Download sss and sas congruence postulates

4.3 Proving Triangles Congruent:
SSS and SAS
GOAL: PROVE THAT TRIANGLES ARE CONGRUENT USING THE SSS AND SAS
CONGRUENCE POSTULATES
.
Activities:
1. Open SSS. Complete steps 1 and 2 and answer Q1 and Q2
on the Triangle Congruence activity sheet.
2. Open SAS. Complete step 3 and answer Q3 on the
Triangle Congruence activity sheet.
3. View Lesson 4.3. Work out all examples and take notes.
4. Summary
a. In your notes, explain the two concepts explored in
class today relating measures of sides and angles in
triangles.
b. Would SSA (two sides and a non-included angle of
one triangle congruent to two sides and a nonincluded angle of another) work? Explain.
SSS AND SAS CONGRUENCE POSTULATES
If all six pairs of corresponding parts (sides and angles) are
congruent, then the triangles are congruent.
If
Sides are
congruent
and
Angles are
congruent
1. AB
DE
4.
A
D
2. BC
EF
5.
B
E
3. AC
DF
6.
C
F
then
Triangles are
congruent
 ABC
 DEF
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
S MN
QR
Side
S NP
RS
Side
S PM
SQ
then  MNP
 QRS
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.
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.
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.
If
Side
S
Angle
A
Side
S
PQ
WX
Q
X
QS
XY
then  PQS
WXY
Using the SAS Congruence Postulate
Prove that
 AEB DEC.
1
2
3
1
Statements
Reasons
AE  DE, BE  CE
Given
1 2
 AEB   DEC
2
Vertical angles are congruent
SAS Congruence Postulate
MODELING A REAL-LIFE SITUATION
Proving Triangles Congruent
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.
ARCHITECTURE
Can you conclude that  DRA   DRG ?
D
SOLUTION
GIVEN
PROVE
DR
AG
RA
RG
 DRA
A
 DRG
R
G
Proving Triangles Congruent
GIVEN
DR
AG
RA
RG
 DRA
PROVE
D
 DRG
A
Statements
R
G
Reasons
Given
1
DR
AG
2
DRA and DRG
are right angles.
If 2 lines are , then they form
4 right angles.
3
DRA 
4
RA  RG
Given
5
DR  DR
Reflexive Property of Congruence
6
 DRA   DRG
SAS Congruence Postulate
DRG
All right angles are congruent
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
Congruent Triangles in a Coordinate Plane
Use the distance formula to find lengths BC and GH.
d=
BC =
(x 2 – x1 ) 2 + ( y2 – y1 ) 2
(– 4 – (– 7)) 2 + (5 – 0 ) 2
d=
GH =
(x 2 – x1 ) 2 + ( y2 – y1 ) 2
(6 – 1) 2 + (5 – 2 ) 2
=
32 + 52
=
52 + 32
=
34
=
34
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.