Similar triangles are triangles that have the same shape but not necessarily the same size.

A

C

B

D

F

E

ABC DEF

When we say that triangles are similar we know:

A D

B E

C F

ABDE

BCEF

ACDF = =

2. SSS Similarity Theorem 3 pairs of proportional sides

Six of those statements are true as a result of the similarity of the two triangles. However, if we need to prove that a pair of triangles are similar how many of those statements do we need? Because we are working with triangles and the measure of the angles and sides are dependent on each other. We do not need all six. There are three special combinations that we can use to prove similarity of triangles.

3. SAS Similarity Theorem 2 pairs of proportional sides and

congruent angles between them

1. AA Similarity Postulate 2 pairs of congruent angles

8.4 – Similar Triangles (AA)

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

M

N O

Q

P R

70

7050

50

mN = mR mO = mP

MNO QRP

Example:Given: ABC and DEF are isosceles right trianglesProve: DEFABC ~

Example:

Given:

Prove:

JSOH //

SM

HM

JM

OM

JO

M

HS

8.5 – Proving that Triangles are Similar

(SSS and SAS)

SSS Similarity Theorem If corresponding sides of two triangles are proportional, then the two triangles are similar.

A

B C

E

F D

25145

.DFm

ABm

25169

12.

.FEm

BCm

251410

13.

.DEm

ACm

5

412

9.613

10.4

ABC DFE

Proof of SSS Similarity Theorem

Given:

Prove:

Draw so that and

Then because ____________.

So by _______.

So

Since PS = ML, then SQ = MN and PQ = LN

Then by ________.

Use CPCTC and AA sim postulate to show

R

S

T

M

L

N

PQ RTPQ //

P

Q

LMPS STRSQP

RTSPQS ~SS

QP

TR

SQ

ST

PS

RS

NL

TR

MN

ST

LM

RS

LMNRST ~

LMNPSQ LMNRST ~

SAS Similarity Theorem If an angle of one triangle is congruent to an angle of another triangle, and then lengths of the sides including these angles are proportional, then the triangle are similar.

G

H I

L

J K

660575

..LKm

GHm

660510

7.

.KJmHIm

5 7.5

710.5

7070

mH = mK GHI LKJ

The SAS Similarity Theorem does not work unless the congruent angles fall between the proportional sides. For example, if we have the situation that is shown in the diagram below, we cannot state that the triangles are similar. We do not have the information that we need.

G

H I

L

J K

5 7.5

7

10.5

50

50

Angles I and J do not fall in between sides GH and HI and sides LK and KJ respectively.

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