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4.4 – Prove Triangles Congruent by SAS Geometry Ms. Rinaldi Included Angles The included angle is the angle between two sides. Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. B C A S T If Side AB RS Angle A R Side CA TR Then ABC RST R EXAMPLE 1 Use SAS Decide whether the congruence statement is true. Explain your reasoning. ABC DEF SOLUTION (S) (A) (S) AC DF C F CB FE So, by SAS, ABC DEF EXAMPLE 2 Use SAS Decide whether the congruence statement is true. Explain your reasoning. NJH DFR SOLUTION Although there are two pairs of congruent sides, the congruent angles are not included (between) the congruent sides. Therefore SAS does not apply and the triangles are not congruent. EXAMPLE 3 Use SAS Decide whether the congruence statement is true. Explain your reasoning. EPF ERF Hint: This is possible! Do not forget the side they both share! EXAMPLE 4 Use SAS Decide whether the congruence statement is true. Explain your reasoning. PTF GTS Hint: This is possible! You have the sides, you only need a pair of congruent angles in between… EXAMPLE 5 Use SAS Decide whether the congruence statement is true. Explain your reasoning. ABC CDA Hint: This is possible! Look for alternate interior angles as well as a side that both triangles share. EXAMPLE 6 Use SAS and properties of shapes In the diagram, QS and RP pass through the center M of the circle. What can you conclude about MRS and MPQ? SOLUTION Because they are vertical angles, PMQ RMS. All points on a circle are the same distance from the center, so MP, MQ, MR, and MS are all equal. ANSWER MRS and MPQ are congruent by the SAS Congruence Postulate. EXAMPLE 7 Use SSS or SAS State the third congruence that must be given to prove that ABC DEF using the indicated postulate. B Given: AB DE E CB FE Use the SSS Congruence Postulate. C A D SOLUTION To use SSS, you need to know that AC DF F EXAMPLE 8 Use SSS or SAS State the third congruence that must be given to prove that ABC DEF using the indicated postulate. B Given: E A D CA FD Use the SAS Congruence Postulate. C A D SOLUTION To use SAS, you need to know that BA ED F EXAMPLE 9 Use SSS or SAS State the third congruence that must be given to prove that ABC DEF using the indicated postulate. B Given: B E E AB DE Use the SAS Congruence Postulate. C A D F