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5-5

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5-5

Similar Figures and Proportions

Course 2

Warm Up

Problem of the Day

Lesson Presentation

5-5

Similar Figures and Proportions

138 = 144; not equal

Course 2

Warm Up

Find the cross products, then tell whether the ratios are equal.

16

6

40

15

,

1.

240 = 240; equal

3

8

18

46

,

2.

8

9

24

27

,

3.

216 = 216; equal

28

12

42

18

,

4.

504 = 504; equal

5-5

Similar Figures and Proportions

Course 2

Problem of the Day

Every 8th telephone pole along a road has a red band painted on it. Every 14th pole has an emergency call phone on it. If pole 1 has neither, what is the number of the first pole with both a red band and a call phone?

pole 56

5-5

Similar Figures and Proportions

Course 2

Learn to use ratios to determine if two figures are similar.

5-5

Similar Figures and Proportions

Course 2

Insert Lesson Title Here

Vocabulary

similar

corresponding sides

corresponding angles

5-5

Similar Figures and Proportions

Course 2

Octahedral fluorite is a crystal found in nature. It grows in the shape of an octahedron, which is a solid figure with eight triangular faces. The triangles in different-sized fluorite crystals are similar figures. Similarfigures have the same shape but not necessarily the same size.

5-5

Similar Figures and Proportions

Corresponding

sides

E

B

D

F

A

C

Corresponding angles

Course 2

Matching sides of two or more polygons are called correspondingsides, and matching angles are called corresponding angles.

5-5

Similar Figures and Proportions

Course 2

To find out if triangles are similar, determine whether the ratios of the lengths of their corresponding sides are proportional. If the ratios are proportional, then the corresponding angles must have equal measures.

5-5

Similar Figures and Proportions

Reading Math

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

AB

Without the bar, the letters indicate the length of the side.

Course 2

5-5

Similar Figures and Proportions

AB corresponds to DE.

AC corresponds to DF.

BC corresponds to EF.

?

?

?

?

?

?

=

=

=

=

=

=

Course 2

Additional Example 1: Determining Whether Two Triangles Are Similar

Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar.

E

16 in

10 in

A

C

28 in

D

4 in

7 in

40 in

F

B

AB

DE

BC

EF

AC

DF

Write ratios using the corresponding sides.

4

16

10

40

7

28

Substitute the length of the sides.

1

4

1

4

1

4

Simplify each ratio.

Since the ratios of the corresponding sides are equivalent, the triangles are similar.

5-5

Similar Figures and Proportions

AB corresponds to DE.

AC corresponds to DF.

BC corresponds to EF.

?

?

?

?

?

?

=

=

=

=

=

=

Course 2

Try This: Example 1

Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar.

E

9 in

9 in

A

C

21 in

D

3 in

7 in

27 in

F

B

AB

DE

BC

EF

AC

DF

Write ratios using the corresponding sides.

3

9

9

27

7

21

Substitute the length of the sides.

1

3

1

3

1

3

Simplify each ratio.

Since the ratios of the corresponding sides are equivalent, the triangles are similar.

5-5

Similar Figures and Proportions

Course 2

In figures with four or more sides, it is possible

for the corresponding side lengths to be

proportional and the figures to have different

shapes. To find out if these figures are similar,

first check that their corresponding angles have

equal measures.

8 m

10 m

4 m

4 m

5 m

5 m

8 m

10 m

10 m

8 m

5 m

4 m

=

5-5

Similar Figures and Proportions

Course 2

Additional Example 2: Determining Whether Two Four-Sided Figures are Similar

Use the properties of similarity to determine whether the figures are similar.

The corresponding angles of the

figures have equal measure.

Write each set of

corresponding sides as a ratio.

5-5

Similar Figures and Proportions

MN corresponds to QR.

NO corresponds to RS.

OP corresponds to ST.

MP corresponds to QT.

Course 2

Additional Example 2 Continued

MN

QR

NO

RS

OP

ST

MP

QT

5-5

Similar Figures and Proportions

?

?

?

?

?

?

OP

ST

MN

QR

NO

RS

MP

QT

=

=

=

=

=

=

8

12

6

9

4

6

10

15

?

?

?

2

3

2

3

2

3

2

3

=

=

=

Course 2

Additional Example 2 Continued

Determine whether the ratios of the lengths of the corresponding sides are proportional.

Write ratios using corresponding

sides.

Substitute the length of the

sides.

Simplify each ratio.

Since the ratios of the corresponding sides are equivalent, the figures are similar.

5-5

Similar Figures and Proportions

M

P

100 m

80°

65°

60 m

47.5 m

125°

90°

O

80 m

N

Q

T

400 m

80°

65°

240 m

190 m

125°

90°

S

R

320 m

Course 2

Try This: Example 2

Use the properties of similarity to determine whether the figures are similar.

The corresponding angles of the

figures have equal measure.

Write each set of

corresponding sides as a ratio.

5-5

Similar Figures and Proportions

M

P

100 m

80°

65°

MN corresponds to QR.

60 m

47.5 m

125°

90°

O

NO corresponds to RS.

80 m

N

Q

T

400 m

OP corresponds to ST.

80°

65°

MP corresponds to QT.

240 m

190 m

125°

90°

S

R

320 m

Course 2

Try This: Example 2 Continued

MN

QR

NO

RS

OP

ST

MP

QT

5-5

Similar Figures and Proportions

M

P

100 m

80°

65°

?

?

?

?

?

?

OP

ST

MN

QR

NO

RS

MP

QT

60 m

=

=

=

47.5 m

=

=

=

125°

90°

O

80 m

N

80

320

60

240

47.5

190

100

400

Q

T

400 m

80°

65°

?

?

?

1

4

1

4

1

4

1

4

=

=

=

240 m

190 m

125°

90°

S

R

320 m

Course 2

Try This: Example 2 Continued

Determine whether the ratios of the lengths of the corresponding sides are proportional.

Write ratios using corresponding

sides.

Substitute the length of the

sides.

Simplify each ratio.

Since the ratios of the corresponding sides are equivalent, the figures are similar.

5-5

Similar Figures and Proportions

NO corresponds to QR;

PN corresponds to SQ; similar

PO corresponds to SR;

Course 2

Insert Lesson Title Here

Lesson Quiz: Part 1

1. Identify the corresponding sides in the pair of triangles, and use ratios to determine whether the triangles are similar.

5-5

Similar Figures and Proportions

Course 2

Insert Lesson Title Here

Lesson Quiz

2. Use properties of similarity to determine whether the figures are similar.

not similar