8.5 Proving Triangles are Similar
Side-Side-Side (SSS) Similarity
Theorem

If the lengths of the corresponding sides of
two triangles are proportional, then the
triangles are similar.
P
A
If AB = BC = CA
PQ QR RP,
Q
B
then, ΔABC ~ΔPQR
C
R
Exercise

Determine which two of the three given triangles are
similar. Find the scale factor for the pair.
K
N
R
6
J
9
12
6
4
L
M
8
6
P
Q
10
14
S
Which triangles are similar to ΔABC?
Explain.
B

4
6
A
C
8
K
2.5
J
N
3.75
5.3
2
L
M
3
P
4
Solve for h.

A
60
B
h
C
12
E
10
D
SAS Similarity Theorem
X
M
N
Y
Z
XY
XZ
IfX  M , and

MN MO
then ΔXYZ ~ ΔMNO
O
Determine whether the triangles are
similar. If they are, write a similarity
statement and solve for the variable.
B
10
DIVIDE BY 4
15
2
2

3
3
12
A )
8
D
C
2 12

3
p
8
10

12
15DIVIDE BY 5
p
2p = 3(12)
2p=36
Yes, ΔABC ~ ΔBDC
p=18
Prove Triangles Similar by AA
Triangle Similarity
Two triangles are similar if two pairs of
corresponding angles are congruent. In other
words, you do not need to know the measures of the
sides or the third pair of angles.
Prove Triangles Similar by AA
Example 1:
Determine whether the triangles are similar. If they
are, write a similarity statement, explain your
reasoning.
Prove Triangles Similar by AA
Example 2:
Determine whether the triangles are similar. If they
are, write a similarity statement, explain your
reasoning.
Prove Triangles Similar by AA
a.
Example 3:
Show that the two triangles are similar.
Triangle ABE and Triangle ACD
b.
Triangle SVR and Triangle UVT
Prove Triangles Similar by AA
Example 5:
A school building casts a shadow that is 26 feet
long. At the same time a student standing
nearby, who is 71 inches tall, casts a shadow
that is 48 inches long. How tall is the building to
the nearest foot?