Similar Triangle Properties

The student is able to (I can):

Use properties of similar triangles to find segment lengths.

Apply proportionality and triangle angle bisector theorems.

Apply triangle angle bisector theorems

Use triangle similarity to solve problems.

Triangle Proportionality Theorem

If a line parallel to a side of a triangle intersects the other two sides then it divides those sides proportionally.

S

P

A

C

E

>

>

PC SE

AP AC

PS CE=

Note: This ratio is not the same as the ratio between the third sides!

AP PC

PS SE

Triangle Proportionality Theorem Converse

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

S

P

A

C

E

>

>

PC SE

AP AC

PS CE=

Two Transversal Proportionality

If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.

G

O

D

T

A

C>

>

>

CA DO

AT OG=

Examples Find PE

10x = (4)(14)

10x = 56

S

C

O

P

E

10101010 14141414

4444

10 14

4 x=

xxxx

28 3x 5 5.6

5 5= = =

>

>

Example Verify that

(15)(8) = (10)(12)?

120 = 120 Therefore,

H

O

RSE

HE OS

15

10

12 8

=15 10

?12 8

HE OS

Example Solve for x.

6x = (10)(9)

6x = 90

x = 15

>

>

>

x

96

10

10 x

6 9=

Triangle Angle Bisector Theorem

An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

=CD CA

DB AB

Example: Solve for x.

=AD AB

DC BC

=

=

= =

3.5 5

x 125x 42

42x 8.4

5

Angle-Angle Similarity (AA~)

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

M P

A O

Therefore, MAC ~ POD by AA~

M

A C

P

O

D

Side-Side-Side Similarity (SSS~)

If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar.

W H

Y

N

O

T

= =WH HY WY

NO OT NT

Therefore, WHY ~ NOT by SSS~

1230

18

16

40

24

Side-Angle-Side Similarity (SAS~)

If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.

E

T X

U

L V

=LU LV

TE TX L T

Therefore, LUV ~ TEX by SAS~

4

5

2

2.5

Example Explain why the triangles are similar and write a similarity statement.

90 56 = 34

Therefore mV = mX, thus V X.

Since mU = mE = 90, U E

Therefore, LUV ~ TEX by AA~

56

34L

U V

T

E

X

Example Verify that SAT ~ ORT

A

ST

R

O

12

15

20

16

ATS RTO (Vertical angles )

12 15?

16 20=

240 = 240Therefore, SAT ~ ORT by SAS~