Geometry Definitions, Postulates, and Theorems

Chapter 6: Similarity

Section 6.1: Ratios, Proportions, and the Geometric Mean

Standards: Prepare for 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.

Ratio – Two quantities that are measured in the same units. They are usually

expressed in simplest form. Two ratios that have the same simplified form are called equivalent ratios.

Ratios can be written three ways: b

a, ba : , or btoa .

The denominator, b, cannot be zero. Ex. Simplify the following ratios.

a) 6m:64m b) 20in.

5ft. c)

12oz.

4lb.

Ex. Find the ratio of the width to the length of the rectangle. Ex. The perimeter of a rectangle is 108 feet. The ratio of the length to the width is 1:8 . Find

the length and the width. Ex. The perimeter of the isosceles triangle is 56 in. The ratio of LM:MN is 5:4. Find the lengths of the sides and the base of the triangle. Ex. The measures of the angles of a triangle are in the extended ratio of 3:4:8. Find the measures of the angles of the triangle. (over)

160cm

6m

L

M N

Proportion – An equation that states that two ratios are equal.

d

c

b

a

Extremes – The values on the top left and bottom right in a proportion. Means – The values on the top right and bottom left in a proportion. Cross Product Property – In a proportion,

the product of the extremes equals

the product of the means. Ex. Solve the proportions.

a) x

6

14

9 b)

104

5 xx

� c)

147

3 yy

�

EX. The ratio of BC:DC is 2:9. Find x .

Geometric Mean – IF a & b are positive numbers, THEN the positive number, x,

represents the geometric mean in the proportion: b

x

x

a

Ex. Find the geometric mean of the two numbers. a) 4 and 9 b) 5 and 25 c) 7 and 20

IF d

c

b

a ,

THEN ad = bc

A B

D C 27

x

Section 6.2: Use Proportions to Solve Geometry Problems

Standards: 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.

Additional Properties of Proportions

IF d

c

b

a , THEN

d

b

c

a IF

d

c

b

a , THEN

d

dc

b

ba �

�

Reciprocal Property – If two ratios are equal, then their reciprocals are also equal.

IF d

c

b

a , THEN

c

d

a

b

Scale Drawing – A drawing that is the same shape as the object it represents.

Scale – A ratio that describes how the dimensions in the drawing are related

to the actual dimensions of the object. Ex. Complete the statements:

a) If ,4

3

b

a then

3

a b) If ,

8

5

q

p then

�q

qp

Ex. Decide whether the statement is true or false.

a) If ,5

4

n

m then

5

4

m

n b) If ,

4

3

g

d then

4

1�

�g

gd

Ex. Use the diagram and the given information to find the unknown length.,

a) Given LP

LQ

MN

MQ , find LQ . b) Given DE

BC

AE

AC , find DE .

M

L Q

N

P

15

5

6

A

C B

E D

4

10

15

P S

R

X

Y

Z

6

6

8

8

20

T

Section 6.3: Use Similar Polygons

Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. 11.0 Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.

Similar Polygons – Two or more polygons where: 1. The corresponding angles are congruent and 2. Corresponding sides are proportional. *Similar is not the same as congruent. (shrunk vs. blown up) Ex. Given PQRJKL '' ~ . List all pairs of congruent angles and write the ratios of the

corresponding side lengths in a statement of proportionality. Scale Factor – The ratio of the corresponding sides of two similar polygons. Ex. Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor.

***Theorem 6.1 – Perimeters of Similar Polygons If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

Ex. In the diagram to the right, ABCDE~FGHJK

a) Find the scale factor of ABCDE to FGHJK b) Find the value of x.

c) Find the perimeter of ABCDE.

Corresponding Lengths in Similar Polygons – If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons.

Ex. In the diagram, XPZTPR '' ~ . Find the length of the altitude PS .

A

B C

D E

F G

H

ABCD~EFGH

L

K

M

E F

D

16 9

20

12

15

12

A B

C

D E

x

10 F G

K

H

J

15

18

15

12

9

Section 6.4: Prove Triangles Similar by AA

Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.

Angle-Angle (AA) Similarity Postulate

IF two angles of one triangle are congruent to two angles of another triangle, THEN the two triangles are similar.

Ex. Determine whether the triangles are similar. If they are, write a similarity statement.

a) b) Ex. In the diagram to the right, LMN' ~ PQN' .

a. Write the proportionality statement. b. Find �Mm

�Pm

c. Find MN

Ex. Use a proportion to find the value of x in the diagram. (over)

L M

Q P

N 1060

33

20

30

360

J

K

L

Y

J

X

Z

450

A

B C

440 E

D F

520

520 V

W

X

Y

Z

4

6 5

x

Shared angle

Section 6.4

FactorScaleIIA

IA

IItorAngle Bi

ItorBiAngle

IIMedian

IMedian

IIPerimeter

IPerimeter

IISide

ISide

ltitude

ltitude

sec

sec

FactorScaleIID

ID

IIPerimeter

IPerimeter

IISide

ISide

iagonal

iagonal

Section 6.5: Prove Triangles Similar by SSS and SAS

Standards: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.

***Theorem 6.2 – Side-Side-Side (SSS) Similarity Theorem

IF the corresponding side lengths of two triangles are proportional, THEN the triangles are similar.

*no information about angles is given

***Theorem 6.3 – Side-Angle-Side (SAS) Similarity Theorem

IF an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional,

THEN the triangles are similar.

*one pair of angles are congruent and adjacent sides are proportional Ex. Which of the following three triangles are similar? Write a similarity statement. (over)

A

C B

10

14

12 5 6

7

D

F E

1000

4

8

A

B C

1000

3

6

D

E F

30

10

25

R

Q S

6

18

15

K

M

L 7

3

5 P

O

N

Reduce

Biggest

SmallestMedium

Ex. Are these two triangles similar? Ex. Are these two triangles similar? Ex. Use the diagram below to complete the statements. 1. �DGEm

2. �GEDm

3. �EDGm

4. FD 5. GD

6. EG

7. Name the three pairs of s'' that are similar ~.

12

14

16

18 10

X

Y Z 4

6

A

B C

9

6

2 A B C

D

E F G

450

450 400

500

2

13

10

All sides are proportional so the triangles are similar.

Sides are not proportional, so triangles aren't similar.

40 45

50 45

45; alt. int. angles r congruent

50; alt. int. angles r congruent

40 + 45 = 95; vertical angles

Section 6.6: Proportions and Similar Triangles

Standards: 4.0 Students prove basic theorems involving congruence and similarity. 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal.

***Theorem 6.4 – Triangle Proportionality Theorem

IF a line parallel to one side of a triangle intersects the other two sides, THEN it divides the two sides proportionally.

***Theorem 6.5 – Converse of the Triangle Proportionality Theorem

IF a line divides two sides of a triangle proportionally, THEN it is parallel to the third side.

***Theorem 6.6 – IF three parallel lines intersect two transversals, THEN they divide the transversals proportionally.

***Theorem 6.7 – IF a ray bisects an angle of a triangle,

THEN it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

Ex. (over)

A

B C

E D

x

16

3

18

P

Q

R T

S

9

9.75

24 26

A

D C B

p

3

4

6

A

B

C

D

E

F

EX. True or False

1. MK

DH

BM

BD

2. HK

DM

DH

BD

3. MB

DB

KM

HD

4. BK

BM

BH

BD

EX. Find x and y.

H

K

D

M B

10

15

18

9

x

y

Section 6.7: Perform Similarity Transformations

Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. 11.0 Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.

Dilation – A transformation that stretches or shrinks a figure to create a similar figure.

Center of Dilation – In a dilation, a figure is enlarged or reduced with respect to a

fixed point.

Scale Factor of a Dilation – The ratio of a side length of the image to the corresponding side length of the original figure.

Coordinate Notation for a Dilation – You can describe a dilation with respect to the

origin with the notation (x, y) Æ (kx, ky), where k is the scale factor.

Reduction – If 0 < k < 1, the dilation is a reduction.

Enlargement – If k > 1, the dilation is an enlargement.

Ex. Draw a dilation of the polygon with the given vertices using the given scale factor k.

2);1,3(),2,2(),1,1( � kCBA

Ex. Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find its scale factor.

y

x

y

x

A B