4.2: Triangle Congruency by SSS and SAS Objectives: To prove two triangles congruent using the SSS and SAS Postulates. Side-Side-Side (SSS)Postulate If 3 sides of one triangle are to 3 sides of another triangle, then the triangles are . ABC DEF (Notice the order in which the congruency statement is given) B A D C 1. Which other side do we know is congruent? Why? 2. Which two triangles are congruent? 3. How do you know? Included Angles and Sides N B BX X is included between Angle B and Angle X N is included between NB and NX. Which side is included between angle N and angle B? Which angle is included between BX and NX? Side-Angle-Side (SAS) Postulate If 2 sides and the included angle of one triangle are to 2 sides and the included angle of another triangle, then the 2 triangles are . ABC DEF Again, notice the order of the congruency statement. Name the triangle congruence postulate, if any, that you can use to prove each pair of triangles congruent. Then write a congruency statement. Q 1. K P M 2. O N J C M N A 3. L S R P What other information do you need to prove ABC CDA by SSS? 7 A 6 8 D B C 7 From the information given, can you prove AEB CDB ? Explain. D E B A C ASA Postulate Angle-Side-Angle • If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. AAS Theorem Angle-Angle-Side • If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. Can you prove the triangles are congruent? HL Hypotenuse-Leg If the hypotenuse and a leg of one right triangle are congruent To the hypotenuse and a leg of another right triangle, then The triangles are congruent. Can you prove the triangles are congruent? • • • • • • • • • • • • History According to legend, one of Napoleon’s officers used congruent triangles to estimate the width of a river. On the riverbank, the officer stood up straight and lowered the visor of his cap until the farthest thing he could see was the edge of the opposite bank. He then turned and noted the spot on his side of the river that was in line with his eye and the tip of his visor.