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Chapter 4.4 Proving triangle congruence SSS,SAS Objective: Use the SSS and SAS postulates for proving angle congruence. Check.4.35 Prove that two triangles are congruent by applying the SSS, SAS, ASA, AAS, and HL congruence statements. CLE 3108.4.4 Develop geometric intuition and visualization through performing geometric constructions with straightedge/compass and with technology. CLE 3108.4.8 Establish processes for determining congruence and similarity of figures, especially as related to scale factor, contextual applications, and transformations. Forget past mistakes. Forget failures. Forget everything except what you're going to do now and do it." William Durant Side-Angle-Side (SAS) Congruence Side-Side-Side (SSS) Congruence • If the sides of one triangle are congruent to the sides of a second triangle then the triangles are congruent. • If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent • ABC FDE • ABC FDE F D F A A D E B C B C E Constructions – Congruent Triangles Using Sides • • • • • • Draw a triangle, label the vertices X, Y, and Z Elsewhere on the paper, use a straight edge to construct segment RS Such that RS XZ Using R as the center, draw and arc with radius equal to XY Using S as the center draw and arc with a radius equal to YZ. Let T be the point of intersection of the two arcs. T Draw RT and ST to form RST R Y Z X S Constructions – Congruent Triangles using 2 sides and an included angle • • • • • Draw a triangle, label the vertices A, B, and C Elsewhere on the paper, use a straight edge to construct segment KL Such that KL BC Construct and angle congruent to B using KL as a side of the angle and K as the vertex. Construct JK such that JK BA. Draw JL to complete KJL C A B J K L Copy the two Triangles using the requested methodology B Using 2 sides and an included angle create XYZ DEF Using Sides create STV ABC E D A C F Can you prove Congruence? Side, Side, Side - SSS Can you prove Congruence? Side, Angle, Side - SAS Can you prove Congruence? Not enough Information Can you prove Congruence? Not enough Information Can you prove Congruence? Either SSS or SAS Page 197 34. B 4 C 3 $ Given: ABCD, ADCB, ADDC, 2 ABBC, AD|| BC, AB|| CD 1 A D Prove: ACD CAB Statement Reason 1. ABCD, ADCB 1. Given 2. AC CA 2. Transitive Property 3. ADDC, ABBC 3. Given 4. AD|| BC, AB|| CD 4. Given 5. D and B is a right Angle 5. Definition of lines 6. 1 4, 2 3 6. Alternate Interior ’s are 7. D B 7. Definition of Right Angles 8. ACD CAB 8. Def of congruent triangles Given: ABAC,BYCY Prove: BYA CYA B Y A Statement 1. ABAC,BYCY 2. AY AY 3. BYA CYA Reason 1. Given 2. Reflexive Property 3. SSS C Given: X is the midpoint of BD, X is the midpoint of AC Prove: DXC BXA Statement 1. X is the midpoint of BD, X is the midpoint of AC 2. DX XB, CX XA 3. DXC BXA 4. DXC BXA Reason 1. Given A D X C 2. Definition of Midpoint 3. Vertical Angles are 4. SAS B Practice Assignment • Block Page 267, 10 – 18 even • Honors; Page 267, 10 – 28 even