#### Proving Triangles are Congruent: SSS and SAS

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Proving Triangles are Congruent: SSS and SAS

Proving Triangles are
Congruent:
SSS and SAS
Chapter 4.3
Goal 1: SSS & SAS Congruence Postulates
Postulate 19: (SSS) Side-Side-Side
Congruence Postulate
If three sides of one triangle are congruent to
three sides of a second triangle, then the two
If Side
PQ WX,
triangles
are congruent.
Angle S Z, and
Side Q S XY,
If Side AB DE,
IfSide
Side
AB
DE,
DE,
Side
BC
ED,
and
If
AB
then Side
PQ
S EF,
WXY.
BC
and
Side
BC
ED,
and
SideBC
CAED,
FD,and
Side
Side CA
CA FD,
FD,
Side
then ABC DEF
then ABC
ABC DEF
DEF
then
B
C
F
D
A
E
Proof
Given : Diagram
Pr ove : PQW TSW
T
P
Q
W
Statement
1) PQ TS
2) PW TW
3) QW SW
4)PQW TSW
S
Re ason
1) Given
2) Given
3) Given
4) Side Side Side(SSS)
Postulate 20: (SAS) Side-Angle-Side
Congruence Postulate
If two sides and the included angle of one triangle are
congruent to two sides and the included of a second
triangle,
then the two triangles are congruent.
If Side
PQ WX,
Angle S Z, and
Side Q S XY,
If Side PQ WX,
then
PQ
YZ,
WXY.
P
If Side
QS
Angle
S
SZ,
and
IfIf Side
Side PQ
PQ WX,
WX,
Side
Q S XY,
Angle
Angle S
S Z,
Z,and
and
Side
Side Q
QSS XY,
XY,
then Side
PQ
S
WXY.
PS XZ,
then
then
PQS
PQ
SS
PQ
XYZ
WXY.
WXY.
Q
S
X
Y
Z
Proof
E
G iven: C is the midpoint of AE and BD.
Prove: ABC EDC
B
C
D
A
Statement
C is the midpoint of AE and BD.
Reason
G iven
AC EC
Def. of Midpoint
BC DC
Def. of Midpoint
ACB ECD
ABC
EDC
Vertical Angles
SA S
Goal 2: Modeling a Real Life Situation
Example 3: Choosing Which Congruence
Postulate to Use
Q
G iven: RQ RS and Q P SP
Prove: RQ P RSP
S
P
Paragraph Proof
R
The marks on the diagram show that PQ PS and QR
SR. By the Reflexive Property of Congruence, RP RP.
Because the sides of ΔPQR are congruent to the
corresponding sides of ΔPSR, you can use the SSS
Congruence Postulate to prove that the triangle are
congruent.
Example 6: Congruent Triangles
in a Coordinate Plane
Use the SSS Congruence Postulate to show
that
ABC
FGH.
AC = FH = 3
AB = FG = 5
AB
FG
A(-7,5)
H(6,5)
C(-4,5)
**Use the Distance
Formula to find the lengths
BC and GH**
Who remembers the
distance formula?
F(6,2)
G(1,2)
B(-7,0)
x2 x1 y2 y1
2
BC = GH = √34
All sides congruent
2