Five-Minute Check (over Lesson 7–2)

NGSSS

Then/Now

Postulate 7.1: Angle-Angle (AA) Similarity

Example 1: Use the AA Similarity Postulate

Theorems

Proof: Theorem 7.2

Example 2: Use the SSS and SAS Similarity Theorems

Example 3: Standardized Test Example

Theorem 7.4: Properties of Similarity

Example 4: Parts of Similar Triangles

Example 5: Real-World Example: Indirect Measurement

Concept Summary: Triangle Similarity

Over Lesson 7–2

A. A

B. B

A. Yes, corresponding angles are congruent and corresponding sides are proportional.

B. No, corresponding sides are not proportional.

Determine whether the triangles are similar.

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Over Lesson 7–2

A. A

B. B

C. C

D. D A B C D

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A. 5:3

B. 4:3

C. 3:2

D. 2:1

The quadrilaterals are similar. Find the scale factor of the larger quadrilateral to the smaller quadrilateral.

Over Lesson 7–2

A. A

B. B

C. C

D. D A B C D

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A. x = 5.5, y = 12.9

B. x = 8.5, y = 9.5

C. x = 5, y = 7.5

D. x = 9.5, y = 8.5

The triangles are similar.Find x and y.

Over Lesson 7–2

A. A

B. B

C. C

D. D A B C D

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A. 12 ft

B. 14 ft

C. 16 ft

D. 18 ft

__Two pentagons are similar with a scale factor of .The perimeter of the larger pentagon is 42 feet. What is the perimeter of the smaller pentagon?

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MA.912.G.4.6 Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles.

MA.912.G.4.8 Use coordinate geometry to prove properties of congruent, regular, and similar triangles.

Also addresses MA.912.G.2.3 and MA.912.G.4.4.

You used the AAS, SSS, and SAS Congruence Theorems to prove triangles congruent. (Lesson 4–4)

• Identify similar triangles using the AA Similarity Postulate and the SSS and SAS Similarity Theorems.

• Use similar triangles to solve problems.

Use the AA Similarity Postulate

A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Use the AA Similarity Postulate

Since mB = mD, B D

By the Triangle Sum Theorem, 42 + 58 + mA = 180, so mA = 80.

Since mE = 80, A E.

Answer: So, ΔABC ~ ΔDEC by the AA Similarity.

Use the AA Similarity Postulate

B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Use the AA Similarity Postulate

QXP NXM by the Vertical Angles Theorem.

Since QP || MN, Q N.

Answer: So, ΔQXP ~ ΔNXM by the AA Similarity.

A. A

B. B

C. C

D. D A B C D

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A. Yes; ΔABC ~ ΔFGH

B. Yes; ΔABC ~ ΔGFH

C. Yes; ΔABC ~ ΔHFG

D. No; the triangles are not similar.

A. Determine whether the triangles are similar. If so, write a similarity statement.

A. A

B. B

C. C

D. D A B C D

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A. Yes; ΔWVZ ~ ΔYVX

B. Yes; ΔWVZ ~ ΔXVY

C. Yes; ΔWVZ ~ ΔXYV

D. No; the triangles are not similar.

B. Determine whether the triangles are similar. If so, write a similarity statement.

Use the SSS and SAS Similarity Theorems

A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Answer: So, ΔABC ~ ΔDEC by the SSS Similarity Theorem.

Use the SSS and SAS Similarity Theorems

B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Answer: Since the lengths of the sides that include M are proportional, ΔMNP ~ ΔMRS by the SAS Similarity Theorem.

By the Reflexive Property, M M.

A. A

B. B

C. C

D. D A B C D

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A. ΔPQR ~ ΔSTR by SSS Similarity Theorem

B. ΔPQR ~ ΔSTR by SAS Similarity Theorem

C. ΔPQR ~ ΔSTR by AAA Similarity Theorem

D. The triangles are not similar.

A. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data.

A. A

B. B

C. C

D. D A B C D

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A. ΔAFE ~ ΔABC by SSS Similarity Theorem

B. ΔAFE ~ ΔACB by SSS Similarity Theorem

C. ΔAFE ~ ΔAFC by SSS Similarity Theorem

D. ΔAFE ~ ΔBCA by SSS Similarity Theorem

B. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data.

If ΔRST and ΔXYZ are two triangles such that

= which of the following would be sufficient

to prove that the triangles are similar?

A B

C R S D

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___RSXY

Read the Test Item

You are given that = and asked to identify which

additional information would be sufficient to prove that

ΔRST ~ ΔXYZ.

__23

___RSXY

__23

Solve the Test Item

Since = , you know that these two sides are

proportional at the scale factor of . Check each

answer choice until you find one that supplies sufficient

information to prove that ΔRST ~ ΔXYZ.

__23

___RSXY

__23

Choice A

If = , then you know that the other two sides are

proportional. You do not, however, know whether that

scale factor is as determined by . Therefore, this

is not sufficient information.

___RTXZ

___STYZ

___RSXY

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Choice B

If = = , then you know that all the sides are

proportional by the same scale factor, . This is

sufficient information by the SSS Similarity Theorem to

determine that the triangles are similar.

___RSXY

___RTXZ

___RTXZ

Answer: B

A. A

B. B

C. C

D. D A B C D

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Given ΔABC and ΔDEC, which of the following would be sufficient information to prove the triangles are similar?

A. =

B. mA = 2mD

C. =

D. =

___ACDC

___ACDC

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___BCDC

__45

___BCEC

Parts of Similar Triangles

ALGEBRA Given , RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10, find RQ and QT.

Parts of Similar Triangles

Substitution

Cross Products Property

Since

because they are alternate interior angles. By AA

Similarity, ΔRSQ ~ ΔTUQ. Using the definition of similar

polygons,

Parts of Similar Triangles

Answer: RQ = 8; QT = 20

Distributive Property

Subtract 8x and 30 from each side.

Divide each side by 2.

Now find RQ and QT.

A. A

B. B

C. C

D. D A B C D

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A. 2

B. 4

C. 12

D. 14

ALGEBRA Given AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find AC.

Indirect Measurement

SKYSCRAPERS Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 p.m. The length of the shadow was 2 feet. Then he measured the length of the Sears Tower’s shadow and it was 242 feet at the same time. What is the height of the Sears Tower?Understand Make a sketch of

the situation.

Indirect Measurement

Plan In shadow problems, you can assume that the angles formed by the Sun’s rays with any two objects are congruent and that the two objects form the sides of two right triangles. Since two pairs of angles are congruent, the right triangles are similar by the AA Similarity Postulate.

So the following proportion can be written.

Indirect Measurement

Solve Substitute the known values and let x be the height of the Sears Tower.

Substitution

Cross Products Property

Simplify.

Divide each side by 2.

Indirect Measurement

Answer: The Sears Tower is 1452 feet tall.

Check The shadow length of the Sears Tower is

or 121 times the shadow length of the light pole.

Check to see that the height of the Sears Tower

is 121 times the height of the light pole.

= 121

______2422

______145212

A. A

B. B

C. C

D. D A B C D

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A. 196 ft B. 39 ft

C. 441 ft D. 89 ft

LIGHTHOUSES On her tripalong the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina.At that particular time of day, Jennie measures her shadow to be 1 foot 6 inches in length and the length of the shadow of the lighthouse to be 53 feet6 inches. Jennie knows that her heightis 5 feet 6 inches. What is the height ofthe Cape Hatteras lighthouse to the nearest foot?