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Geometry Notes Section 4-4 What you’ll learn. . . . How to use the SSS Postulate to test for triangle congruence How to use the SAS Postulate to test for triangle congruence Vocabulary Included angle Remember the Definition of Congruent Triangles Two triangles are congruent if and only if their corresponding parts are congruent. According to this definition we would have to show all six pairs of corresponding parts congruent to prove two triangles are congruent. There has to be a shorter way. . . What is the minimum number of corresponding congruent parts necessary to show to triangles are congruent? Would knowing all three pairs of sides congruent be enough? This is called the Side-SideSide (or SSS) Postulate Side-Side-Side Postulate: If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. You can prove two triangles are congruent by showing they have sides of the same length. Example #1: Identify the congruent triangles in each figure. Example #2: Determine whether ΔDEF ΔPQR given the coordinates of the vertices. Explain. D(-6, 1), E(1, 2), F(-1, -4), P(0, 5), Q(7, 6), R(5, 0) You can show all 3 pairs of corresponding sides are congruent by using the distance formula. DE ( 6 1) (1 2) 50 2 2 EF (1 ( 1)) ( 2 ( 4 )) 40 2 2 DF ( 6 ( 1)) (1 ( 4 )) 50 2 2 Example #2 (continued): Now for PQ, QR, and PR D(-6, 1), E(1, 2), F(-1, -4), P(0, 5), Q(7, 6), R(5, 0) PQ ( 0 7) ( 5 6 ) 50 2 2 QR ( 7 ( 5 )) ( 6 0 ) 40 2 2 PR ( 0 5 ) ( 5 0 ) 50 2 2 Since DE PQ , EF QR , and DF PR , DEF PQR by SSS. Example 3: . Example #4: Example #5: Write a twocolumn proof. Might there be another shortcut combination of parts? Possibly mixing pairs of congruent sides with pairs of congruent angles? Would knowing all two pairs of sides congruent and the angles between them be enough? This is called the SideAngle-Side (or SAS) Postulate Side-Angle-Side Postulate: If the 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. You can prove two triangles are congruent by showing they have 2 sides and their included angles equal. Example #6: Identify the congruent triangles in each figure. Example #7: For each diagram, determine which pairs of triangles can be proved congruent by the SAS Postulate. Example #8: For each diagram, determine which pairs of triangles can be proved congruent by the SAS Postulate. Example #9: For each diagram, determine which pairs of triangles can be proved congruent by the SAS Postulate. Example #10: Write a twocolumn proof Have you learned. . . . How to use the SSS Postulate to test for triangle congruence How to use the SAS Postulate to test for triangle congruence Assignment: Non-Proof: Worksheet 4-4 Proof: p.204(10-24Even, 30-47)