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Similar triangles are triangles that have the same
D
shape and the same size.
A
E
B
F
C
ABC  DEF
When we say that triangles are congruent there are
several repercussions that come from it.
A  D
B  E
C  F
AB  DE
BC  EF
AC  DF
Six of those statements are true as a result of the
congruency of the two triangles. However, if we need
to prove that a pair of triangles are congruent, how
many of those statements do we need? Because we are
working with triangles and the measure of the angles
and sides are dependent on each other. We do not
need all six. There are three special combinations that
we can use to prove congruency of triangles.
1. SSS Congruency Theorem
 3 pairs of congruent sides
2. SAS Congruency Theorem
 2 pairs of congruent sides and congruent
angles between them
3. ASA Congruency Theorem
 2 pairs of congruent angles and a pair of
congruent sides
E
1. SSS Congruency Theorem
 3 pairs of congruent sides
A
F
mAB  mDF  5
mBC  mFE  12
mAC  mDE  13
C
5
12
D
B
12
5
ABC  DFE
2. SAS Congruency Theorem
 2 pairs of congruent sides and congruent
angles between them
G
L
70
70
H
7
I
J
7
K
mGH  mLK  5
mH = mK = 70°
mHI  mKL  7
GHI  LKJ
The SAS Congruency Theorem does not work unless
the congruent angles fall between the congruent
sides. For example, if we have the situation that is
shown in the diagram below, we cannot state that the
triangles are congruent. We do not have the
information that we need.
G
L
50
H
7
50
I
J
7
K
Angles I and J do not fall in between sides GH and HI and
sides LK and KJ respectively.
3. ASA Congruency Theorem
 2 pairs of congruent angles and one pair of
congruent sides.
M
Q
70
N
50
7
50
O
mN = mR
mO = mP
mNO  mRP  7
R
7
70
P
MNO  QRP
It is possible for two triangles to be congruent when they have a pair
of congruent angles and a pair of congruent sides given but another
pair of angles that are not congruent. It is possible that the noncongruent angles are not corresponding and if you calculate the third
angle of one of the triangles, you may find that it is congruent to the
angle
T
X
Y
34
34
59
59
87 59
S
U
mS = 180- (34 + 87)
mS = 180- 121
mS = 59
Z
mT = mX
mTS  mXZ  13
mS = mZ
TSU  XZY