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Proving Triangles Congruent The Idea of a Congruence Two geometric figures with exactly the same size and shape. F B A C E D How much do you need to know. . . . . . about two triangles to prove that they are congruent? Corresponding Parts you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. 1. AB DE 2. BC EF 3. AC DF 4. A D 5. B E 6. C F ABC DEF Do you need all six ? NO ! SSS SAS ASA AAS Side-Side-Side (SSS) 1. AB DE 2. BC EF ABC DEF 3. AC DF If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent Side-Angle-Side (SAS) 1. AB DE 2. A D 3. AC DF ABC DEF included side If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent Included Angle The angle between two sides G I H Included Angle Name the included angle: E Y S YE and ES E ES and YS S YS and YE Y Angle-Side-Angle (ASA) 1. A D 2. AB DE 3. B E ABC DEF included side 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. Included Side The side between two angles GI HI GH Included Side Name the included angle: E Y S Y and E YE E and S ES S and Y SY Angle-Angle-Side (AAS) 1. A D 2. B E 3. BC EF ABC DEF Non-included side If 2 angles and a non-included side of 1 triangle are congruent to 2 angles and the corresponding nonincluded side of another triangle, then the 2 triangles are congruent Warning: No SSA Postulate There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C D NOT CONGRUENT F The Congruence Postulates SSS correspondence ASA correspondence SAS correspondence AAS correspondence SSA correspondence AAA correspondence Name That Postulate (when possible) SAS SSA ASA SSS Name That Postulate (when possible) AAA SAS ASA SSA Name That Postulate (when possible) Reflexive Property SAS Vertical Angles SAS Vertical Angles SAS Reflexive Property SSA HW: Name That Postulate (when possible) HW: Name That Postulate (when possible) Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AC FE For AAS: A F Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS: