4.3 Proving Triangles Congruent: SSS and SAS Goal: Prove that triangles are congruent using the SSS and SAS Congruence Postulates. Activities: Open SSS. Complete steps 1 and 2 and answer Q1 and Q2 on the Triangle Congruence activity sheet. Open SAS. Complete step 3 and answer Q3 on the Triangle Congruence activity sheet. View Lesson 4.3. Work out all examples and take notes. Summary In your notes, explain the two concepts explored in class today relating measures of sides and angles in triangles. Would SSA (two sides and a non-included angle of one triangle congruent to two sides and a non-included angle of another) work? Explain.

 ABC  DEF SSS AND SAS CONGRUENCE POSTULATES If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. Sides are congruent Angles are congruent Triangles are congruent If and then  ABC  DEF 1. AB DE 4. A D 2. BC EF 5. B E 3. AC DF 6. C F

S S S SSS AND SAS CONGRUENCE POSTULATES POSTULATE POSTULATE Side - Side - Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If Side MN QR S then  MNP  QRS Side NP RS S Side PM SQ S

SSS AND SAS CONGRUENCE POSTULATES The SSS Congruence Postulate is a shortcut for proving two triangles are congruent without using all six pairs of corresponding parts.

The marks on the diagram show that PQ  TS, PW  TW, and QW  SW. Using the SSS Congruence Postulate Prove that  PQW  TSW. SOLUTION Paragraph Proof The marks on the diagram show that PQ  TS, PW  TW, and QW  SW. So by the SSS Congruence Postulate, you know that  PQW   TSW.

S A S SSS AND SAS CONGRUENCE POSTULATES POSTULATE POSTULATE Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. Side PQ WX If S then  PQS WXY Angle Q X A Side QS XY S

1  2 Vertical angles are congruent Using the SAS Congruence Postulate Prove that  AEB  DEC. 2 1 Statements Reasons AE  DE, BE  CE Given 1 1  2 Vertical angles are congruent 2  AEB   DEC SAS Congruence Postulate 3

MODELING A REAL-LIFE SITUATION Proving Triangles Congruent ARCHITECTURE You are designing the window shown in the drawing. You want to make  DRA congruent to  DRG. You design the window so that DR AG and RA  RG. Can you conclude that  DRA   DRG ? D G A R SOLUTION GIVEN DR AG RA RG PROVE  DRA  DRG

If 2 lines are , then they form 4 right angles. DRA and DRG Proving Triangles Congruent D GIVEN PROVE  DRA  DRG DR AG RA RG A R G Statements Reasons 1 Given DR AG If 2 lines are , then they form 4 right angles. DRA and DRG are right angles. 2 3 All right angles are congruent DRA  DRG 4 Given RA  RG 5 Reflexive Property of Congruence DR  DR 6 SAS Congruence Postulate  DRA   DRG

Use the SSS Congruence Postulate to show that  ABC   FGH. Congruent Triangles in a Coordinate Plane Use the SSS Congruence Postulate to show that  ABC   FGH. SOLUTION AC = 3 and FH = 3 AC  FH AB = 5 and FG = 5 AB  FG

Use the distance formula to find lengths BC and GH. Congruent Triangles in a Coordinate Plane Use the distance formula to find lengths BC and GH. d = (x 2 – x1 ) 2 + ( y2 – y1 ) 2 d = (x 2 – x1 ) 2 + ( y2 – y1 ) 2 BC = (– 4 – (– 7)) 2 + (5 – 0 ) 2 GH = (6 – 1) 2 + (5 – 2 ) 2 = 3 2 + 5 2 = 5 2 + 3 2 = 34 = 34

All three pairs of corresponding sides are congruent, Congruent Triangles in a Coordinate Plane BC = 34 and GH = 34 BC  GH All three pairs of corresponding sides are congruent,  ABC   FGH by the SSS Congruence Postulate.