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Proving Similarity and Congruence of Triangles
Proving Triangles Are Congruent
Many times triangles may look like they are congruent. However, you cannot
assume two triangles are congruent just because they look congruent.
Congruent triangles have the following characteristics:
Corresponding sides have the same length.
Corresponding angles have the same measures.
But, you do not have to show that ALL sides and ALL angles are congruent to
prove that triangles are congruent. The following are the four congruency theorems
that prove two triangles are congruent.
5 Congruency Theorems
Side– Side–Side (SSS)
Side–Angle–Side (SAS)
Angle–Side–Angle (ASA)
Angle–Angle–Side (AAS)
Hypotenuse–Leg (H-L)
Side-Side-Side
If each side of one triangle has the same length as the corresponding side of
another triangle, then those triangles are congruent.
Example
B
5
A
4
5
C
6
E
D
ΔABC  ΔDEF
4
F
6
Side–Angle–Side
If two sides and the angle created by those two sides have the same measures as
two sides and the angle in between them of another triangle, then those triangles
are congruent.
Example
B
5
A
50o
4
5
C
E
50o
D
© LaurusSoft, Inc.
4
F
Angle–Side–Angle
If two angles and the side shared by those two angles have the same measures as
two angles and the side shared by those two angles of another triangle, then those
triangles are congruent.
B
5
A
70o
40o
E
Example
5
C
D
ΔABC  ΔDEF
70o
40o
F
Angle–Angle–Side
If two angles and the side not shared by those two angles have the same measures
as two angles and the side not shared by those two angles of another triangle, then
those triangles are congruent.
B
70o
A
40o
6
E
Example
70o
C
D
40o
6
ΔABC  ΔDEF
F
Hypotenuse-Leg
If the hypotenuse and leg of one right triangle have the same measure of a leg and
the hypotenuse of another right triangle, then the right triangles are congruent.
Example
E
ΔABC  ΔDEF
B
11
6
6
A
C
D
11
F
Proving Triangles Are Similar
Many times triangles may look like they are similar. However, you cannot assume
two triangles are similar just because they look similar.
Similar triangles have the following characteristics:
Corresponding sides are proportional
Corresponding angles have the same measures.
But, you do not have to show that ALL corresponding sides are proportional and
ALL corresponding angles are congruent to prove that triangles are similar. The
following are the three similarity theorems that prove two triangles are similar.
© LaurusSoft, Inc.
3 Similarity Theorems
Side–Side–Side (SSS)
Side–Angle–Side (SAS)
Angle–Angle (AA)
Side-Side-Side
If each side of one triangle has a length that is proportional to the corresponding
side of another triangle, then those triangles are similar.
Example
B
3
A
2
E
ΔABC ~ ΔDEF
4
6
C
3
F
6
D
Side–Angle–Side
If two pairs of corresponding sides have lengths that are proportional and the
angles created by those two pairs of sides have the same measure, then those
triangles are similar.
E
B
Example
4
70o
2
8
70o
ΔABC ~ ΔDEF
4
C
A
F
D
Angle–Angle
If two angles of one triangle have the same measure as two angles of another
triangle, then those triangles are similar. By the way, the third angle of both
triangles is congruent as well.
Example
E
ΔABC ~ ΔDEF
B
70o
o
70
A
40o
C
D
40o
© LaurusSoft, Inc.
F