2

5

?

2

2

2

2

2

2

5 5 5

Similar Shapes and ProportionsSimilar shapes have the same shape but not necessarily the same size. You can

use square tiles to model similar figures.

A rectangle made of square tiles measures 5 tiles long and 2 tiles wide.

Find the length of a similar rectangle that measures 6 tiles wide.

Using the square tiles, make a rectangle 5 tiles long and

2 tiles wide.

Add tiles to increase the width of the rectangle to 6 tiles.

Add tiles to also increase the length of

the rectangle.

Reflect1. Justify Reasoning If one dimension is changed, why does the other

dimension have to change to create a similar figure?

STEP 1

STEP 2

STEP 3

There are now sets of the original tiles along

the width of the rectangle because × = 6.

The width of the rectangle is times the width of the

original rectangle.

The width of the new rectangle is times the

width of the original. To keep the lengths of the sides

proportional, the length must also be times

the length of the original. The length of the similar

rectangle is × = 15.

The length of the similar rectangle is tiles.

L E S S O N

4.1Similar Shapes and Proportions

ESSENTIAL QUESTIONHow can you use ratios to determine if two figures are similar?

EXPLORE ACTIVITY 7.5.A

Proportionality—7.5.A Generalize the critical attributes of similarity, including ratios within and between similar shapes.

115Lesson 4.1

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12 in.8 in.

7 in.

34˚

106˚

40˚E

F

D

34˚

40˚

24 in.36 in.

21 in.

106˚R Q

S

Math On the Spotmy.hrw.com

A C

B

D F

E

Determining Whether Two Triangles Are Similar Similar shapes have the same shape, but not necessarily the same size.

Corresponding angles and corresponding sides of two or more similar

shapes are in the same relative position.

The symbol ~ means “is similar to.” In

the figure shown, sides and angles that

are the same color correspond to each

other. △ABC ~ △DEF.

? ?

??

? ?

Explain whether the triangles are similar.

Check that the

corresponding angles of

the triangles have equal

measures.

m∠E = m∠R = 106 °

m∠F = m∠S = 34 °

m∠D = m∠Q = 40 °

Check that the corresponding side lengths are proportional._

DE corresponds to _

QR . _

EF corresponds to _

RS .

_

DF corresponds to _

QS .

DE ___ QR = EF __ RS = DF ___ QS

7 __

21 = 8 __

24 = 12

__ 36

1 _

3 = 1 _

3 = 1 _

3 ✓

Since the measures of the corresponding angles are equal and the

corresponding sides are proportional, the triangles are similar.

EXAMPLE 1

STEP 1

STEP 2

Math TalkMathematical Processes

7.5.A

Are all equilateral triangles similar?

Explain.

∠E corresponds to ∠R

∠F corresponds to ∠S

∠D corresponds to ∠Q

Write ratios using the lengths of corresponding sides.

Substitute the lengths of the sides.

Simplify.

A side of a figure can be named by its endpoints with a bar above.

Similar Figures

In two similar figures:

• the measures of their corresponding angles are equal, and

• the lengths of their corresponding sides are proportional.

Unit 2116

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129̊A C

B

10 in.

7 in.4 in.

33̊ 18̊

129̊

E

F

D16 in.

28 in.

40 in.33̊

18̊

102̊

49̊29̊

22 ft

11 ft17 ft

E D

F99̊

9 ft 5 ft

11 ftAB

C

27̊ 54̊

101̊

86̊

86̊

48̊

125̊

O

N P

M Q

R

S

T

101̊

125̊

48̊

18 mm

15 mm6 mm

6 mm 10.8 mm

9 mm10 mm

10 mm

Math Trainer

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Animated Math

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Math On the Spot

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Explain whether the triangles are similar.

2. 3.

YOUR TURN

Determining Whether Two Four-Sided Figures Are Similar Shapes with four or more sides can also be similar if the corresponding angles

have equal measures and the corresponding side lengths are proportional.

Diana makes two sizes of earrings.

Explain whether the shapes of

the earrings are similar.

Check that the

corresponding angles have equal measures. The angle

measures are 48°, 86°, 101°, and 125° in both earrings.

Check that the corresponding side lengths are proportional._

MN corresponds to _

QR _

NO corresponds to _

RS _

OP corresponds to _

ST _

PM corresponds to _

TQ

MN ___ QR =  NO ___ RS =  OP ___ ST =  PM ___ TQ

10 __ 6

= 10 __ 6

= 18 ____

10.8 = 15 __

9

1. _

6 = 1. _

6 = 1. _

6 = 1. _

6

The measures of the corresponding angles are equal and the

corresponding sides are proportional, so the earrings are similar.

EXAMPLEXAMPLE 2

STEP 1

STEP 2

? ? ?

???

7.5.A

Write ratios using the lengths of corresponding sides.

Substitute the lengths of the sides.

Divide.

Note that each ratio in simplest form is 5 __ 3 .

117Lesson 4.1

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48 ft

18 ft

6 ft6 ft 150˚ 110˚

54˚46˚

8 ft

3 ft

6 ft 6 ft

110˚

150˚

46˚ 54˚

Math Trainer

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12 cm

18 cm18 cm70̊

40̊

70̊J

K

L

48 cm48 cm

28 cm

73̊73̊

Q

RP

34̊

21 ft

21 ft 21 ft

21 ft

90˚ 90˚

90˚ 90˚

14 ft

14 ft

14 ft 14 ft

90˚ 90˚

90˚ 90˚

Explain whether the shapes are similar.

4. rectangle ABCD with sides of 7 and 5 and rectangle MNOP with sides of

21 and 15

5.

YOUR TURN

Guided Practice

1. A rectangle made of square tiles measures 7 tiles long and 3 tiles

wide. What is the length of a similar rectangle whose width is 9 tiles? (Explore Activity)

Explain whether the shapes are similar. (Examples 1 and 2)

2. 3.

4. Describe how to use ratios to determine whether two shapes are similar.

ESSENTIAL QUESTION CHECK-IN??

Unit 2118

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Personal Math Trainer

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Interventionmy.hrw.com

Painting

MonaLisa

The DanceClass

The BlueVase

Artist

Leonardoda Vinci

Edgar Degas

PaulCézanne

OriginalSize (in.)

30 x 21

33 x 30

28 x 18

Name Class Date

Independent Practice4.1

Determine if each statement is true or false. Justify your answer.

5. All squares are similar. 6. All right triangles are similar.

Art For 7–10, use the table. Assume all angle measures are

equal to 90°.

7. Hugo has a small print of one of the paintings in the table.

It is similar in size to the original. The print measures

11 in. × 10 in. Of which painting is this a print? Explain.

8. A local artist painted a copy of Cezanne’s painting. It measures

88 in. × 74 in. Is the copy similar to the original? Explain.

9. A company made a poster of da Vinci’s painting. The poster is 5 feet long and

3.5 feet wide. Is the poster similar to the original Mona Lisa? Explain.

10. The same company made a poster of The Blue Vase. The poster is 36 inches

long and 26 inches wide. Is the poster similar to the original The Blue Vase?

Explain.

7.5.A

119Lesson 4.1

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Work Area

A B

C D

4 ft15 ft

12 ft

5 ft

T S

RQ

10 cm

5 cm5 cm

10 cm

5 cm5 cm

10 cm

10 cmW X

YZ4 cm4 cm

D C

BA

8 cm

8 cm

Problem Solving The figure shows a 12 ft by 15 ft garden divided into

four rectangular parts, each planted with a different vegetable. Explain

whether the rectangles in each pair are similar and why.

11. rectangle A and the original rectangle

12. the original rectangle and rectangle D

13. rectangle C and rectangle B

14. Analyze Relationships Which of these four-sided shapes are similar?

15. Communicate Mathematical Ideas Describe the two tests two polygons

must pass to be proven similar.

16. Make a Conjecture Using what you know of similar figures, explain

whether you believe all rectangles are similar. Give an example or a

counterexample.

FOCUS ON HIGHER ORDER THINKING

Unit 2120

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