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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. ABC  DEF. When we say that triangles are similar there are several repercussions that come from it. A  D.

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SIMILAR

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Similar

SIMILAR

TRIANGLES


Similar

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.

ABCDEF

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

AD

BE

CF


Similar

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


Similar

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


Similar

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

mH = mK

GHI  LKJ


Similar

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.


Similar

Q

M

70

50

N

50

70

O

P

R

3. AA Similarity Theorem

 2 pairs of congruent angles

mN = mR

MNO  QRP

mO = mP


Similar

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.

mT = mX

mS = mZ

mS = 180- (34 + 87)

TSU  XZY

mS = 180- 121

mS = 59


Similar

Yes

Yes

No

Yes

Yes

Yes


Practice 2

A

D

E

C

B

Practice 2:

  • 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


Similar

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


And then

…and then…

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


And then1

…and then…

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

?


Similar

la fin

КОНЕЦ

τέλοσ

final

The end

KATAPUSAN

finito

ﭙﺎﻴﺎﻥ

sof


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