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Proving Triangles
Congruent
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Corresponding Parts
In Lesson 4.1, you learned that if all
six pairs of corresponding parts (sides
and angles) are congruent, then the
triangles are congruent.
1. AB  DE
2. BC  EF
3. AC  DF
4.  A   D
5.  B   E
6.  C   F
ABC   DEF
How much do you
need to know. . .
. . . about two triangles
to prove that they
are congruent?
Do you need to show all sides
and all angles?
NO !
SSS
SAS
ASA
AAS
HL
Side-Side-Side (SSS)
All three pairs of corresponding sides are
congruent.
1. GH  PQ
2. HF  QR
3. GF  PR
GHF   PQR
Side-Angle-Side (SAS)
Uses two sides and the included angle.
1. BC  FD
2. C   D
3. AC  ED
ABC   EFD
included
angle
Included Angle
The angle between two sides
G
I
H
Included Angle
Name the included angle:
E
Y
S
YE and ES
E
ES and YS
S
YS and YE
Y
Included Side
Name the included side:
E
Y
S
Y and E
YE
E and S
ES
S and Y
SY
Included Side
The side between two angles
GI
HI
GH
Angle-Side-Angle (ASA)
Uses two angles and the included side.
1. G   K
2. GB  KP
3.  B   P
GBH   KPN
included
side
Angle-Angle-Side (AAS)
Uses two angles and a non-included side.
1. D   G
2.  C   X
3. CM  XT
DMC   GTX
Non-included
side
Hypotenuse-Leg (HL)
•Needs a right triangle
•Uses the hypotenuse and
one other side (leg).
1.AC  ZY (leg)
ABC   ZXY
2. AB  ZX (hypotenuse)
•To use hypotenuse-leg, you MUST have a right
triangle!!
•The hypotenuse is ALWAYS opposite the right
angle.
Methods that
DO NOT
WORK!
The following methods cannot
be used to prove triangles
congruent.
Warning: No A-S-S
(“Donkey Theorem”)
There is no such
thing as an ASS
(or SSA)
postulate!
E
B
F
A
C
D
NOT CONGRUENT
Warning: No AAA Postulate
There is no such
thing as an AAA
postulate!
E
B
A
C
D
NOT CONGRUENT
F
Not enough information
Sometimes you won’t have enough
information given to determine if 2
triangles are congruent.
The Congruence Postulates
 SSS
correspondence
 ASA
correspondence
 SAS
correspondence
 AAS
correspondence
 ASS
correspondence
 AAA
correspondence
Name That Postulate
(when possible)
SAS
ASS
Not 
ASA
SSS
Name That Postulate
(when possible)
AAA
SAS
ASA
ASS
Name That Postulate
(when possible)
Reflexive
Property
Vertical
Angles
SAS
SAS
ASS
Vertical
Angles
SAS
Reflexive
Property
HW: Name That Postulate
(when possible)
SSS
No donkey
theorem!
ASS not allowed
Not enough
information
AAA not allowed
HW: Name That Postulate
(when possible)
Let’s Practice
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
For AAS:
A  F
HW
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
For SAS:
For AAS: