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GEOMETRY MODULE 1 LESSON 22 CONGRUENECE CRITERIA FOR TRIANGLES β SAS OPENING EXERCISE ο· Notebook check ο· Homework Due Dates DISCUSSION Side-Angle-Side Triangle Congruence Criteria (SAS): Given two triangles βπ΄π΅πΆ and βπ΄β²π΅β²πΆβ² so that π΄π΅ = π΄βπ΅β (Side), πβ π΄ = πβ π΄β² (Angle), and π΄πΆ = π΄βπΆβ (Side). Then the triangles are congruent. Consider the two triangles to the right. From earlier lessons, we know that basic rigid motions preserve segment and angle measurements. From this, we can say corresponding sides and angles are congruent. We will determine the transformations taking βπ΄β²π΅β²πΆβ² back to βπ΄π΅πΆ. First, use a translation T to map a common vertex. ο· What two points determine the appropriate vector for our translation? π΄ πππ π΄β ο· Considering the given triangles, can any other pair of points be used? Explain. No, We use π΄ πππ π΄β because only these angles are given as congruent. MOD1 L22 1 ο· What rigid motion will take side π΄βπΆβ to π΄πΆ? How can we be sure that vertex πΆβ maps to πΆ? A rotation maps π΄βπΆβ to π΄πΆ. We can be sure that vertex πΆβ maps to πΆ because rotations preserve segment lengths. ο· What is the last rigid motion mapping βπ΄β²π΅β²πΆβ² to βπ΄π΅πΆ? A reflection maps βπ΄β²π΅β²πΆβ² to βπ΄π΅πΆ. What transformations are needed to demonstrate congruence? PRACTICE 1. Given: πβ πΏππ = πβ πΏππ and ππ = ππ. Do βπΏππ and βπΏππ meet the SAS criteria? STEP JUSTIFICATION πβ πΏππ = πβ πΏππ Given ππ = ππ Given LN =LN Reflexive Property βπΏππ β βπΏππ SAS What rigid motion(s) would map βπΏππ onto βπΏππ? Reflection MOD1 L22 2 Μ Μ Μ Μ and π π Μ Μ Μ Μ bisect each other. 2. Given: ππ Do βπππ and βπππ meet the SAS criteria? STEP JUSTIFICATION Μ Μ Μ Μ and π π Μ Μ Μ Μ bisect each other. ππ Given ππ = ππ Definition of Segment Bisector π π = ππ Definition of Segment Bisector πβ πππ = πβ πππ Vertical Angles βπππ β βπππ SAS What rigid motion maps βπππ onto βπππ? 180° π ππ‘ππ‘πππ 3. Given: πβ πππ = πβ πππ. βπππ and βπππ do not meet the SAS criteria. Why? Not enough information is given to show that the triangles meet SAS. We need information to show that ππ β ππ. SUMMARY Two triangles, βπ΄π΅πΆ and βπ΄β²π΅β²πΆβ², meet the Side-Angle-Side criteria when π΄π΅ = π΄βπ΅β (Side), πβ π΄ = πβ π΄β² (Angle), and π΄πΆ = π΄βπΆβ (Side). The SAS criteria implies the existence of congruence that maps one triangle onto the other. EXIT TICKET If two triangles satisfy the SAS criteria, what rigid motion(s) would map one onto the other whenβ¦ ο· The two triangles share a single common vertex. Rotation, Reflection ο· The two triangles share a common side. Reflection ο· The two triangles are distinct from each other. Translation, Rotation, Reflection MOD1 L22 3