# PROVING TRIANGLES CONGRUENT by lanyuehua

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```									PROVING TRIANGLES
CONGRUENT
FOUR WAYS TO PROVE
TRIANGLES ARE CONGRUENT
   Side-Side-Side Postulate (SSS)

   Side-Angle-Side Postulate (SAS)

   Angle-Side-Angle Postulate (ASA)

   Angle-Angle-Side Theorem (AAS)
Side-Side-Side (SSS)
    If the sides of a triangle are
congruent to the sides of a
second triangle, then the two
O              triangles are congruent.

Since the following information is true:
A
BO  MA; OY  AN ; BY  MN
Y
B                           Then the two triangle must be congruent.

BOY  MAN
M               N
Side-Angle-Side (SAS)
    If two sides and the included
angle are congruent to two
sides and the included angle of
O       another triangle, then the two
triangles are congruent.
A

Y
B
WHAT IS AN INCLUDED ANGLE?

M            N
Included Angles
   An included angle is the angle formed
at the intersection of two sides of a
triangle
O                       So, the included angle
between BO and BY would
A       be angle B.

Y           The included angle
B                           between MA and MN
would be angle M.
M           N
Are these two triangles
congruent?
O                   A

Y     M            N
B

   If angle B is congruent to
angle M, then the two
triangles are congruent
because of SAS.
Angle-Side-Angle (ASA)
   If two angles and the included side
of a triangle are congruent to two
angles and the included side of
another triangle, then the two
triangles are congruent.
O                             A

WHAT IS AN
INCLUDED
SIDE?
Y                  M            N
B
Included Sides
   An included side between two angles is
the side that is adjacent to both angles.

O          So, BY is the included side
between angle B and angle Y

Y
B
So are these two triangles
congruent?
O                   A

Y       M          N
B

   Yes, because two angles
and the included side are
congruent which makes
the two triangles
congruent by ASA.
Angle-Angle-Side (AAS)
   If two angles and a nonincluded side
of a triangle are congruent are
congruent to two angles and a
nonincluded side of another triangle,
then the two triangles are congruent.
O                 A

Y
B                    M          N
Are these two triangles
congruent?
O               A

Y
B                M         N

   Since OY is not included
between B and Y and AN is
not included between M and
N, these two triangles are
congruent because of AAS.
Angle-Side-Side
   THERE IS NO SUCH THING AS
ANGLE SIDE SIDE BECAUSE
YOU CAN’T USE THAT KIND OF
LANGUAGE AT SCHOOL.
Angle-Angle-Angle
   This cannot prove two triangles
congruent. It just shows that they are
similar.
 If both of these

triangles are
equiangular, they have
three congruent
angles. Are they
congruent triangles?

```
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