GEOMETRY Unit 6

GEOMETRY6Unit

HONORS GEOMETRY

Unit Unit Unit Unit 6666 –––– Similar TrianglesSimilar TrianglesSimilar TrianglesSimilar Triangles: : : : Sample Unit OutlineSample Unit OutlineSample Unit OutlineSample Unit Outline TOPIC HOMEWORK

DAY 1 Ratios and Proportions HW #1

DAY 2 Similar Figures;

Using Proportions to Solve for Missing Sides HW #2

DAY 3 More Practice with Finding Missing Sides

DAY 4 Quiz 6-1 None

DAY 5 Proving Triangles are Similar: SSS, SAS, and AA HW #3

DAY 6

HW #4

DAY 7 Quiz 6-2 None

DAY 8 Parallel Lines & Proportional Parts

HW #5 DAY 9 Parts of Similar Triangles

DAY 10 Unit 6 Review Study

for Test

DAY 11 UNIT 6 TEST None

Main Ideas/Questions Notes/Examples

What is a RATIO?

• A __________________________ of _________________________ quantities.

• Ways to represent a ratio: ________________; ____________; ___________

• Ratios can be ______________________!

Example: A music store has 40 trumpets, 39 clarinets, 24 violins, 51 flutes, and 16 trombones in stock. Write each ratio in simplest form.

1. trumpets to violins 2. flutes to clarinets

3. trombones to trumpets 4. violins to total instruments

EXTENDED RATIOS

• A _________________________ of _________________________ quantities.

• Extended ratios are written as __________________________.

USING EXTENDED RATIOS FOR ANGLE AND SIDE MEASURES:

5. The ratio of two complementary

angles is 3:7. Find the measures ofboth angles.

6. The ratio of two supplementary

angles is 4:1. Find the measures ofboth angles.

7. The ratio of the measures of the

angles in a triangle is 4:7:9. Findthe measures of the angles.


angles in a triangle is 11:2:5. Findthe measure of the largest angle.

_________________________________________________

_________________________________

_

__________________________________

___________________________________________________

________________

x y

Section 6.1 Ratios & Proportions

1


sides of a triangle is 2:8:9. If the

perimeter of the triangle is 76inches, find the length of each

side.


sides of a triangle is 10:15:6. If the

perimeter of the triangle is 217meters, find the length of the

shortest side.

What is a PROPORTION?

• An ______________________ that states two ________________ are equal.

• A proportion is written as __________________________

• Cross Product Property: For any proportion, _________________________

Directions: Solve each proportion using the Cross Product Property

11. 4 2

7x= 12.

19

10 12

x=

13. 1 13

6 19

x −= 14.

5 19

17 4x=

+

15. 10 20

2 9 9x=

−16.

12 3 4

18 15

x +=

17. 20 11

3 18

x x− −= 18.

6 7

16 3 3x x=

+ +

19. 5 5

1 27

x

x

+=

−20.

2 5 7

6 6

x

x

+=

−

2

Name: _____________________________________ Unit 6: Similar Triangles

Date: _________________________ Per: _________ Homework 1: Ratio & Proportion

Directions: Write the ratio in simplest form.

1. 28 elementary schools to 16 middle schools 2. 30 treadmills to 36 elliptical machines

3. 18 buses to 66 cars 4. 180 red marbles to 145 blue marbles

5. The hockey team played 82 regular-season

games last year. If they won 44 games, what

is the ratio of wins to losses?

6. In the word FLASHLIGHT what is the ratio of

vowels to total letters?

Directions: Use the given ratios to solve each problem.

7. The ratio of the measures of two

complementary angles is 7:8. What is the

measure of the smaller angle?

8. The ratio of the measures of the three angles

in a triangle is 2:9:4. Find the measures of the

angles.

9. The ratio of the measures of the three angles

in a triangle is 10:3:7. Find the measure of the

largest angle.

10. The ratio of the measure of the vertex angle

to the base angle of an isosceles triangle is

8:5. Find the measure of the vertex angle.

11. The ratio of the measures of the sides of a

triangle is 21:8:14. If the perimeter of the

triangle is 215 feet, find the length of each

side.

12. The ratio of the measures of the sides of a

triangle is 4:7:5. If the perimeter of the

triangle is 128 yards, find the length of the

longest side.

** This is a 2-page document! **

3

Directions: Solve each proportion.

13. 9

16 12

x= 14.

3 12

18 9

x −=

15. 7 18

11 1x=

+16.

3 4 9

14 10

x −=

17. 17 10

15 2 2x=

−18.

16 3

6 5

x

x

−=

+

19. 6 12

19 2 2

x

x

−=

−20.

9 2 9

15 10

x x− −=

21. 9 56

3 4

x

x

−=

+22.

7 2 1

1 36

x

x

−=

+

© G

ina

Wils

on

(A

ll Th

ing

s A

lge

bra

®,

LLC

)

4


SIMILAR POLYGONS

• Polygons with the same ______________ but different __________.

• Polygons are similar if:

(1) __________________________________________________________________

(2) __________________________________________________________________

• The ratio of corresponding sides is called the ____________ ______________.

• If polygons are similar, then their ____________________are also proportional.

SCALE FACTOR

(order matters!)

What is the scale factor

of ABC to DEF?

What is the scale factor

of DEF to ABC?

What is the ratio of the

perimeter of DEF to ABC?

Similarity Statements

Symbol for Similar:

A valid similarity statement must match corresponding angles and sides!

Write a similarity statement for the triangles above:

_____________________________________

Directions: List all congruent angles and write a proportion that relates the

corresponding sides.

1. JKL ~ PMN

________________

_________________________________

_

__________________________________

________________

x y

A

B

C

8 10

14 E

F

D

21

15 12

Angles

Sides J

K

L

M N

P

Section 6.2 Similar Figures

5

2. XYZ ~ RYS

3. JKLM ~ QRSP

FINDING SIDE LENGTHS

(using proportions!)

4. If the figures below are similarwith a scale factor of 2:3, find thevalue of x.

5. If the figures below are similarwith a scale factor of 6:5, find thevalue of x.

6. If ABC ~ DEF, find the valueof x.

7. If PQRS ~ WXYZ, find the value

of x.

Angles

Sides

X

Y

Z

R

S

Angles

Sides P Q

R S

J

K L

M

42 x

x

28

25

20

E D

F

x

28

A

B C 75

x P

Q

R

S

W Z

Y X

32

40

6

8. If DEF ~ HJG, find the values of x and y.

9. If JKL ~ NML, find the values of x and y.

10. If AGM ~ KXD, find the value of x.

11. If TLY ~ CHK, find LY.

12. If BNV ~ BRG, BR = 4x + 7, and BG = 18, find the value of x.

X

D

K

x + 3

x + 6

G

M

A

32

36

10

x + 5 K

C

H 25

4x – 1

T

Y

L

10 12

8 B

N

R

G V

J

K

L

M

N x

14

40 16

y

45

D

E

F

9 14

12

G

H

J

15

x

y

7

USE THE SIMILARITY RELATIONSHIP TO FIND THE INDICATED VALUE. 1. QRS ~ TUV; find x 2. ABC ~ ADE; find x

3. MNP ~ QRP; find PR 4. BCD ~ FGE; find FE

5. KLMN ~ PQRS; find x 6. RST ~ YSZ; find YZ

Q

R

S

T

U

V24

54

36

x + 5

24

28x + 8

3x – 9

M

N

P

Q

R

A

B

C

D

E

20

8

15 2x + 3

39

42B

C

D

4x + 2 5x – 2

E

F

G

PQ

R S

48 60

K

L M

N

4x + 4

7x – 9

R

S

T

Y Z2x + 2

3x – 7

5

40

SIMILAR FIGURESSIMILAR FIGURESSIMILAR FIGURESSIMILAR FIGURES more practice with

8

7. CDE ~ FGE; find CE 8. KMZ ~ KDH; find x

9. AGN ~ FLS; find x 10. CSE ~ YJE; find EY

11. If KLM ~ PQR with a scale factor of 3:5,

find the perimeter of PQR.12. If TSR ~ TFE, find the perimeter of TFE.

x – 2

10

18

x + 1

C

S

E

J

Y

6x + 3

8x – 1

21

17

C

D

E

F

G

K

L

M

P

Q

R

6

12

15

K M D

H

Z

32

x + 8 21

28

A

G

N

SF

L

x + 7

246

x

T

F E

SR

40

25

22

54

9

Name: ________________________________________ Unit 6: Similar Triangles

Date: ___________________________ Per: _________ Homework 2: Similar Figures

Directions: List all congruent angles and write a proportion that relates the corresponding sides.

1. FGH ~ JKH

Directions: The pairs of polygons below are similar. Give the scale factor of figure A to figure B.

2. 3.

4. If the scale factor of Figure A to Figure B is4:5, find the value of x.

5. If the scale factor of Figure A to Figure B is7:2, find the perimeter of Figure A.

6. If ABC ~ DEC, find the values of x and y.

7. If JKL ~ NMP, find the value of x. 8. If DGH ~ DEF, find the value of x.


F

G

H

J

K Angles Sides

4

16 2

8

A B

A

10

15

18 B 4 6

7.2

x

A

15 B

A B 9 9

5

12

6x

10 21

28

A

BC

D

E

D

E

F

G

H

91

52x + 3

2x – 1

J

K

L9x + 1

49

M

NP x + 5

14

10

9. If XYZ ~ RST, find RS. 10. If ABC ~ EDC, find AC.

11. If JKL ~ MKN, find the value of x. 12. If BCD ~ GEF, find BD.

13. If ∆PQR ~ ∆SQT, find the value of x. 14. If CDE ~ GDF, find ED.

R

S

T3x + 2

40

X

Y

Z60

5x – 3

A

B

C

D

E

56

323x – 5

5x – 5

x + 1

3x – 2

1220

J

K

LM

N

B

C

D

x + 4

2x – 7

E

F

G

57

51

Q

P R

S T

21

8

x + 5x – 9 x + 3

4

3x + 1

15 C

D

E

F

G

11

Triangle Similarity

3

4

5

P

Q

R

S

T

U

6 10

8A

B

C75°31°

D

E

F75° 31°

Determine if the examples below are similar by AA~. If yes, write

a similarity statement.

1

2

V

W

X

Y

Z

AA~Angle-Angle Similarity

If two corresponding angles are congruent, then

the triangles are similar.

Determine if the examples below are similar by SSS~. If yes, write


3

4

SSS~Side-Side-Side Similarity

If all corresponding sides are proportional, then the

triangles are similar.

SAS~Side-Angle-Side Similarity

If two corresponding sides are proportional and the included

angles are congruent, then the triangles are similar.

Determine if the examples below are similar by SAS~. If yes, write


5

6

J

K

L

M

N

P

68

1520

12

15

A

B

C

21

25D

E

C

D

E

F

G

H

30 24

35 28

3225

12

42

30

31.5

20.25


AA~ (Angle-Angle Similarity)

Directions: Determine whether the triangles are similar by Angle-Angle

Similarity. If yes, write a similarity statement.

1. 2.

3.

4.

5. 6.

SSS~ (Side-Side-Side Similarity)

Directions: Determine whether the triangles are similar by Side-Side-Side


7. 8.

C

D

E F

G L

Q

P

M

N

39°

S

Y

E

51° H

C

W

68° 54°

A

B

D

68°

63°

E

F

G

67°

25°

M

N

L B

C

91° P

Q

X

Y

Z

D

S E

14

21

K

N R

9

13.5

.5

15

8

20

U

V

W X Y6

Section 6.3 Proving Triangles are Similar

13

4

9. 10.

11. 12.

SAS~ (Side-Angle-Side Similarity)

Directions: Determine whether the triangles are similar by Side-Angle-Side


13. 14.

15. 16.

17. 18.

S

T

M 16

12 5

5 18 12

43.2 15

54

P

A

R

T

Y

8

12 9

24

16C

A

B

D E

6

D J

Z 7

10

9

21.6 W

R

V

5

13

X

Y Z

R S

E

F G

48

12

24

6 40

25 44

27.5 J

F

P

S

Y

36 22

54 33

C

M P Q

R

50

45 E

F

G 35

31.5

H

J

K

15

14 30

28

S

W

Z

Q

R

30 27

G B

L

X 9

Y7

14

ARE WE SIMILAR ?Directions: Determine whether the triangles are similar. If similar, state how (AA~, SSS~,

or SAS~), and write a similarity statement.

3

5

E

F

G

9

15

R

S

T

22

28

35

W

V

U

16

2025

L

M

N

42°53°

85°A B

C

D

E

18

6

155

1216

P

Q

R

X

Y

J

K

L

M

N

30

3524

20

F

T

D

S

E

61°

X

Y

Z

34°

G

H

J

84

52

63

39

Q

R

S

T U

1 2

3 4

5 6

7 8

15

62°

93°

31°

F

G

H

P

Q

YT

Z

XS

64

54

60 45P

L E

A

R

54

49.540.5

K

N

B

24

18

22

D

F

P

36

40

45

90R

S

T

G

H

V

U

T

C

D

24

1532

20

L

M

N

O

P

42

42

35

S

T

W

30

30

25

D

E

A

G

H

I

BC

44

46.75

H

L

V

9 10

1 1 12

13 14

15 16

17 18

16

Directions: Determine whether the triangles are similar by AA~, SSS~, SAS~, or not similar.

If the triangles are similar, write a valid similarity statement.

1. 2.

3. 4.

5. 6.

7. 8.

Name: ______________________________________ Unit 6: Similar Triangles

Date: __________________________ Per: _________ Homework 3: Proving Triangles are Similar

55

44

37.4 S

T

R

N

17 20

25

P

Q

F

E

G H

J


X

Y

Z

T 28 8

30

7

W

H

E

S 29° 45°

A

N

F 29°

106°

Q

R

N

M

L

56 70

48

60

C D

E

96

64

80 L

M

N

54

45

36

J

K

L

M

N

21

6

10

4

A B C

D

E

17

9. 10.

11. 12.

13. 14. Given: 42 ;m QPT = PR bisects QPT

15. 16. M

B R

L

P

Z

K

L

J

G

H

87

72

95

68

3514

12

3025

35

P

Q

R S

T

A

B

C

D E

16

21 32

26

S

T

U

V

W

72°

72°

K

D

H 38°

15

20

38°

A

B

D 21

28

8

15

30.6 27

J K

L

M

N

P

Q

R S

T

58°

102°

18


TRIANGLE

PROPORTIONALITY

Theorem

Triangle Proportionality Theorem:

If a line is parallel to one side of a triangle and

intersects the other two sides, then it divides the

sides into segments of proportional lengths.

If _____________________, then __________________.

Converse of the Triangle Proportionality Theorem:

If _____________________, then __________________.

EXAMPLES

Directions: Find the value of x.

1. 2.

3. 4.

5. 6.

B

D E

A C

12

14

15 x

x 18

56

21

6 x – 1

21 3x + 1

x + 7

30

15

25

27 6

7

x

x

36

45

55

Section 6.4 Parallel Lines & Proportional Parts

19

Directions: Determine if MN is parallel to .JL7. 8. 9.

PARALLEL LINES

& Proportional Parts

If three or more parallel lines intersect two transversals, then theycut off the transversals proportionally.

EXAMPLES

Directions: Find the value of x.

10. 11.

12. 13.

14. 15.

If ______________________,

then ______________________.

A B

C D

E F

J

K

L M

N 11

15.4

10 14

L

K

JM

N

4 15

5

18

K

J L

N24 30

37.5 54 M

45

18

52 x 30

12

x 14

18

27 x

32

12

24

18 12

x + 3

21

2x + 2

28 31.5 92

x

3x + 1

20

Directions: Solve for x.

1. 2.

3. 4.

5. 6.

7. Find CE.

Name: ______________________________________ Unit 6: Similar Triangles

Date: _________________________ Per: _________ Homework 4: Parallel Lines & Proportional Parts

9

x

24

32

20

30

x + 5

36

4

3

2x + 4

x + 7

28 2x + 8

5x – 4 82

x + 8 2x – 5

20

22.5

14

20 33

x


A

B C

D

E

x – 5 6

14

2x + 3

21

Directions: Determine if ST is parallel to PR .

8. 9. 10.

Directions: Solve for x.

11. 12.

13. 14.

15. 16.

P

Q R

S

T

7

10

11.2

16

S

P

R Q

T

45

33 41.8

102 19

38

24

52

Q

P

R

S T

4

x – 7

x – 3

35

20

4x – 2

7x – 11

35

25 30

48 x 28

36.4

x

49

2x + 6 32

60 52.5

21

x – 3

x – 1

27

22


Parts ofSIMILAR

TRIANGLES

If two triangles are similar, then the following corresponding parts

are also proportional to the corresponding sides.

Altitudes

Angle

Bisectors

Medians

EXAMPLES

Directions: Given the similar triangles, solve for x.

1. STU ~ DEC 2. GHI ~ MLK

3. ABC ~ EGF 4. PQR ~ UTS

____________________________________________________________________________________

A

B

C D E

F

G H

P

Q

R S T

U

V W

K

L

M N G

H

I J

S

T

U

W

x

16

19.2

C

D

E F

20 24 15

K

L

M N

8 14

G H

I

J 10

x

A

B

C

D

32

38

E

F

G

H 24

x

P

Q

R

12 x

S

T

U

42 15

Section 6.5 Parts of Similar Triangles

23

5. KYC ~ BLH

TRIANGLE ANGLE

BISECTOR

Theorem

An angle bisector in

a triangle separates

the opposite sides into

two segments that are

proportional to the lengths

of the other two sides.

EXAMPLES

Directions: If KM represents an angle bisector, solve for x.

6. 7.

8. 9.

10. Find KL.

K

Y

C

M

20 2x – 9 B

L

X

H

x – 5

15

A

B

C D

J

K

L M 4.8

12 18

x K

M

J

L

40

36 54

x

J K

L

M

3x – 9

x + 11

27

45

J

L

M

15 39

36

x L

J

K

M

18

30

12 x

24

P

Directions: Given each pair of similar triangles, find the missing value.

1. If ABD ~ HGE and AC and HF are angle

bisectors, find x.

2. If LJK ~ PQN, find x.

3. If WVX ~ RTS, find x.

4. If DEF ~ JHI, find EF.

5. If MNP ~ TSR and NQ and SU are angle bisectors, find TS.

Name: _______________________________________ Unit 6: Similar Triangles

Date: ___________________________ Per: _________ Homework 5: Parts of Similar Triangles

27

3x – 21

V

X

W

Y

M

N

Q

39

5x – 2

24x + 13

UR

S

T

R

S

T

U

263x + 1

9

x + 8

D

E

F

G14

3x – 2

H

I

J

K

35

24

A

B C

D

14

x

E

F

G

H

30

x

J

K

L M 39

42.9N

O

P

Q


25

Directions: If QS represents an angle bisector, solve for x.

6. 7.

8. 9.

10.

11.

12.

18

P

Q R

S

2045

x P

Q R

S7

12.6

5

x

Q

P RS 14

6

22 x

Q

S

P

R

7042

x

35

P

QR

S

15

3x – 4 21

5x – 16

S

Q

P

R

5

x

26

31.2

S R

Q

P

30 45

40

x

26

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GEOMETRY Unit 6