6.4-6.5: Similarity Shortcuts

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:

𝐢𝑂𝑅𝑁~𝑀𝐴𝐼𝑍

C

O

R

Z

C

∠𝐢 ≅ ∠𝑀 ∠𝑂 ≅ ∠𝐴

∠𝑅 ≅ ∠𝐼 ∠𝑁 ≅ ∠𝑍

M

𝐢𝑂

𝑀𝐴

=

𝑂𝑅

𝐴𝐼

=

𝑅𝑁

R

=

𝐼𝑍

𝑁𝐢

𝑍𝑀

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

Δ𝐴𝐡𝐢 where π‘š∠𝐴 and π‘š∠𝐡 equal sensible values of your choosing.

C

A

50

ο‚°

40

ο‚°

B

Investigation 1

Step 1:

Draw

Δ𝐴𝐡𝐢 where π‘š∠𝐴 and π‘š∠𝐡 equal sensible values of your choosing.

Step 2:

Draw

Δ𝐷𝐸𝐹 where π‘š∠𝐷 = π‘š∠𝐴 and π‘š∠𝐸 = π‘š∠𝐡 and 𝐴𝐡 ≠ 𝐷𝐸.

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.

Share your results and make a conjecture based on your findings.

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 𝑦.

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

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