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Exploring Congruent Triangles Congruent triangles: Triangles that are the same size and shape – Each triangle has six parts, three angles and three sides – If the corresponding six parts of one triangle are congruent to the six parts of another triangle, then the triangles are congruent • This is abbreviated by CPCTC (corresponding parts triangles are congruent) of congruent – Orientation of the triangles is not important. This means that the triangles can be flipped, slid and turned around, and if the corresponding parts are congruent, the triangles are congruent D B E C A If ABC @ F DE F, then the following parts are congruent: Segments · AB @ DE Segments: · BC @ EF 1. · AC @ DF 2. 3. Angles: 1. 2. 3. Angles · РA @ РD · РB @ РE · РC @ РF Note: The order matters. If ABC @ DEF, it is not the same as saying ABC @ FED. EXAMPLE 1: If ABC @ RQC, name the corresponding congruent sides and angles. B Congruent Sides Q C – – – Congruent Angles – – – A R Example 2 Write the correct congruency statement. Compare the sides and the angles. 3. 4. 5 • Do worksheet Parts of congruent triangles and exploring congruent triangles. Congruence of triangles is: Reflexive: • ABC @ ABC A B B' C C' A' Congruence of triangles is: Symmetric • If ABC @ DEF, then DEF @ ABC A C E B D F Congruence of triangles is: Transitive • If ABC @ DEF and DEF @ LMN, then ABC @ LMN. A C E B D H F G I Proving Triangle Congruency There are 4 ways to prove that two triangles are congruent to each other. Remember, once you know that two triangles are congruent, then Corresponding Parts of Congruent Triangles are Congruent. Side-Side-Side Postulate (SSS): If all 3 sides of one triangle are congruent to all 3 sides of another triangle, then the two triangles are congruent. Side-Angle-Side Postulate (SAS): If two sides & the included angle (the angle between the two sides) of one triangle are congruent to two sides & the included angle of another triangle, then the two triangles are congruent. Ex: • Do worksheet ways to prove triangle congruence SAS SSS A C E B D F Angle-Side-Angle Postulate (ASA): If 2 angles & the side between them in one triangle are congruent to 2 angles & the side between them in another triangle, then the 2 triangles are congruent. Angle-Angle-Side Postulate (AAS): If 2 angles & a side not between them in one triangle are congruent to 2 angles & the corresponding side not between them in another triangle, then the 2 triangles are congruent. Determine which Postulate or theorem can be used to prove the 2 triangles are congruent. If it’s not possible, write Not Possible. Remember, you can choose from SSS, SAS, ASA, or AAS. Hypotenuse-Leg (HL) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangle are congruent. Proofs: F I G H E Given : FE @ HI & G is the midpoint of FHand EI Prove: FEG @ HIG if EI @ FH Statements 1) FE @ HI Is the midpoint of FHand EI 2) 3) FG @ HGand EG @ IG FEG @ HIG Reasons R S Q Given: T RQ TS RQ @ TS Prove: QRT @ STR Statement 1) RQ TS RQ @ TS 2) РQRT @ РSTR 3) RT @ TR 4) QRT @ STR Reasons R E L D W Given: L is the Midpoint of WE,WR ED Prove: ΔWRL @ ΔEDL Statement 1. L is the Midpoint of WE,WR ED 2. РW @ РE 3. 4. WL @ EL РWLR @ РELD 5. ΔWRL @ ΔEDL Reason J K F L M N Given: РNKL @ РNJM , KL @ Prove: LN @ MN Statement 1. 2. 3. РNKL @ РNJM , KL @ JM РN @ РN JMN @ LKN 4. LN @ MN JM Reason L M J K Given: JK KM , JM @ KL, ML JK Prove: ML @ JK Statement 1. JK KM , JM @ KL, ML JK 2. РJKM 3. KM ML 4. РLMK 5. Is a right angle Is a right angle MK MK 6. JMK @ LMK 7. ML @ JK Reason