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Name: ______________________________________________________ Date: __________________ Per: _______ LC Math 2 Adv β Triangle Congruence: ASA and SSS Congruence (LT 7) ANGLE-SIDE-ANGLE TRIANGLE CONGRUENCE CRITERIA (ASA): Given two triangles β³ π΄π΅πΆ and β³ π΄β²π΅β²πΆβ². If πβ πΆπ΄π΅ = πβ πΆ β² π΄β² π΅β² (Angle), π΄π΅ = π΄β²π΅β² (Side), and πβ πΆπ΅π΄ = πβ πΆ β² π΅β² π΄β² (Angle), then the triangles are congruent. Proof: Assuming βπ΄π΅πΆ and βπ΄β²π΅β²πΆβ² are distinct triangles where πβ πͺπ¨π© = πβ πͺβ² π¨β² π©β² , π¨π© = π¨β²π©β², and πβ πͺπ©π¨ = πβ πͺβ² π©β² π¨β², what transformations are needed to yield the diagram to the right? Be specific. What is the final isometry needed to prove that βπ΄π΅πΆ β βπ΄π΅πΆβ²? Do we know that π(πΆβ²β²β²) = πΆ? Explain. We have shown that β____________ β β____________. This means that two triangles that have a pair of congruent sides, and a pair of congruent included angles must be congruent. The ASA criteria implies the existence ofβ¦ SIDE-SIDE-SIDE TRIANGLE CONGRUENCE CRITERIA (SSS): Given two triangles β³ π΄π΅πΆ and β³ π΄β²π΅β²πΆβ², if π΄π΅ = π΄β²π΅β² (Side), π΄πΆ = π΄β²πΆβ² (Side), and π΅πΆ = π΅β²πΆβ² (Side), then the triangles are congruent. Proof: Assuming βπ΄π΅πΆ and βπ΄β²π΅β²πΆβ² are distinct triangles where π¨π© = π¨β²π©β², π¨πͺ = π¨β²πͺβ², and π©πͺ = π©β²πͺβ² , what transformations are needed to yield the diagram to the right? Be specific. Without any information about the angles, we cannot perform a reflection like we have for SAS and ASA. So instead, we Μ Μ Μ Μ Μ and label the angles formed as r, s, t and u. will draw an auxiliary line π΅π΅β² How can β³ π¨π©π΅β² and β³ π΅π©β²πͺ be used to show that β³ π¨π©πͺ and β³ π¨β²π©β²πͺβ²? (Hint: Can you use a previous congruence criteria?) We have shown that β____________ β β____________. This means that two triangles that have three pairs of congruent sides must be congruent. The SSS criteria implies the existence ofβ¦ Exercises Based on the information provided, determine whether a congruence exists between triangles. If a congruence exists between triangles or if multiple congruencies exist, state the congruencies and the criteria used to determine them. 1. Given: Μ Μ Μ Μ , πβ π» = πβ π π is the midpoint of π»π 2. Given: Rectangle π½πΎπΏπ with diagonal Μ Μ Μ Μ Μ πΎπ 3. Given: π π = π π΅, π΄π = ππ 4. Given: πβ π΄ = πβ π·, π΄πΈ = π·πΈ 5. Given: π΄π΅ = π΄πΆ, π΅π· = 4 π΄π΅, πΆπΈ = 4 π΄πΆ 6. Given: Circles with centers π΄ and π΅ intersect at πΆ and π· 1 1