Similar Polygons

Two polygons are similar polygons iff the

corresponding angles are congruent and the

corresponding sides are proportional.

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MACO

ZNIR

AOMC

C

OR

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A

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C

OR

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Similarity Statement:

Corresponding Angles:

Statement of Proportionality:

Example 1

Triangles ABC

and ADE are

similar. Find the

value of x. 6 cm

8 cm

9 cm

x

E

D

A

B

C

Example 1

Triangles ABC and ADE

are similar. Find the

value of x.

8

6=𝑥 + 8

9

6𝑥 + 48 = 72

6𝑥 = 24

𝑥 = 4

6 cm

8 cm

9 cm

x

E

D

A

B

C

Example 2

Are the triangles below similar?

3

5

4 6

8

1053

37

Do you REALLY have to check all the sides and angles?

Investigation 1

In this Investigation we will check the first

similarity shortcut. If the angles in two

triangles are congruent, are the triangles

necessarily similar?

4050

C

A B 50 40

F

ED

Investigation 1

Step 1: Draw ΔABC where m<A and m<B

equal sensible values of your choosing.

4050

C

A B

Investigation 1

Step 1: Draw ΔABC where m<A and m<B

equal sensible values of your choosing.

Step 2: Draw ΔDEF where m<D = m<A and

m<E = m<B and AB ≠ DE.

4050

C

A B 50 40

F

ED

Investigation 1

Now, are your triangles similar? What would

you have to check to determine if they are

similar?

4050

C

A B 50 40

F

ED

6.4-6.5: Similarity Shortcuts

Objectives:

1. To find missing measures in similar

polygons

2. To discover shortcuts for determining that

two triangles are similar

Angle-Angle Similarity

AA Similarity

Postulate

If two angles of one

triangle are

congruent to two

angles of another

triangle, then the two

triangles are similar.

Example 3

Determine whether the triangles are similar.

Write a similarity statement for each set of

similar figures.

Example 3

Yes, from AA.

∆𝐶𝐷𝐸~∆𝐾𝐺𝐻

Yes, from AA.

(They share angle A)

∆𝐴𝐵𝐸~∆𝐴𝐶𝐷

Thales

The Greek mathematician

Thales was the first to

measure the height of a

pyramid by using

geometry. He showed

that the ratio of a

pyramid to a staff was

equal to the ratio of one

shadow to another.

Example 4

If the shadow of the pyramid is 576 feet, the

shadow of the staff is 6 feet, and the height

of the staff is 5 feet, find the height of the

pyramid.

Example 5

Explain why Thales’ method worked to find

the height of the pyramid?

Example 6

If a person 5 feet tall casts a 6-foot shadow

at the same time that a lamppost casts an

18-foot shadow, what is the height of the

lamppost?

Example 6

5

6=

𝑥

18

6𝑥 = 90

𝑥 = 15′

Side-Side-Side Similarity

SSS Similarity Theorem:

If the corresponding side lengths of two

triangles are proportional, then the two

triangles are similar.

Side-Angle-Side Similarity

SAS Similarity Theorem:

If two sides of one triangle are proportional to two

sides of another triangle and the included angles

are congruent, then the two triangles are similar.

Example 8

Are the triangles below similar? Why or why

not?

Example 8

Are the triangles below similar? Why or why not?

Yes, from SSS since

all the corresponding

sides have the

same ratio 2/3.

Yes, from SAS since the two pairs of corresponding

sides have the ratio 6/7 and they

have a congruent angle

between them.

Example 9

Use your new conjectures to find the missing

measure.

18

24

x

24

28

y

Example 9

Use your new conjectures to find the missing measure.

24

18=

𝑥

24

𝑥 = 32

24

18=28

𝑦

𝑦 = 21

18

24

x

24

28

y

Example 10

Find the value of x that makes ΔABC ~

ΔDEF.

Example 10

Find the value of x that makes ΔABC ~ ΔDEF.

4

12=𝑥 − 1

18

12𝑥 − 12 = 72

12𝑥 = 84

𝑥 = 7