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.

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AOMC

C

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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 cm9 cm

xE

D

A

B

C

HINT: separate the diagram into 2 distinct triangles.

x=4

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?

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.

Answer in your notebook

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

In your notebook:

1) Find the missing value of the pyramid

2) 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?

15’

Example 7

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?

840

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 triangle pairs below similar? Why or why not?

Yes, corresponding sides are in the same proportion.

Yes, 2 corresponding sides are in proportion and the included angles are congruent.

Example 9

Use your new conjectures to find the missing measure.

18

24

x

24

28

y

x=32y=21

Example 10

Find the value of x that makes ΔABC ~ ΔDEF.

x=7