# User manual | 6.4-6.5: Similarity Shortcuts

### 6.4-6.5: Similarity Shortcuts

Objectives:

1.

To discover and use shortcuts for determining that two triangles are similar

2.

To find missing measures in similar polygons

Assignment:

P. 384-387: 1-4, 7, 8,

10, 12, 14-17, 20, 30,

31, 32, 36, 41, 42

P. 391-395: 4, 6-8,

10-14, 33, 39, 40

Challenge Problems

O

BJECTIVE

1

You will be able to discover and use shortcuts for determining that two triangles are similar

### Warm-Up

Since they are polygons, what two things must be true about triangles if they are similar?

### Similar Polygons

Two polygons are similar polygons iff the corresponding angles are congruent and the corresponding sides are proportional.

N

Similarity Statement:

πΆπππ~ππ΄πΌπ

C

O

R

Z

C

∠πΆ ≅ ∠π ∠π ≅ ∠π΄

∠π ≅ ∠πΌ ∠π ≅ ∠π

M

πΆπ

ππ΄

=

ππ

π΄πΌ

=

ππ

R

=

πΌπ

ππΆ

ππ

A

I

A

M

I

Z

### Example 1

Triangles ABC and ADE are similar. Find the value of x.

D

9 cm

B

6 cm

E

x

C

8 cm

A

### Example 2

Are the triangles below similar?

8

3

4

37

ο°

6

53

ο°

10

5

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?

F

C

A

50

ο°

40

ο°

B

D

50

ο°

40

ο°

E

### Investigation 1

Step 1:

Draw

Δπ΄π΅πΆ where π∠π΄ and π∠π΅ equal sensible values of your choosing.

C

A

50

ο°

40

ο°

B

### Investigation 1

Step 1:

Draw

Δπ΄π΅πΆ where π∠π΄ and π∠π΅ equal sensible values of your choosing.

Step 2:

Draw

Δπ·πΈπΉ where π∠π· = π∠π΄ and π∠πΈ = π∠π΅ and π΄π΅ ≠ π·πΈ.

F

C

A

50

ο°

40

ο°

B

D

50

ο°

40

ο°

E

### Investigation 1

Now, are your triangles similar? What would you have to check to determine if they are similar?

F

C

A

50

ο°

40

ο°

B

D

50

ο°

40

ο°

E

### Angle-Angle 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.

### Investigation 3

Each group will be given one of the three candidates for similarity shortcuts. Each group member should start with a different triangle and complete the steps outlined for the investigation.

### Side-Side-Side Similarity Theorem

If the corresponding side lengths of two triangles are proportional, then the two triangles are similar.

### Side-Angle-Side 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 4

Are the triangles below similar? Why or why not?

### Objective 2

You will be able to find missing measures in similar polygons

### Indirect Measurement

Indirect measurement

involves measuring distances that cannot be easily measured directly.

This often involves using properties of similar triangles.

### 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 5

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 6

Explain why Thales’ method worked to find the height of the pyramid?

### Example 7

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?

### Investigation 3

What if you decide to indirectly measure a height on a day when there are no shadows?

The following GSP

Animation will help you discover an alternate method of indirect measurement using a mirror.

### Example 8

Your eye is 168 centimeters from the ground and you are 114 centimeters from the mirror.

The mirror is 570 centimeters from the flagpole. How tall is the flagpole?

### Example 9

Find the values of π₯ and π¦.

28

24

x y

24

18

### 6.4-6.5: Similarity Shortcuts

Objectives:

1.

To discover and use shortcuts for determining that two triangles are similar

2.

To find missing measures in similar polygons

Assignment:

P. 384-387: 1-4, 7, 8,

10, 12, 14-17, 20, 30,

31, 32, 36, 41, 42

P. 391-395: 4, 6-8,

10-14, 33, 39, 40

Challenge Problems