one triangle are nonincluded side anglesand a ______________ congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent. S B Theorem 5-4 AAS Theorem A C R If A R and C T and CB TS then ΔABC ΔRST T Warm-Up Exercises AAS • AAS – If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, the two triangles are congruent ABC DEF AAS/SAA HIJ KIJ AAS/SAA ASA and AAS ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to show congruence. What other pair of angles must be marked so that the two triangles are congruent by AAS? If F and M are marked congruent, then FE and MN would be Included sides. However, AAS requires the nonincluded sides. D Therefore, D and L must be marked. L M F E N Summary: Proving Triangles are Congruent • Ways that do not work: • Ways that work: – – – – SSS SAS ASA AAS/SAA – AAA – SSA (The bad word doesn’t work!) Warm-Up1Exercises EXAMPLE Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. AAS No Congruence ASA Warm-Up Exercises GUIDED PRACTICE 1. for Examples 1 and 2 In the diagram at the right, what postulate or theorem can you use to RST VUT ? Explain. prove that SOLUTION STATEMENTS REASONS S U Given RS UV Given RTS UTV RST VUT Vertical angles are congruent AAS Warm-Up3Exercises EXAMPLE Write a flow proof In the diagram, CE BD and CAB Write a flow proof to show GIVEN PROVE CE BD, CAB ABE ADE ABE CAD CAD. ADE Warm-Up4Exercises EXAMPLE Standardized Test Practice Warm-Up4Exercises EXAMPLE Standardized Test Practice The locations of tower A, tower B, and the fire form a triangle. The dispatcher knows the distance from tower A to tower B and the measures of A and B. So, the measures of two angles and an included side of the triangle are known. By the ASA Congruence Postulate, all triangles with these measures are congruent. So, the triangle formed is unique and the fire location is given by the third vertex. Two lookouts are needed to locate the fire. Warm-Up4Exercises EXAMPLE Standardized Test Practice ANSWER The correct answer is B. Daily Homework Quiz Warm-Up Exercises Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. 1. ANSWER ASA . Daily Homework Quiz Warm-Up Exercises Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. 2. ANSWER not necessarily congruent . Daily Homework Quiz Warm-Up Exercises Write flow proof. Given : BD bisects ABC, Prove : ABD CBD 3. A C Daily Homework Quiz Warm-Up Exercises ANSWER