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.

If AB = BC = CA

PQ QR RP,

then, ΔABC ~ΔPQR

A

B C

P

Q R

Exercise Determine which two of the three given triangles are

similar. Find the scale factor for the pair. K

J L

6 9

12

4

N

M P8

6R

Q S

6 10

14

Which triangles are similar to ΔABC? Explain.

A

B

C

4 6

8

J

K

L

2.5 3.75

5.3M

N

P

2 3

4

Solve for h.

A B

C

DE12

10

60

h

SAS Similarity Theorem

X

Y Z

M

N O

)

) then ΔXYZ ~ ΔMNOMOXZ

MNXYandMXIf ,

Determine whether the triangles are similar. If they are, write a similarity statement and solve for the variable.

1510

128

32

32

A

C

B

D8

10

1215

p

)

)p12

32

Yes, ΔABC ~ ΔBDC

DIVIDE BY 4 DIVIDE BY 5

2p = 3(12) 2p=36 p=18

Prove Triangles Similar by AATriangle 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 AAExample 1:

Determine whether the triangles are similar. If they are, write a similarity statement, explain your

reasoning.

Prove Triangles Similar by AAExample 2:

Determine whether the triangles are similar. If they are, write a similarity statement, explain your

reasoning.

Prove Triangles Similar by AAExample 3:

Show that the two triangles are similar. a. Triangle ABE and Triangle ACD

b. Triangle SVR and Triangle UVT

Prove Triangles Similar by AAExample 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?