similarity theorems

SIMILARITY THEOREMS

Similarity in Triangles

Angle-Angle Similarity Postulate (AA~)- If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

W

R

SV

B

4545

WRS BVS because of the AA~ Postulate.

Similarity in Triangles

Side-Angle-Side Similarity Postulate (SAS~)- If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the angles are proportional, then the triangles are similar.

TEA CUP because of the SAS~ Postulate.

C

U P

T

E A

3216 12

3228 21

The scale factor is 4:3.

Similarity in Triangles

Side-Side-Side Similarity Postulate (SSS~)- If the corresponding sides of two triangles are proportional, then the triangles are similar.

C

A

BQ

R S

3

4

6

1530

20

ABC QRS because of the SSS~ Postulate.

The scale factor is 1:5.

Are the following triangles similar? If so, what similarity statement can be made. Name the postulate or theorem you used. F

G

H

K

J

Yes, FGH KJH because of the AA~ Postulate

Are the following triangles similar? If so, what similarity statement can be made. Name the postulate or theorem you used.M

O R

G

H I6

10

3

4

No, these are not similar because

Are the following triangles similar? If so, what similarity statement can be made. Name the postulate or theorem you used. A

X Y

B C

20

25

25

30

No, these are not similar because

Are the following triangles similar? If so, what similarity statement can be made. Name the postulate or theorem you used.

Yes, APJ ABC because of the SSS~ Postulate.

A

P J

B C

3

5

2

3

8

3

Explain why these triangles are similar. Then find the value of x.

3

5

4.5

x

These 2 triangles are similar because of the AA~ Postulate. x=7.5

Explain why these triangles are similar. Then find the value of x.

These 2 triangles are similar because of the AA~ Postulate. x=2.5

5

70 1103 3

x

Explain why these triangles are similar. Then find the value of x.

22

1424

x

These 2 triangles are similar because of the AA~ Postulate. x=12

Explain why these triangles are similar. Then find the value of x.

These 2 triangles are similar because of the AA~ Postulate. x= 12

x

6

29

Explain why these triangles are similar. Then find the value of x.

These 2 triangles are similar because of the AA~ Postulate. x=8

15

4

x

5

Explain why these triangles are similar. Then find the value of x.

These 2 triangles are similar because of the AA~ Postulate. x= 15

18

7.5 12

x

Side Splitter Theorem - If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

Similarity in Triangles

T

S U

R V

x 5

16 10

You can either use

or

Theorem

Triangle Angle Bisector Theorem -If a ray bisects an angle of a triangle, then it divides the opposite side on the triangle into two segments that are proportional to the other two sides of the triangle. A

BC D