Untitled Lesson

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Topic 9 Lesson 3

I CAN…
use dilation and rigid motion to establish triangle similarity theorems.

Proving Triangles Similar

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Topic 9 Lesson 3

The measurements of two triangles are shown.

A. Are the triangles similar? Explain.

B. Construct Arguments Would any triangle with 40°- and 60°-angles be similar to

ABC? Explain.

EXPLORE & REASON

Fill in the Blank

1.) Are the triangles similar? Explain your answer?

Fill in the Blank

B.) Construct Arguments Would any triangle with $$40\degree$$ - and - angels be similar to $$\Delta ABC$$

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Topic 9 Lesson 3

ESSENTIAL QUESTION

How can you use the angles and sides of two triangles to determine whether they
are similar?

Establish the Angle-Angle Similarity (AA~) Theorem

If A

R and B

S, is

ABC

RST? Explain.

To show that the triangles are similar, determine whether there is a similarity transformation that maps
△ABC to △RST.

EXAMPLE 1

Determine the center of dilation and the scale factor that map △ABC to image △A′B′C′ such that
A′B′ = RS.

Let the scale factor k be .

Then, A′B′ = k • AB = RS.

CONCEPTUAL UNDERSTANDING

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Topic 9 Lesson 3

Establish the Angle-Angle Similarity (AA~) Theorem

If A

R and B

S, is

ABC

RST? Explain.

The dilation D(k, A) maps △ABC to △A′B′C′, and △A′B′C′ ≅ △RST by ASA, so there is a rigid motion that
maps △A′B′C′ to △RST. Thus, the composition is a similarity transformation that maps △ABC to △RST.
So, △ABC ∼ △RST.

EXAMPLE 1
CONCEPTUAL UNDERSTANDING

REASON
Think about what it means for figures to be similar. Are congruent triangles similar to the same triangle?

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Topic 9 Lesson 3

Establish the Angle-Angle Similarity (AA~) Theorem

Try It!

1.

If A is congruent to R, and C is congruent to T, how would you prove the triangles
are similar?

EXAMPLE 1

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Topic 9 Lesson 3

Angle-Angle Similarity (AA ) Theorem

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

PROOF: SEE EXERCISE 10.

If...

∠A ≅ ∠D and ∠B ≅ ∠E

Then... △ABC ∼ △DEF

THEOREM 7-1

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Topic 9 Lesson 3

Establish the Side-Side-Side Similarity (SSS ) Theorem

If , is there a similarity transformation that maps

PQR to

LMN?

Explain.

Dilate △PQR by scale factor to map △PQR to △P′Q′R′.

Because P′R′ = LN, P′Q′ = LM, and Q′R′ = MN, △P′Q′R′ ≅ △LMN by SSS.
By the definition of congruence, there is a rigid motion that maps △P′Q′R′ to △LMN.

So, △PQR was mapped to △LMN by a similarity transformation and △PQR is similar to △LMN.

EXAMPLE 2

STUDY TIP
Remember, two figures are similar if there is a similarity transformation between them.

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Topic 9 Lesson 3

Establish the Side-Side-Side Similarity (SSS ) Theorem

Try It!

2.

If and F

J, is there a similarity transformation that maps

DEF to

GHJ?

Explain.

EXAMPLE 2

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Topic 9 Lesson 3

Side-Side-Side Similarity (SSS ) Theorem

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

PROOF: SEE EXERCISE 20.

If...

Then... △ABC ∼ △DEF

THEOREM 7-2

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Topic 9 Lesson 3

Side-Angle-Side Similarity (SAS ) Theorem

If an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two
angles are proportional, then the triangles are similar.

PROOF: SEE EXERCISE 13.

If...

Then... △ABC ∼ △DEF

THEOREM 7-3

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Topic 9 Lesson 3

Verify Triangle Similarity

A. Are

ABC and

DEF similar?

The ratios are equal, so the corresponding side lengths are proportional. △ABC ∼ △DEF by SSS ∼.

EXAMPLE 3

Determine whether the ratios of the corresponding side lengths are equal.

COMMON ERROR
When setting up ratios to check if triangles are similar, be sure you place sides from the same triangle in the
same position in each ratio.

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Topic 9 Lesson 3

Verify Triangle Similarity

B. Are

PQS and

RQP similar?

Since the lengths of the sides that include ∠Q are proportional.

By SAS ∼, △PQS ∼ △RQP.

EXAMPLE 3

The two triangles share an included angle, ∠Q. Separate the triangles and see whether the lengths
of the corresponding sides are in proportion.

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Topic 9 Lesson 3

Verify Triangle Similarity

Try It!

3.

a. Is

ADE

ABD? Explain.

b. Is

ADE

BDC? Explain.

EXAMPLE 3

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Topic 9 Lesson 3

Find Lengths in Similar Triangles

What is MN?

The sides that include ∠JLK and ∠MLN are proportional.

By SAS ∼, △JLK ∼ △MLN.

Write a proportion using corresponding sides, and solve for MN.

EXAMPLE 4

STUDY TIP
The measures of the two pairs of corresponding legs of any two isosceles triangles
will always be proportional, even if the triangles are not similar.

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Topic 9 Lesson 3

Find Lengths in Similar Triangles

Try It!

4. a. In Example 4, if the measure of were 150 instead of 75, how would the value

of MN be different?

b. In Example 4, if the measure of were 20 instead of 10, how would the value

of MN be different?

EXAMPLE 4

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Topic 9 Lesson 3

Since ∠BAC ≅ ∠SRT and ∠ACB ≅ ∠RTS, you can apply the AA ∼ Theorem.

Solve Problems Involving Similar Triangles

Avery puts up a radio antenna tower in his yard. Ella tells him that their city has a law limiting towers to
50 ft in height. How can Avery use the lengths of his shadow and the shadow of the tower to show that
his tower is within the limit without directly measuring it?

EXAMPLE 5
APPLICATION

MODEL WITH MATHEMATICS
Think about how the situation is
modeled with triangles. What
information do you need to be
able to apply a similarity theorem?

The antenna tower is 48 ft high. Avery’s tower is within the 50-ft limit.

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Topic 9 Lesson 3

Solve Problems Involving Similar Triangles

Try It!

5.If the tower were 50 ft tall, how long would the shadow of the tower be?

EXAMPLE 5

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Topic 9 Lesson 3

Side-Angle-Side Similarity

If...

Then... △ABC ∼ △DEF

Side-Side-Side Similarity

If...

Then... △ABC ∼ △DEF

Angle-Angle Similarity

If...

∠A ≅ ∠D and ∠B ≅ ∠E

Then... △ABC ∼ △DEF

CONCEPT SUMMARY

Triangle Similarity Theorems

THEOREM 7-1

THEOREM 7-2

THEOREM 7-3

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Topic 9 Lesson 3

Do You UNDERSTAND?

1.

ESSENTIAL QUESTIONHow can you use the angles and sides of two triangles to
determine whether they are similar?

2.

Error Analysis Allie says

JKL

XYZ. What is Allie’s error?

3.

Make Sense and Persevere Is any
additional information needed to show

DEF

RST? Explain.

4.

Construct Arguments Explain how
you can use triangle similarity to show
that ABCD

WXYZ.

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Topic 9 Lesson 3

Do You KNOW HOW?

For Exercises 5 and 6, explain whether the two triangles are similar.

5.

6.

For Exercises 7 and 8, find the value of each variable such that the triangles are
similar.

7.

a

8.

b

9. When Esteban looks at the puddle, he

sees a reflection of the top of the cactus.
How tall is the cactus?