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 Develop geometric intuition and visualization through performing geometric constructions with straightedge/compass and with technology. CLE Establish processes for determining congruence and similarity of figures, especially as related to scale factor, contextual applications, and transformations.

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.  ABC   FDE Forget past mistakes. Forget failures. Forget everything except what you're going to do now and do it." William Durant A B C D E F Side-Angle-Side (SAS) Congruence 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 A B C F D 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. Draw RT and ST to form  RST X Y Z RS T

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 A B C K L J

Copy the two Triangles using the requested methodology Using Sides create  STV   ABC Using 2 sides and an included angle create  XYZ   DEF A B C D E 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 Statement 1.AB  CD, AD  CB 2.AC  CA 3.AD  DC, AB  BC 4.AD || BC, AB || CD 5.  D and  B is a right Angle 6.  1   4,  2   3 7.  D   B 8.  ACD   CAB Reason 1.Given 2.Transitive Property 3.Given 4.Given 5.Definition of  lines 6.Alternate Interior  ’s are  7.Definition of Right Angles 8.Def of congruent triangles Given: AB  CD, AD  CB, AD  DC, AB  BC, AD || BC, AB || CD Prove:  ACD   CAB $ A BC D

Given: AB  AC,BY  CY Prove:  BYA   CYA Statement 1.AB  AC,BY  CY 2.AY  AY 3.  BYA   CYA Reason 1.Given 2.Reflexive Property 3.SSS A B CY

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 2.Definition of Midpoint 3.Vertical Angles are  4.SAS A B C D X

Practice Assignment Block Page 267, 10 – 18 even Honors; Page 267, 10 – 28 even