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Transcript
Aim #24: How do we prove triangles congruent by AAS and HL? CC Geometry H Do Now: Given: DE ≅ DG, EF ≅ GF Prove: DF is the angle bisector of≮EDG Statement Reason Let's examine threepossible triangle congruence criteria: Angle-Angle-Side (AAS) and Side-Side-Angle (SSA), and Angle-Angle-Angle (AAA) Angle-Angle-Side triangle congruence criteria (AAS): Given two Δs ABC and A'B'C'. If m≮B = m≮B' (angle) and m≮C = m≮C' (angle) andAB = A'B' (side), then the triangles are congruent. Mark the triangles to show AAS: In the diagram below, consider a pair of triangles that meet the AAS criteria. If you knew that two angles of one triangle corresponded to and were equal in measure to two angles of the other triangle, what conclusion can you draw about the third angles of each triangle? _________________________ Given this conclusion, which formerly learned triangle congruence criteria can we use to determine if the pair of triangles are congruent? Therefore, the AAS criterion is actually an extension of the triangle congruence criterion. Hypotenuse-Leg triangle congruence criteria (HL) : Given two right Δs ABC and A'B'C' with right angles B and B'. If AB = A'B' (leg) OR BC = B'C' (leg) and AC = A'C'(hypotenuse), then the triangles are congruent. Below, ΔABC and ΔA'B'C' have brought together with corresponding angles ≮A = ≮A' and ≮C = ≮C'. The hypotenuse acts as a common side to the triangles. B B A C A' C' B' A C B B' We draw auxiliary line BB': A Proof of the HL Theorem: Statements Reasons 1. AB = AB' 2. ≮______ = ≮_______ 3. ≮ CBB' and ≮ ABB'are complementary; ≮ CB'B and ≮ AB'B are complementary 4. ≮ CBB' = ≮ ______ 5. ____ = _____ C B' 1. Given 2. If two sides of a Δ are =, the angles opposite are =. 3. _________________________________________ 4. Complements of equal angles are equal. 5. _________________________________________ 6. ____ = _____ 6. Reflexive Property 7. ΔABC ≅ ΔA'B'C' 7. ______________________ *When using HL in a proof, you must state as a reason the triangles are rightΔs.* Criteria that do NOT determine two triangles as congruent: SSA and AAA Side-Side-Angle (SSA): Observe the diagrams below. Each triangle has a set of adjacent sides of measures 11 and 9, as well as the non-included angle of 23˚. Yet, the triangles are not congruent Examine the diagram below which is made of both triangles. The sides of lengths 9 each have been dashed to show their possible locations. SSA cannot guarantee congruence criteria. T wo triangles under SSA criteria might be congruent, but they might not be. W e cannot use SSA in proofs! Angle-Angle-Angle (AAA): ΔABC is an isosceles right triangle. B ΔDEF is also an isosceles right triangle. E D F C What does this mean about the angles of the two triangles? A Why can‛t we categorize AAA as a congruence criteria? Even though the angle measures may be the same, the sides can be proportionally larger; you can have ___________triangles which may or may not be congruent. 1) Given: BC Τ CD, AB Τ AD , m≮1 = m≮2 Prove: ΔBCD ≅ ΔBAD Statements Reasons 3 4 Τ Τ 2) Given: AD BD, BD BC, AB ≅ CD Prove ΔABD ≅ ΔCDB Reasons Statements B 3) Given: BD is an angle bisector of ≮ABC , ≮BAD ≅≮BCD. Prove: ΔADC is isosceles. Statements D A Reasons Let's Sum it Up!! Criteria that can be used for triangle congruence: SAS ASA SSS AAS Criteria that CANNOT be used for triangle congruence: SSA AAA HL C Name_________________________ Date ________________ Τ Τ 1) Given: AB BC, DE EG, BC ll EF, AF = DC Prove: ΔABC ≅ ΔDEF Statements 1) Reasons 1) Givens o o 2) ≮B = 90 , ≮E = 90 3) ≮B = ≮E 4) ≮BCA = ≮EFD 5) FC = FC 6) AF + FC = DC + FC 7) AC = AF + FC DF = DC + FC 8) AC = DF 9) ΔABC ≅ ΔDEF 2) 3) 4) 5) 6) 7) 8) 9) Τ Τ 2) Given: PA AR and PB BR and R is equidistant from PA and PB. Prove: a) ΔPAR ≅ ΔPRB b) PR bisects ≮APB Statements Reasons CC Geometry H HW #24 3) D C ≅ EA, CE ≅ DE Given: EB Prove: Δ ACB ≅ Δ BDA E ,≮CBA ≅ ≮DAB B A Statements Reasons Review: 1) Parallel lines x and y are cut by transversal z. Ray w is perpendicular to line z. o If the m ≮1 = 58 , find the remaining numbered angles. State the geometric z w reason for each step. 2 x y 1 3 4 5