Course 3 5-5

Class Notes

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In geometry, two polygons are similar when one is a replica (scale model) of the other.

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Consider Dr. Evil and Mini Me from Mike Meyers’ hit movie Austin Powers. Mini Me is supposed to be an exact replica of Dr. Evil.

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The following are similar figures.

I

II

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The following are non-similar figures.

I

II

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Feefee the mother cat, lost her daughters, would you please help her to find her daughters. Her daughters have the similar footprint with their mother.

Feefee’s footprint

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A B C

1.

Which of the following is similar to the above triangle?

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 there are several repercussions that come from it.

A D

B E

C F

ABDE

BCEF

ACDF = =

1. PPP 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.

2. PAP Similarity Theorem 2 pairs of proportional sides and

congruent angles between them

3. AA Similarity Theorem 2 pairs of congruent angles

1. PPP Similarity Theorem 3 pairs of proportional sidesA

B C

E

F D

25145

.DFm

ABm

25169

12.

.FEm

BCm

251410

13.

.DEm

ACm

5

412

9.613 1

0.4

ABC DFE

2. PAP Similarity Theorem 2 pairs of proportional sides and

congruent angles between them

G

H I

L

J K

66057

5.

.LKmGHm

660510

7.

.KJmHIm

5 7.5

7

10.5

70

70

mH = mK

GHI LKJ

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.

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.

3. AA Similarity Theorem 2 pairs of congruent angles

M

N O

Q

P R

70

7050

50

mN = mR mO = mP

MNO QRP

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.

87

34

34

S

T

U

XY

ZmT = mX

mS = 180- (34 +

87) mS = 180- 121mS = 59

mS = mZ

TSU XZY

59

5959

34

34

Note: One triangle is a scale model of the other triangle.

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How do we know if two triangles are similar or

proportional?

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Triangles are similar (~) if corresponding angles are equal and the ratios of the lengths of corresponding sides are equal.

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A

B

C

The sum of the measure of the angles of a triangle is

C

Interior Angles of Triangles

Determine whether the pair of triangles is similar. Justify your answer.

Answer: Since the corresponding angles have equal measures, the triangles are similar.

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If the product of the extremes equals the product of the means then a proportion exists.

d

c

b

a

adbc

 

  

AB=K

XY

BC=K

YZ

AC=K

XZ

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26

12 2

4

8 2

5

10

This tells us that ABC and XYZ are similar and proportional.

Q: Can these triangles be similar?

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Answer—Yes, right triangles can also be similar but use the criteria.

AB =

XY

BC =

YZ

AC = K

XZ

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6 8 10 = = = K

4 6 8

AB =

XY

BC =

YZ

AC = K

XZ

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6 8 10 = = = K

4 6 8

6 8 = 1.5 but = 1.3

4 6

This tells us our triangles are not similar. You can’t have two different scaling factors!

Do we have equality?

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If we are given that two triangles are similar or proportional what can we determine about the triangles?

The two triangles below are known to be similar, determine the missing value X.

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x

5.4

5

5.7

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x

5.4

5

5.7

x5.75.45

x5.75.22

x3

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A

B

C

P

Q

R10

6

c

5

4 d

In the figure, the two triangles are similar. What are c and d ?

45

10 c c540 c8

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A

B

C

P

Q

R10

6

c

5

4 d

In the figure, the two triangles are similar. What are c and d ?

d

6

5

10 d1030 d3

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Sometimes we need to measure a distance indirectly. A common method of indirect measurement is the use of similar triangles.

h

6

17102

h

6

102

17

h36

Algebra 1 Honors 4-2

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