Warm-UpWarm-Up

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

Similar PolygonsSimilar Polygons

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

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IZRN

AIOR

MACO

ZNIR

AOMC

C

OR

N

C

OR

NM

A

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Similarity Statement:Similarity Statement:

Corresponding Angles:Corresponding Angles:

Statement of Proportionality:Statement of Proportionality:

Example 1Example 1

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

6 cm

8 cm9 cm

xE

D

A

B

C

Example 2Example 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 Shortcuts6.4-6.5: Similarity Shortcuts

Objectives:

1. To find missing measures in similar polygons

2. To discover shortcuts for determining that two triangles are similar

Investigation 1Investigation 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 1Investigation 1

Step 1: Draw ΔABC where m<A and m<B equal sensible values of your choosing.

4050

C

A B

Investigation 1Investigation 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 1Investigation 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

Angle-Angle SimilarityAngle-Angle Similarity

AA Similarity AA Similarity PostulatePostulate

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Example 3Example 3

Determine whether the triangles are similar. Write a similarity statement for each set of similar figures.

ThalesThales

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 4Example 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 5Example 5

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

Example 6Example 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?

Investigation 2Investigation 2

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 7Example 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?

Investigation 3Investigation 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 SimilaritySide-Side-Side Similarity

SSS Similarity Theorem:SSS Similarity Theorem:

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

Side-Angle-Side SimilaritySide-Angle-Side Similarity

SAS Similarity Theorem: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 8Example 8

Are the triangles below similar? Why or why not?

Example 9Example 9

Use your new conjectures to find the missing measure.

18

24

x

24

28

y

Example 10Example 10

Find the value of x that makes ΔABC ~ ΔDEF.

AssignmentAssignment

• 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