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SIMILAR

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SIMILAR

TRIANGLES

A

D

E

B

F

C

= =

BC

AB

AC

EF

DF

DE

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

ABCDEF

When we say that triangles are similar there are several repercussions that come from it.

AD

BE

CF

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.

1. PPP Similarity Theorem

3 pairs of proportional sides

2. PAP Similarity Theorem

2 pairs of proportional sides and congruent angles between them

3. AA Similarity Theorem

2 pairs of congruent angles

E

A

13

9.6

10.4

5

B

C

12

D

F

4

1. PPP Similarity Theorem

3 pairs of proportional sides

ABC DFE

L

G

7.5

5

70

H

70

I

7

J

K

10.5

2. PAP Similarity Theorem

2 pairs of proportional sides and congruent angles between them

mH = mK

GHI LKJ

L

G

7.5

5

50

H

50

I

7

J

K

10.5

The PAP 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.

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

Q

M

70

50

N

50

70

O

P

R

3. AA Similarity Theorem

2 pairs of congruent angles

mN = mR

MNO QRP

mO = mP

T

X

Y

34

34

34

34

59

59

87

59

Z

S

U

It is possible for two triangles to be similar when they have 2 pairs of angles given but only one of those given pairs are congruent.

mT = mX

mS = mZ

mS = 180- (34 + 87)

TSU XZY

mS = 180- 121

mS = 59

Yes

Yes

No

Yes

Yes

Yes

A

D

E

C

B

- Triangle ABC is similar to triangle ADE.
- DE is parallel to BC.
- AE = 3 cm, EC = 6 cm, DE = 4cm
- Calculate the length of BC

3

4

E

D

Answer :

A

9

6

C

12

A

B

AC = 9 = 3

AE 3

BC = 3

DE

BC = 3 x DE

BC = 3 x 4 = 12

AB & DE are parallel

Explain why ABC is similar to CDE

<CED = <BAC Alternate Angles

5

A

B

<EDC = <ABC Alternate Angles

<ECD = <ACB Vert Opp Angles

3

C

6

E

D

?

Triangle ABC is similar to Triangle CDE

Calculate the length of DE

AC corresponds to CE

Scale Factor = 2

5

A

B

AB corresponds to DE

DE = 2 x AB

3

C

DE = 10cm

6

E

D

?

la fin

КОНЕЦ

τέλοσ

final

The end

KATAPUSAN

finito

ﭙﺎﻴﺎﻥ

sof