Similarity

Similar Polygons

1

• Dilating a figure produces a figure that is the same ________________ as the

original figure, but a different _________________ . Like ____________ motions,

dilations preserve ______________ measures. Unlike rigid motions, dilations do

not preserve the __________________ of line segments. Instead, they produce a

figure with sides that are ____________________ to the sides of the pre-image.

So, the original figure and its __________________ image are

_________________ figures.

• A _________________ polygon is a polygon in which all sides have the

________________ length and all angles have the same ____________________.

Any two regular polygons of the same type – having the same number of sides -

are __________________ similar to each other.

• The symbol for similar is _______________ .

2

MAKING CONNECTIONS

• Dilating a figure produces a figure that is the same ________________ as the

original figure, but a different _________________ . Like ____________ motions,

dilations preserve ______________ measures. Unlike rigid motions, dilations do

not preserve the __________________ of line segments. Instead, they produce a

figure with sides that are ____________________ to the sides of the pre-image.

So, the original figure and its __________________ image are

_________________ figures.

• A _________________ polygon is a polygon in which all sides have the

________________ length and all angles have the same ____________________.

Any two regular polygons of the same type – having the same number of sides -

are __________________ similar to each other.

• The symbol for similar is _______________ .

3

MAKING CONNECTIONS

length

size rigidangle

shape

proportional

resultingsimilar

regular

same measure

always

~

Characteristics of Similar Polygons

1. Corresponding angles are _____________ .

2. Corresponding sides are _______________ .

Similarity Statement

ABC ~ DEF

EXS: Are the pairs of figures similar? EXPLAIN.

2)1)

3) 4)

Corresponding Sides : Corresponding Sides

EXS: Are the pairs of figures similar? EXPLAIN.

2)1)

3) 4)

Corresponding Sides : Corresponding Sides

Setting Up the Ratio of Similar Polygons

1) Find the Scale Factor.

2) Set the scale factor equal to a ratio containing the missing side to form a proportion.

3) Solve for the variable.

8/24/2018

EXS: Each pair of polygons is similar. Write a proportion to find each missing side. Solve for x.

2)1)

3)

EXS: Each pair of polygons is similar. Write a proportion to find each missing side. Solve for x.

2)1)

3)

Angle Relationships Side Relationships

EX 1: Solve for x and y.

~ABC SLT A

B C

S

L

T

x

5 cm

24 cm

10 cm

13 cm

y

EX 1: Solve for x and y.

~ABC SLT A

B C

S

L

T

x

5 cm

24 cm

10 cm

13 cm

y

A B

CD

6

x

E F

GH

18

27

EX 2: ABCD ~ EFGH. Solve for x.

A B

CD

6

x

E F

GH

18

27

EX 2: ABCD ~ EFGH. Solve for x.

EX 3: A tree cast a shadow 18 feet long. At the same time a person who is 6 feet tall cast a shadow 4 feet long. How tall is the tree?

EX 3: A tree cast a shadow 18 feet long. At the same time a person who is 6 feet tall cast a shadow 4 feet long. How tall is the tree?

tree's shadow tree's height

person's shadow person's height

Corresponding Sides : Corresponding Sides

Or

Perimeter : Perimeter

A : B8/24/2018

Setting Up the Ratio of Similar Polygons

Area : Area

A2 : B2

8/24/2018

Volume: Volume

A3 : B3

REMEMBER! The ratio of the perimeters of

two similar polygons equals the ratio of

any pair of corresponding sides.

A

C T

O

D G

6 4

10

y

EX 4: The ratio of the perimeters of CAT to

DOG is 3:2. Find the value of y.

A

C T

O

D G

6 4

10

y

EX 4: The ratio of the perimeters of CAT to

DOG is 3:2. Find the value of y.

12 cm 4 cm

Perimeter = 60 cmPerimeter = x

EX 5: Find the perimeter of the smaller triangle.

12 cm 4 cm

Perimeter = 60 cmPerimeter = x

EX 5: Find the perimeter of the smaller triangle.

U

W

V X

Z

Y

7.5

5

12

61. ~ ____

2.

3. ?

4. ?

5. 50 30 , ?

UVW

What is the scale factor of UVW to XYZ

What is VW

What is XZ

If m U and m Y what is m Z

Warm-Up

U

W

V X

Z

Y

7.5

5

12

61. ~ ____

2.

3. ?

4. ?

5. 50 30 , ?

UVW

What is the scale factor of UVW to XYZ

What is VW

What is XZ

If m U and m Y what is m Z

XYZ

5/6

10

9

100

Warm-Up

Similarity

Proving Triangles Similar

25

Shared angles are congruent in each triangle by the REFLEXIVE property.

∠𝑨 ≅ ∠𝑨

Shared angles are congruent in each triangle by the REFLEXIVE property.

𝐒𝐢𝐧𝐜𝐞 ∠𝑨 ≅ ∠𝑨,

t𝐡𝐞𝐧, ∠𝑪𝑨𝑩 ≅ ∠𝑫𝑨𝑬

VERTICAL ANGLES are the opposite angles created when two lines intersect one another. VERTICAL ANGLES ARE ALWAYS CONGRUENT.

∠𝟏 ≅ ∠𝟑 ∠𝟐 ≅ ∠𝟒

14

32

Before we start…let’s get a few things straight

A B

C

X Z

Y

INCLUDED ANGLEIt’s stuck in between!

• Two triangles are similar if all of their

corresponding angles have ___________

measures and all of their corresponding sides

have _______________________ lengths.

However, you do not need to know every one of

those angle measures and side lengths to

______________ that two triangles are

________________ . 30

KEY IDEAS

equal

proportional

prove

similar

• For instance, if two angles of one triangle are

__________________ to two angles of another triangle, then the

triangles are ___________________ . Also, if the three sides of

one triangle are _______________________ to the three sides

of another triangle, then the triangles are ___________________

. Finally, if two sides of one triangle are ____________________

to two sides of another triangle and the ____________________

angles of those sides are _____________________, then the

triangles are ____________ .31

KEY IDEAS (cont.)

congruentsimilar

proportionalsimilar

proportional

includedcongruent

similar

Angle-Angle Similarity (AA~)

Postulate

If two angles of one triangle

are congruent to two angles

of another triangle, then the

triangles are similar.

Side-Side-Side Similarity (SSS~)

Theorem

If the three sides of one

triangle are proportional in

length to the three sides of

another triangle, then the

triangles are similar.

Side-Angle-Side Similarity (SAS~)

Theorem

If two sides of one triangle have

lengths that are proportional to

two sides of another triangle

and the included angles of

those sides are congruent, then the triangles are similar.

Ex. Determine whether the triangles are similar. If so, tell which similarity test is used and complete the statement.

F

G

H

KL

M

43°

43°68°

68°

W

V

U

7

11X

Y

Z5

3

Prove that RST~ PSQ

R

S

T

P Q

12

4 5

15SS

reflexive

5

20

4

16

1

4

1

4

1. Two sides are proportional

2. Included angle is congruentSAS~

37

Using only the information given, can it be shown that CDE ~ FGE?

A) Yes, by the AA~ Postulate

B) Yes, by the SAS~ Theorem

C) Yes, by the SSS~ Theorem

D) No, not enough information is given.