SimilaritySimilaritySimilaritySimilarity

TESTS FOR SIMILARITY

Two triangles are similar if:

• All corresponding angles are equal• All the corresponding sides are proportional

Triangles are similar if their corresponding angles are equal and their corresponding sides

are proportional

How To Identify？Two triangles are similar if:

• 3 angles of 1 triangle are the same as 3 angles of the other

• 3 pairs of corresponding sides are in the same ratio

• 2 pairs of corresponding sides are in the same ratio and the included angle is equal

Type 1: 3 angles of 1 triangle are the same as 3

angles of the other (AAA)

A

B C

D

E F

In triangle ABC, any 2 angle is equal to any 2 angles of triangle DEF, then they are similar.

g

h i

k

l

Angle g= angle k

Angle h= angle j

triangle ABC and DEF are similar

j

Angle i= angle l

Type 2: 3 pairs of corresponding sides are in

the same ratio

A

B C

z

h

i

D

EF

In triangle ABC and DEF, if

kz

kh

ki

triangle ABC and DEF are similar.

where k is a constant.

kAC

DF

BC

EF

AB

DE

Type 3: 2 pairs of corresponding sides are in the

same ratio and the included angle is equal (SAS) A

B

z

h

i

C

D

E

kz

kh

ki

F

triangle ABC and triangle DEF are similar.

g

h

In triangle ABC and DEF, if k

BC

EF

AB

DE

and angle g = angle h.

1. Are the following triangles similar? Not drawn to scale.

A

B

C

D

E

F

75

40

40

65

Ans: yes

1. Are the following triangles similar? Not drawn to scale.

A

B

C

2

5

6

E

4

10

12

D

F

Ans: yes

B

1

4

3.5

A

B

E

2

5

7.5

F

D

Ans: no

1. Are the following triangles similar? Not drawn to scale.

B

A

C

E

F

D

3.5

5

7

1055 55

Ans: yes

1. Are the following triangles similar? Not drawn to scale.

Think!

1 Which of the following triangles are always similar?

a.  right triangles

b.  isosceles triangles

c.  equilateral triangles

Ans:C

Similar triangles are exactly the same shape and size.

A) True

B) False

Ans:B

2.

The sides of a triangle are 5, 6 and 10.  Find the length of the longest side of a similar triangle whose shortest side is 15.

Ans:D

A)10

B)15

C)18

D)30

3.

4. In the diagram, DE is parallel to AC.  BD = 4, DA = 6 and EC = 8.  Find BC to the nearest tenth.

A) 4.3

B) 5.3

C) 8.3

D) 13.3

Ans: D

5. Find BC.

A) 4

B) 4.5

C) 13.5

D) 17

Ans:C

6. At a certain time of the day, the shadow of a 5' boy is 8' long.  The shadow of a tree at this same time is 28' long.  How tall is the tree?

A) 8.6

B) 16

C) 17.5

D) 20

Ans:C