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Section 4-2
Ways to Prove Triangles
Congruent
Side, Side, Side Postulate
(SSS Postulate)
• If three sides of one triangle…
are congruent to three sides of another
triangle,
then the triangles are congruent.
Example
P
A
T
A
T
Do we have two congruent triangles
R here?
In this picture, there are only two
sides marked congruent.
But the two triangles also share side
AT.
S
S
S
AP = AR
given
PT = RT
given
AT = AT
reflexive prop
▲PAT  ▲RAT SSS Postulate
Side, Angle, Side Postulate
(SAS Postulate)
• 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.
Example
Do we have two congruent triangles
here?
In this picture, there are only two
sides marked congruent.
But the two triangles will have
congruent angles (vertical angles)
A
C
R
1
H
2
S
S
S
A
CR = SR
HR = AR
m<1 = m<2
▲CRH  ▲SRA
given
given
vertical angles
SAS Postulate
Angle, Side, Angle Postulate
(ASA Postulate)
• If two angles and the included side of one
triangle…
are congruent to two angles and the
included side of another triangle,
then the triangles are congruent.
Example
A
Given: AT = YT, and m<A = m<Y
Prove: YP = AR
It would be easy to prove YP= AR if
we had 2 congruent triangles
P
1
2
T
R
Statements
A
S
A
Y
Nice Job!!!
Reasons
m<Y = m<A
given
YT = AT
given
m<1 = m<2
vertical angles
▲YTP  ▲ATR ASA Postulate
YP = AR
CPCTC
It’s all about choices…
The triangles must be congruent because there is only one way to
create a triangle with those specifications
8 cm
38˚
6 cm
An angle between 2 sides
3 sides
67˚
32˚
A side between 2 angles
Think About This…
1
2
Are these triangles congruent?
Are the triangles congruent by ASA?
What can you conclude about <1 and <2? <1  <2
Now are these triangles congruent by ASA?
Try classroom exercises – pg 123-124 (1-9)