1. 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 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.

2.  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
A D
2. BC EF
B E
3. AC DF
C F

3. 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

4. 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.

5. 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.

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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 = = = =

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