high school math solution€¦ · trapezoid. is magda correct? explain. 4. sylvia drew what she...

High SchoolMath Solution

Geometry

Select Skills Practice for Review at Home

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ISBN: 978-1-60972-567-9Skills Practice

Printed in the United States of America1 2 3 4 5 6 7 8 9 CC 21 20 19 18 17


Module 2: Establishing Congruence

Topic 3: Using Congruence Theorems ................................... 85

Module 3: Investigating Proportionality

Topic 1: Similarity ................................................................... 95

Module 4: Connecting Geometric and Algebraic Descriptions

Topic 1: Circles and Volume ................................................. 140

TABLE OF CONTENTS©

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Odd-numbered Answers ........................................... 202

USING CONGRUENCE THEOREMS: Skills Practice • 85

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Skills PracticeName Date

Topic 3

USING CONGRUENCE THEOREMS

I. Right Triangle Congruence and Tangent Segments

A. For each fi gure, determine if there is enough information to prove that the two triangles are congruent. If so, name the congruence theorem used.

1. Given: ‾ GF bisects ∠RGS, and ∠R and ∠S are right angles. Is △FRG ≅ △FSG?

G

S

F

R

2. Given: ‾ DV ⊥ ‾ TU Is △DVT ≅ △DVU?

T V

D

U

3. Given: ‾ NM ≅ ‾ EM , ‾ DM ≅ ‾ OM , and ∠NMD and ∠EMO are right angles.

Is △NMD ≅ △EMO?

O

N

M D

E

4. Given: ‾ RP ≅ ‾ QS, and ∠R and ∠Q are right angles.

Is △SRP ≅ △PQS?

R S

QP

5. Given: ‾ GO ≅ ‾ MI, and ∠E and ∠K are right angles.

Is △GEO ≅ △MKI?

E G I

KMO

6. Given: ‾ HM ≅ ‾ VM , and ∠H and ∠V are right angles.

Is △GHM ≅ △UVM?

GH

M

UV

GEO_SP_M02_T03.indd 85GEO_SP_M02_T03.indd 85 24/04/19 2:43 PM24/04/19 2:43 PM

86 • MODULE 2: Establishing Congruence

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Topic 3


B. Use the given information to answer each question.

1. Two friends are meeting at the library. Maria leaves her house and walks north on Elm Street and then east on Main Street to reach the library. Paula leaves her house and walks south on Park Avenue and then west on Main Street to reach the library. Maria walks the same distance on Elm Street as Paula walks on Main Street, and she walks the same distance on Main Street as Paula walks on Park Avenue. Is there enough information to determine whether Maria’s walking distance is the same as Paula’s walking distance? Explain.

N

Paula’s house

Main Street

Library

Maria’s house

Elm

Str

eet

Park

Ave

nue

EW

S

2. An auto dealership displays one of their cars by driving it up a ramp onto a display platform. Later they will drive the car off the platform using a ramp on the opposite side. Both ramps form a right triangle with the ground and the platform. Is there enough information to determine whether the two ramps have the same length? Explain.

Ground

Displayplatform

3. A radio station erected a new transmission antenna to provide its listeners with better reception. The antenna was built perpendicular to the ground, and to keep the antenna from swaying in the wind, two guy wires were attached from it to the ground on opposite sides of the antenna. Is there enough information to determine if the guy wires have the same length? Explain.

guy wire guy wire

Ground

Tow

er

4. Two ladders resting on level ground are leaning against the side of a house. The bottom of each ladder is exactly 2.5 feet directly out from the base of the house. The point at which each ladder rests against the house is 10 feet directly above the base of the house. Is there enough information to determine whether the two ladders have the same length? Explain.



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Topic 3


C. Write a statement to show the congruent segments.

1. ⟷ AC and

⟷ BC are tangent to circle O.

A

O

B

C

2. ⟷ XZ and

⟷ WZ are tangent to circle O.

O

XW

Z

3. ⟷ RS and

⟷ RT are tangent to circle O.

O

S

R

T

4. ⟷ MP and

⟷ NP are tangent to circle O.

MN

O

P

5. ⟷ DE and

⟷ FE are tangent to circle O.

O

D

E

F

6. ⟷ GH and

⟷ GI are tangent to circle O.

O

H

G

I



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Topic 3


D. Calculate the measure of each angle. Explain your reasoning.

1. If ‾ EF and ‾ GF are tangent segments, what is the measure of ∠EGF?

F

E

G

O64°

2. If HI and JI are tangent segments, what is the measure of ∠HJI?

48°I

H

J

O

3. If KM and LM are tangent segments, what is the measure of ∠KML?

63°

L

M

K

O

4. If NP and QP are tangent segments, what is the measure of ∠NPQ?

OP

Q

N

71°

5. If AF and VF are tangent segments, what is the measure of ∠AVF?

22°

O

A V

F

6. If RT and MT are tangent segments, what is the measure of ∠RTM?

O

R

M

T

33°



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Topic 3


II. Properties of Quadrilaterals

A. Complete each statement for parallelogram MNPL.

1. MN ≅ and ML ≅

2. ∠NML ≅ ∠ and ∠MLP ≅ ∠

3. MN || and ML ||

4. MR ≅ and LR ≅

B. Complete each statement for rhombus UVWX.

1. UV ≅ ≅ ≅

2. ∠UVW ≅ ∠ and ∠XUV ≅ ∠

3. UV || and UX ||

4. UE ≅ and XE ≅

C. Determine the missing statement needed to prove each quadrilateral is a parallelogram by the Parallelogram/Congruent-Parallel Side Theorem.

1. XY || ZW

X Y

WZ

2. PS ≅ QR

S

R

Q

P

M N

PL

R

V

W

E

X

U



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3. MN ≅ PQ

Q

P

M

N

4. CF || DE

F E

DC

5. ST || VU

S T

UV

6. KG ≅ HL

G

K

H

L

D. Complete each statement for trapezoid UVWX.

1. The bases are and .

2. The pairs of base angles are ∠ and ∠ , and ∠ and ∠ .

3. The legs are and .

4. The vertices are , , , and .

U V

WX



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Topic 3


E. Use the given fi gure to answer each question.

1. The fi gure shown is a kite with ∠DAB ≅ ∠DCB. Which sides of the kite are congruent?

D C

B

A

2. The fi gure shown is a kite with FG ≅ FE Which of the kite’s angles are congruent?

G H

E

F

F. Use the given fi gure to answer each question.

1. The fi gure shown is an isosceles trapezoid with AB || CD. Which sides are congruent?

DC

A B

2. The fi gure shown is an isosceles trapezoid with EH ≅ FG. Which sides are parallel?

E F

H G

3. Given that IJLK is a kite, what kind of triangles are formed by diagonal IL?

L

JI

K

4. Given that LMNO is a kite, what is the relationship between the triangles formed by diagonal MO?

M N

O

L

5. Given that PQRS is a kite, which angles are congruent?

P

SR

Q

6. Given that TUVW is a kite, which angles are congruent?

W V

U

T



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3. The fi gure shown is an isosceles trapezoid with IJ ≅ LK. What are the bases?

LK

JI

4. The fi gure shown is an isosceles trapezoid with MP ≅ NO. What are the pairs of base angles?

M

P

ON

5. The fi gure shown is an isosceles trapezoid with PQ || RS. Which sides are congruent?

Q

SR

P

6. The fi gure shown is an isosceles trapezoid with LM || KN . What are the pairs of base angles?

K

N

M

L

G. Use the given information to answer each question.

1. Alice created a kite out of two sticks and some fabric. The sticks were 10 inches and 15 inches long. She tied the sticks together so they were perpendicular and attached the fabric. When she measured the kite, she noticed that the distance from where the sticks met to the top of the kite was 5 inches. What is the area of the kite Alice created?

2. Simon connected a square and two congruent right triangles together to form an isosceles trapezoid. Draw a diagram to represent the isosceles trapezoid.

3. Magda told Sam that an isosceles trapezoid must also be a parallelogram because there is a pair of congruent sides in an isosceles trapezoid. Is Magda correct? Explain.

4. Sylvia drew what she thought was an isosceles trapezoid. She measured the base angles and determined that they measured 81°, 79°, 101° and 99°. Could her drawing be an isosceles trapezoid? Explain.

5. Joanne constructed a kite with a perimeter of 38 centimeters so that the sum of the two shorter sides is 10 centimeters. What is the length of each of the two longer sides?

6. Ilyssa constructed a kite that has side lengths of 8 inches and 5 inches. What are the lengths of the other two sides? Explain.



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Topic 3


III. Chord Relationships

A. Use the given information to answer each question. Explain your answer.

1. If diameter BD bisects AC , what is the angle of intersection?

B

CO

D

A

2. If diameter FH intersects EG at a right angle, how does the length of EI compare to the length of IG?

OI

G

H

E

F

3. How do the measures of ⏜

KL and ⏜

LM compare?

JO

N

K

M

L

4. If KP ≅ LN , how does the length of QO compare to the length of RO?

OQ R

K L

M

NP

J

5. If YO ≅ ZO, what is the relationship between TU and XV?

O

W

VZ

X

T

SY

U

6. If GO ≅ HO and diameter EJ is perpendicular to both chords DF and IK, what is the relationship between GF and HK?

I K

FE

D

J

H

O

G



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Topic 3


B. Determine each measurement.

1. If BD is a diameter, what is the length of EC?

D

C

B

OE

A 5 cm

2. If the length of AB is 13 millimeters, what is the length of CD?

AB

C

D

O

3. If the length of AB is 24 centimeters, what is the length of CD?

A

O

D

C

B

4. If the length of BF is 32 inches, what is the length of CH?

A

BF

E C

D

OG

H

5. If the measure of ∠AOB 5 155°, what is the measure of ∠DOC?

A B

C

DO

6. If segment AC is a diameter, what is the measure of ∠AED?

DB

C

E

O

A


SIMILARITY: Skills Practice • 95

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I. Understanding Similarity

A. Given each pre-image and image, determine the scale factor.

1.

M’L’

P’N’N

P

LM

Z

2.

M’

L’

L

M

P

N

P’

Z

N’


Topic 1

SIMILARITY

3.

P N

L M

Z

L’ M’

N’P’

4.

20

2

4

6

8

10

12

14

16

4 6 8 10 12 14 16 x

y

A

A’

C’

BB’

C


96 • MODULE 3: Investigating Proportionality

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SIMILARITY

5.

20

2

4

6

8

10

12

14

16

4 6 8 10 12 14 16 x

y

L’

L

J

K’J’

K

6.

20

2

4

6

8

10

12

14

16

4 6 8 10 12 14 16 x

y

A’

A B’C’

BC

1. The scale factor is 3 and the center of dilation is the origin.

20

2

4

6

8

10

12

14

16

4 6 8 10 12 14 16 x

y

A

B

C


20

2

4

6

8

10

12

14

16

4 6 8 10 12 14 16 x

y

C

A

B

B. Given the pre-image, scale factor, and center of dilation, use a compass and straightedge to graph the image.



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SIMILARITY


10

1

2

3

4

5

6

7

8

2 3 4 5 6 7 8 x

y

F

E D


40

4

8

12

16

20

24

8 12 16 20 24 x

y

E

FD

5. The scale factor is 1.5 and the center of dilation is the origin.

20

2

4

6

8

10

12

14

16

4 6 8 10 12 14 16 x

y

A

C B

6. The scale factor is 3.5 and the center of dilation is the origin.

20

2

4

6

8

10

12

14

16

4 6 8 10 12 14 16 x

y

F

D E



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SIMILARITY

C. Use coordinate notation to determine the coordinates of the image.

1. △ABC has vertices A (1, 2), B (3, 6), and C (9, 7). What are the vertices of the image after a dilation with a scale factor of 4 using the origin as the center of dilation?

2. △DEF has vertices D (8, 4), E (2, 6), and F (3, 1). What are the vertices of the image after a dilation with a scale factor of 5 using the origin as the center of dilation?

3. △GHI has vertices G (0, 5), H (4, 2), and I (3, 3). What are the vertices of the image after a dilation with a scale factor of 9 using the origin as the center of dilation?

4. △JKL has vertices J (6, 2), K (1, 3), and L (7, 0). What are the vertices of the image after a dilation with a scale factor of 12 using the origin as the center of dilation?

5. △ABC has vertices A (8, 4), B (14, 16), and C (6, 10). What are the vertices of the image after a dilation with a scale factor of 1 __ 2 using the origin as the center of dilation?

6. △DEF has vertices D (25, 25), E (15, 10), and F (20, 10). What are the vertices of the image after a dilation with a scale factor of 1 __ 5 using the origin as the center of dilation?

1.

4 8

36

2.

4 cm12 cm

7 cm 21 cm

4 cm12 cm

D. Explain how you know that each pair of triangles are similar.



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SIMILARITY

3.

60°40° 60°40°

4.

4 in.8 in.

5 in. 10 in.

4 in.8 in.

5.

A B

C

D E

6.

F

G

I J

H

7.

6 in.

8 in. 4 in.

3 in.

8.

B

A C E

D



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Topic 1

SIMILARITY

1. What information would you need to use the Angle-Angle Similarity Theorem to prove that the triangles are similar?

35° 60°

2. What information would you need to use the Side-Angle-Side Similarity Theorem to prove that the triangles are similar?

4 m

6 m

10 m

3. What information would you need to use the Side-Side-Side Similarity Theorem to prove that these triangles are similar?

12 cm

14 cm

6 cm 5 cm

4. What information would you need to use the Side-Angle-Side Similarity Theorem to prove that the triangles are similar?

9 in. 9 in.

5 in.5 in.

5. What information would you need to use the Side-Side-Side Similarity Theorem to prove that these triangles are similar?

4 ft6 ft

6. What information would you need to use the Angle-Angle Similarity Theorem to prove that the triangles are similar?

110°35°

E. Determine what additional information you would need to prove that the triangles are similar using the given theorem.



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SIMILARITY

3. 4.

10 m

18 m

4.5 m

2.5 m

V

A B

C

XW

3 ft5 ft

6 ft

10 ft

7 ft

12 ft

K

JL

I

H G

F. Determine whether each pair of triangles is similar. Explain your reasoning.

1.

10 yd6 yd

18 yd

10.8 yd

Y

T R

SX Z

2.

3 in.

5 in.

2 in.

7.5 in.

60°

150°

N

Q

R

P

M O



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SIMILARITY

5.

15 m 18 m

36 m

20 m

24 m

27 m

D

S

T

U

E F

6.

15 cm

21 cm

10 cm

14 cm

75°

75°

E

S

RQ

D

F

G. Use each similarity statement to write the corresponding sides of the triangles as proportions.

1. △CGL ∼ △MKP 2. △XZC ∼ △YMN

3. △ADF ∼ △GLM 4. △WNY ∼ △CQR

5. △BTK ∼ △ALC 6. △SHM ∼ △NFR



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SIMILARITY

II. Calculating Corresponding Parts of Similar Triangles

A. Calculate the length of the indicated segment in each fi gure.

1. ‾ HJ bisects ∠H. Calculate HF.

15 cm18 cm

21 cm

F

H

G

J

2. ̄ LN bisects ∠L. Calculate NM.

4 in.

5 in.8 in.

L

KN M

3. ̄ BD bisects ∠B. Calculate AD.

3 ft

6 ft

2 ft

C

B

D

A

4. ̄ SQ bisects ∠S. Calculate SP.

9 m

12 m

18 m

P

S R

Q

5. ̄ YZ bisects ∠Y. Calculate YW.

4 cm

8 cm 9 cm

Y

XZ

W

6. ̄ VX bisects ∠V. Calculate XW.

9 ft

10 ft 6 ft

U

V

W

X



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SIMILARITY

1.

62

9

Q S

TxR

P

2.

4

6

6

x

3.

4.4

6.3

14.08

x

4.

4x – 3

3x – 1 5x – 3

8x – 7

A

ED

B C

5.

6 in. 6 in.

D

A GF

x

6.

4 cm12 cm

MZ

N P

x

B. Solve for x.



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SIMILARITY

1. On the map shown, Willow Street bisects the angle formed by Maple Avenue and South Street. Mia’s house is 5 miles from the school and 4 miles from the fruit market. Rick’s house is 6 miles from the fruit market. How far is Rick’s house from the school?

Mia’s house

Fruit market

Rick’s houseRiver Avenue

Maple Avenue

School South Street

Willow Street

7.

9 in.

6 in.

RS

V

Q

x

8.

4 m

10 m

B

D

L

T

x

C. Use the given information to answer each question.



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SIMILARITY

2. The road from Central City on the map shown bisects the angle formed by the roads from Central City to Minville and from Central City to Oceanview. Central City is 12 miles from Oceanview, Minville is 6 miles from the beach, and Oceanview is 8 miles from the beach. How far is Central City from Minville?

Minville

OceanviewCentral City

Beach

3. You are on a fi shing trip with your friends. The diagram shows the location of the river, fi shing hole, campsite, and bait store. The campsite is located 200 feet from the fi shing hole. The bait store is located 110 feet from the fi shing hole. How wide is the river?

Bait store

FishingholeCampsite

Rive

r



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SIMILARITY

4. Marsha wants to walk from the parking lot through the forest to the clearing, as shown in the diagram. She knows that the forest ranger station is 154 feet from the fl agpole and the fl agpole is 350 feet from the clearing. How far is the parking lot from the clearing?

Forestrangerstation

Parking lot

Flagpole

Clearing

Forest

5. Elly and Jeff are on opposite sides of a canyon that runs east to west, according to the graphic. They want to know how wide the canyon is. Each person stands 10 feet from the edge. Then, Elly walks 24 feet west, and Jeff walks 360 feet east.

24 ft

360 ft10 ft

10 ft

What is the width of the canyon?

6. Minh wanted to measure the height of a statue. She lined herself up with the statue’s shadow so that the tip of her shadow met the tip of the statue’s shadow. She marked the spot where she was standing. Then, she measured the distance from where she was standing to the tip of the shadow, and from the statue to the tip of the shadow.

Minh5 ft

12 ft84 ft

What is the height of the statue?



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SIMILARITY

III. Proofs Using Similar Triangles

A. Use the diagram and given information to write a statement that can be justifi ed using the Proportional Segments Theorem, Triangle Proportionality Theorem, or its Converse. State the theorem used.

1.

B

A P C

Q

2.

18

6

24

8

A

C B

D E

3.

x

y b

a

4.

A

P Q

CB



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SIMILARITY

5.

2.6 cm

3.9 cm

3.6 cm2.4 cm

E

F Q D

P

6.

AB

F

G

H

C D

1.

CA

D E

B

Given: ABC is a triangle.D is the midpoint of ‾ AB .E is the midpoint of ̄ BC .

2.

R

V W

ST

Given: RST is a triangle.V is the midpoint of ‾ RT .W is the midpoint of ‾ RS .

B. Use the diagram and given information to write two statements that can be justifi ed using the Triangle Midsegment Theorem.



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Topic 1

SIMILARITY

3.

M YN

X

P

Given: MNP is a triangle.X is the midpoint of ̄ MP .Y is the midpoint of ̄ MN .

4.

ZT

Y

U

X

Given: XYZ is a triangle.T is the midpoint of ̄ YZ .U is the midpoint of ‾ XY .

5.

J

M

K

N

L

Given: JKL is a triangle.M is the midpoint of ̄ JK .N is the midpoint of ̄ KL .

6.

A QC

R

B

Given: ABC is a triangle.Q is the midpoint of ‾ AC .R is the midpoint of ‾ BC .



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Topic 1

SIMILARITY

1.

G

Q

H

P

2.

M P

R

Z

3.

NK

L T

4.

UM

N

W

5.

C

A D B

6.

R

WX

B

C. Use the Right Triangle/Altitude Similarity Theorem to write three similarity statements involving the triangles in each diagram.



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Topic 1

SIMILARITY

IV. Dividing Segments by Ratios

A. Determine the midpoint of a line segment with each set of given endpoints.

1. (8, 0) and (4, 6) 2. (3, 8) and (9, 10)

3. (27, 2) and (3, 6) 4. (6, 23) and (24, 5)

5. (210, 21) and (0, 4) 6. (22, 7) and (28, 29)

1. 1 : 3

–8 –6 –4 –2

2

–2

–4

–6

–8

4

6

8

4 x

y

A

B

20 86

2. 1 : 1

–8 –6 –4 –2

2

–2

–4

–6

–8

4

6

8

4 x

y

A

B

20 86

B. Determine the coordinates of the point that divides each directed segment AB into the given ratio.



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Topic 1

SIMILARITY

3. 3 : 1

–8 –6 –4 –2

2

–2

–4

–6

–8

4

6

8

4 x

y

A

B

20 86

4. 1 : 2

–8 –6 –4 –2

2

–2

–4

–6

–8

4

6

8

4 x

y

A

B

20 86

5. 2 : 1

–8 –6 –4 –2

2

–2

–4

–6

–8

4

6

8

4 x

y

A

B

20 86

6. 1 : 1

–8 –6 –4 –2

2

–2

–4

–6

–8

4

6

8

4 x

y

A

B

20 86


140 • MODULE 4: Connecting Geometric and Algebraic Descriptions

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Topic 1

CIRCLES AND VOLUME

I. Arc Lengths

A. Calculate the ratio of the length of each arc to the circle’s circumference.

1. The measure of ⏜ AB is 40°.

3. The measure of ⏜ EF is 120°.

5. The measure of ⏜ IJ is 105°.

2. The measure of ⏜

CD is 90°.


GH is 150°.

6. The measure of ⏜ KL is 75°.


1. If the measure of ⏜ AB is 45° and the radius is

12 meters, what is the arc length of ⏜ AB ?

3. If the measure of ⏜ EF is 60° and the radius is

8 inches, what is the arc length of ⏜ EF ?

5. If the measure of ⏜ IJ is 80° and the diameter is

10 centimeters, what is the arc length of ⏜ IJ ?

7. If the measure of ⏜

MN is 75° and the diameter is 20 millimeters, what is the arc length of

⏜

MN ?


CD is 120° and the radius is 15 centimeters, what is the arc length of

⏜

CD ?


GH is 30° and the radius is 6 meters, what is the arc length of

⏜

GH ?

6. If the measure of ⏜ KL is 15° and the diameter

is 18 feet, what is the arc length of ⏜ KL ?


OP is 165° and the diameter is 21 meters, what is the arc length of

⏜

OP ?

B. Calculate each arc length. Write your answer in terms of π.


CIRCLES AND VOLUME: Skills Practice • 141

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Name Date

Topic 1

CIRCLES AND VOLUME

C. Calculate each arc length. Write your answer in terms of π.

1. If the measure of ⏜ AB is 135°, what is the arc

length of ⏜

AB ? 2. If the measure of

⏜

CD is 45°, what is the arc length of

⏜

CD ?

B

A

16 cm

CD

18 cm

3. If the measure of ⏜ EF is 90°, what is the arc

length of ⏜ EF ?


GH is 120°, what is the arc length of

⏜

GH ?E

F30 in.

G

H

27 in.

5. If the length of the radius is 4 centimeters, what is the arc length of

⏜ IJ ? 6. If the length of the radius is 7 centimeters,

what is the arc length of ⏜ KL ?

I

J

160°

K L

20°

7. If the length of the radius is 11 centimeters, what is the arc length of

⏜

MN ? 8. If the length of the radius is 17 centimeters,

what is the arc length of ⏜

OP ?M

N100°

O

P75°



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Topic 1

CIRCLES AND VOLUME

B. Convert each degree measure to radians.

II. Using Radians to Measure Arcs and Angles

A. Convert each radian measure to degrees.

1. π __ 3 2. 3π

___ 2

3. π __ 4 4. 18π

5. 0.1π 6. 4π

___ 5

1. 100° 2. 180°

3. 30° 4. 540°

5. 1° 6. 345°

C. Determine each arc length.

1. The radius of a circle is 1 centimeter. What is the length of an arc intercepted by an angle of π radians?

2. The radius of a circle is 4 inches. What is the length of an arc intercepted by an angle of π

__ 2 radians?

1 cm

� 4 in.

�2



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Topic 1

CIRCLES AND VOLUME

3. The radius of a circle is 10 millimeters. What is the length of an arc intercepted by an

angle of 3π

___ 2 radians?

4. The radius of a circle is 2.2 feet. What is the length of an arc intercepted by an angle of π

__ 4 radians?

10 mm

3�2

2.2 ft

�4

5. The radius of a circle is 0.5 meters. What is the length of an arc intercepted by an angle of 4π

___ 3 radians?

6. The radius of a circle is 8 centimeters. What is the length of an arc intercepted by an angle of π

___ 12 radians?

0.5 m

4�3

8 cm

�12

7. The radius of a circle is 24 yards. What is the length of an arc intercepted by an angle of π

__ 6 radians?

8. The radius of a circle is 6 inches. What is the length of an arc intercepted by an angle of 35π

____ 36 radians?

24 yd�6

6 in.35�36

9. The radius of a circle is 30 millimeters. What is the length of an arc intercepted by an

angle of 5π

___ 18 radians?

10. The radius of a circle is 15 feet. What is the length of an arc intercepted by an angle

of 49π

____ 36 radians?

30 mm5�18

15 ft

49�36



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Topic 1

CIRCLES AND VOLUME

11. The radius of a circle is 11.8 centimeters. What is the length of an arc intercepted by an angle of 4π

___ 9 radians?

12. The radius of a circle is 50 yards. What is the length of an arc intercepted by an angle of 19π

____ 18 radians?

11.8 cm4�9

50 yd19�18

III. Sectors and Segments

A. Calculate the area of each sector. Write your answer in terms of π.

1. If the radius of the circle is 9 centimeters, what is the area of sector AOB?

2. If the radius of the circle is 16 meters, what is the area of sector COD?

O

A

B

120° O

C

D45°

3. If the radius of the circle is 15 feet, what is the area of sector EOF?

4. If the radius of the circle is 10 inches, what is the area of sector GOH?

O

E

F30°

O

G H

20°



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Topic 1

CIRCLES AND VOLUME

5. If the radius of the circle is 32 centimeters, what is the area of sector IOJ?

6. If the radius of the circle is 20 millimeters, what is the area of sector KOL?

7. If the radius of the circle is 24 centimeters, what is the area of sector MON?

8. If the radius of the circle is 21 meters, what is the area of sector POQ?

B. Calculate the area of each segment. Round your answer to the nearest tenth, if necessary. Use 3.14 to estimate π.

1. If the radius of the circle is 6 centimeters, what is the area of the shaded segment?

2. If the radius of the circle is 14 inches, what is the area of the shaded segment?

O

A

B90°

O

C

D 90°

3. If the radius of the circle is 17 feet, what is the area of the shaded segment?


O

F

E90°

O

H

G90°

O

I J

18° O L

K

80°

O

M

N

135° O

P

Q

150°



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Topic 1

CIRCLES AND VOLUME

5. If the radius of the circle is 25 meters, what is the area of the shaded segment?


O

JI

90° O

LK

90°

C. In circle O, m ⏜ AB = 90°. Use the given information to determine the length of the radius of circle O.

1. If the area of the segment is 16π 2 32 square feet, what is the length of the radius of circle O?

2. If the area of the segment is 25π 2 50 square inches, what is the length of the radius of circle O?

3. If the area of the segment is π 2 2 square meters, what is the length of the radius of circle O?

4. If the area of the segment is 56.25π 2 112.5 square yards, what is the length of the radius of circle O?

OB

A

5. If the area of the segment is 121π 2 242 square feet, what is the length of the radius of circle O?

6. If the area of the segment is 90.25π 2 180.5 square millimeters, what is the length of the radius of circle O?



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Topic 1

CIRCLES AND VOLUME

IV. Creating Three-Dimensional Figures

A. Write the name of the solid fi gure that would result from rotating the plane fi gure shown around the axis shown.

1.

sphere

2.

cylinder

3.

cylinder

4.

cone

5.

cone

6.

cylinder



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Topic 1

CIRCLES AND VOLUME

B. Relate the dimensions of the plane fi gure to the solid fi gure that results from its rotation around the given axis.

1. 2.

3. 4.

5. 6.



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Topic 1

CIRCLES AND VOLUME

C. Connect the corresponding vertices of the fi gure and the translated fi gure. Name the shape that was translated and name the resulting solid fi gure.

1.

2.

3.

4.

5.

6.



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Topic 1

CIRCLES AND VOLUME

D. Name the solid formed by stacking 1000 of the congruent shapes shown.

E. Name the solid formed by stacking similar shapes so that each layer of the stack is composed of a slightly smaller shape than the previous layer.

1. 2.

3. 4.

5. 6.

1. 2.



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CIRCLES AND VOLUME

3. 4.

1. 2.

5. 6.

3. 4.

5. 6.

V. Volume

A. Calculate the volume of each cylinder. Use 3.14 for π. Round decimals to the nearest tenth, if necessary.

7 m 30 yd

22 yd

20 m

5 m

10 ft

4.5 ft

4 mm

6 mm

16 ft

5 ft



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Topic 1

CIRCLES AND VOLUME

1. 2.

3. 4.

7. 8.

5. 6.

B. Calculate the volume of each cone. Use 3.14 for π. Round answers to the nearest hundredth, if necessary.

5 cm

4 cm

7 cm2 cm

6 in.

3 in.

4 in.13 in.

10 m

15 m 14 mm

5 m

m

9 m

12 m

3.5 cm

13 cm



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CIRCLES AND VOLUME

7. 8.

5 cm

6.5 cm

1 cm

3.2 cm

1. 2.

10 in.10 in.

9 in.

12 ft12 ft

9 ft

3. 4.

11 cm

7 cm7 cm 25 m

25 m

20 m

C. Calculate the volume of each square pyramid. Round answers to the nearest hundredth, if necessary.

5. 6.

30 ft30 ft

22 ft

28 mm

21 mm21 mm



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CIRCLES AND VOLUME

7. 8.

42 in.42 in.

34.5 in.

90 cm90 cm

75 cm

1. r 5 7 meters 2. r 5 6 inches

r

D. Calculate the volume of each sphere. Use 3.14 for π. Round decimals to the nearest tenth, if necessary.

r

3. d 5 20 inches 4. d 5 16 meters

d d

5. r 5 2.5 centimeters 6. r 5 11.25 millimeters

r r



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Topic 1

CIRCLES AND VOLUME

VI. Surface Area

A. Determine the total surface area of each right prism. Bases with a single label on one edge are regular polygons. Round your answer to the nearest hundredth, where appropriate.

B. Determine the total surface area of each right pyramid. Bases with a single label on one edge are regular polygons. All other bases are rectangles. Round your answer to the nearest hundredth, where appropriate.

1.

4

73

2.

5

134

1.

8 in.6 in.

7 in.

2.

12 cm9 cm

15 cm

3.

6 in.

8 in.9 in.

4.

12 cm10 cm

16 cm



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Topic 1

CIRCLES AND VOLUME

C. Determine the total surface area of each right cylinder. Round your answer to the nearest hundredth.

1.

5 cm

8 cm

2.

9 cm

23 cm

3.

7.5 cm

6 cm

4.

86 m

35 m

3.

7 m

8 m

4.

4 mm

10 mm

15 mm



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Topic 1

CIRCLES AND VOLUME

D. Determine the total surface area of each right cone. Round your answer to the nearest hundredth.

E. Determine the total surface area of each sphere. Round your answer to the nearest hundredth.

1.

12 ft

5 ft

2.

15 cm

12 cm

1.

r = 7.5 cm

2.

r = 6 cm

3.

14 in.

18 in.

4.

21 cm

18 cm

3.

d = 22 in.

4.

d = 4 cm


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Topic 3


Topic 3

Using Congruence TheoremsI. A.

1. Yes. There is enough information to concludethat △FRG ≅ △FSG by HA.

3. Yes. There is enough information to concludethat △NMD ≅ △EMO by LL.

5. No. △GEO might not be congruent to △MKI.There is not enough information.

I. B.

1. Yes. Maria’s walking distance to the libraryis equal to Paula’s walking distance. Thetriangles formed are right triangles. Thecorresponding legs of the triangles arecongruent. So, by the Leg-Leg CongruenceTheorem, the triangles are congruent. If thetriangles are congruent, the hypotenuses arecongruent.

3. No. There is not enough information todetermine whether the guy wires have thesame length. The triangles formed by theantenna, the guy wires, and the ground areright triangles. But the lengths of the legs,the lengths of the hypotenuses, and themeasures of the acute angles are unknown.

I. C.

1. AC ≅ BC 3. RS ≅ RT 5. DE ≅ FE

1. The measure of angle EGF is 58 degrees. I knowtriangle EGF is isosceles and its base angles arecongruent. Let x represent the measure of angleEGF and the measure of angle FEG. m∠F 1 x 1 x 5 180

64 1 2x 5 180

2x 5 116

x 5 58

3. The measure of angle KML is 54 degrees.I know triangle KML is isosceles and its baseangles are congruent.

m∠KML 1 m∠MLK 1 m∠LKM 5 180

m∠KML 1 63 1 63 5 180

m∠KML 1 126 5 180

m∠KML 5 54

I. D.


GEO_SP_AK_ODD.indd 232GEO_SP_AK_ODD.indd 232 24/04/19 2:42 PM24/04/19 2:42 PM

Answer Key - ODD Answers

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USING CONGRUENCE THEOREMS: Skills Practice Answers • 233

Topic 3


5. The measure of angle AVF is 79 degrees.I know triangle AVF is isosceles and its baseangles are congruent. Let x represent themeasure of angle AVF and the measure ofangle VAF.

m∠AFV 1 x 1 x 5 180

22 1 2x 5 180

2x 5 158

x 5 79

II. A.

1. MN ≅ LP and ML ≅ NP 3. MN || LP and ML || NP

II. B.

1. UV ≅ VW ≅ WX ≅ XU 3. UV || XW and UX || VW

1. XY ≅ ZW 3. MN || PQ 5. ST ≅ VU

II. C.

1. The bases are ‾ UV and WX . 3. The legs are ‾ UX and VW .

II. D.

1. AB and CB are congruent. AD and CDare congruent.

3. Triangle IKL and triangle IJL are both isoscelestriangles.

5. ∠QPS and ∠QRS are congruent. ∠RQS and∠PQS are congruent. ∠RSQ and ∠PSQ arecongruent.

II. E.


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Topic 3



1. AC ≅ BD 3. The bases are IL and JK . 5. PR ≅ QS

II. F.

1. The area of the kite is 75 square inches. 3. Magda is incorrect. Parallelograms have twopairs of parallel sides. Trapezoids have at leastone pair of parallel sides. A parallelogramis always a trapezoid, but a trapezoid is notalways a parallelogram.

5. The length of each of the two longer sides is14 centimeters.

II. G.

III. A.

1. The angle of intersection is 90° becausediameters that bisect chords areperpendicular bisectors.


KL is equal to the measure of⏜

LM because a diameter that intersects a chordat a right angle bisects the arc formed bythe chord.

5. Chords TU and XV are congruent becausechords that are the same distance from thecenter of the circle are congruent.

III. B.

1. EC 5 EA 5 5 cm

5. m∠DOC 5 m∠AOB 5 155°

3. CD 5 AB 5 24 cm


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SIMILARITY: Skills Practice Answers • 235

Topic 1

SIMILARITY

MODULE 3Topic 1

SimilarityI. A.

1. The scale factor is 1.5. 3. The scale factor is 2.2.

5. The scale factor is 1 __ 2 .

1.

0

4

8

12

16

4 8 12 16 x

y

B C'

B'

A'

A C

3.

0

2

4

6

8

2 4 6 8 x

y

F

E D

D'

F'

E'

5.

0

4

8

12

16

4 8 12 16 x

y

A

B'

A'

C'C B

I. B.


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SIMILARITY


I. C.

I. D.

1. A’ (4, 8), B ’ (12, 24), C ’ (36, 28) 3. G’ (0, 45), H ’ (36, 18), I ’ (27, 27)

1. The triangles are congruent by theSide-Angle-Side Similarity Theorem. Two pairsof corresponding sides are proportional andthe included angles are congruent.

3. The triangles are congruent by the Angle-Angle Similarity Theorem. Two correspondingangles are congruent.

5. The angles formed by ̄ AE and ̄ BD are verticalangles, so angles ∠ACB and ∠ECD arecongruent. Angles ABC and CDE are both rightangles, so they are also congruent. So by theAngle-Angle Similarity Theorem, the trianglesformed are similar.

7. The known corresponding sides of thetriangles are proportional: 6 __ 3 5 2 __ 1 and 8 __ 4 5 2 __ 1 .The angle between the known sides is a rightangle for both triangles, so those angles arecongruent. Therefore, by the Side-Angle-SideSimilarity Theorem, the triangles are similar.

5. A’ (4, 2), B ’ (7, 8), C ’ (3, 5)

I. E.

1. To prove that the triangles are similar usingthe Angle-Angle Similarity Theorem, the fi rsttriangle should have a corresponding 60degree angle and the second triangle shouldhave a corresponding 35 degree angle.

3. To prove that the triangles are similar usingthe Side-Side-Side Similarity Theorem, allcorresponding side lengths between the twotriangles should be proportional.

5. To prove that the triangles are similar usingthe Side-Side-Side Similarity Theorem, allcorresponding side lengths between the twotriangles should be proportional.


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SIMILARITY: Skills Practice Answers • 237

I. F.

1. The triangles are similar by the Side-Angle-Side Similarity Theorem because the includedangles in both triangles are congruent andthe corresponding sides are proportional.

3. The triangles are similar by the Side-Angle-Side Similarity Theorem. The correspondingsides are proportional and the includedangles are congruent.

5. The triangles are similar by the Side-Side-Side Similarity Theorem because all of thecorresponding sides are proportional.

Topic 1

SIMILARITY

I. G.

1. CG ____ MK 5 GL ___ KP 5 CL ____MP 3. AD ___ GL 5 DF ___ LM 5 AF ____

GM

5. BT ___ AL 5 TK ___ LC 5 BK___AC

II. A.

II. B.

1. The length of segment HF is17.5 centimeters.

3. The length of segment AD is 4 feet.

5. The length of segment YW is4.5 centimeters.

1. x 5 3 3. x 5 20.16

5. x 5 6 in. 7. x 5 7.5 in.

II. C.

5. The distance across the canyon is 140 feet.

1. Rick’s house is 7.5 miles from the school. 3. The river is 60.5 feet wide.


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SIMILARITY


III. A.

III. B.

III. C.

1. AQ

____ QB 5 AP ___ PC , Triangle Proportionality Theorem 3. x __ y 5 a __ b , Proportional Segments Theorem

1. ̄ DE � ̄ AC , DE 5 1 __ 2 AC 3. ̄ XY � ̄ NP , XY 5 1 __ 2 NP

5. ̄ MN � ̄ JL , MN 5 1 __ 2 JL

1. △HPG ∼ △PQG, △HPG ∼ △HQP, △PQG ∼ △HQP 3. △NLT ∼ △NKL, △NLT ∼ △LKT, △NKL ∼ △LKT

5. △ACB ∼ △ADC, △ACB ∼ △CDB, △ADC ∼ △CDB

5. ̄ EF � ̄ PQ , Converse of TriangleProportionality Theorem

Answers may vary.

IV. A.1. The midpoint is (6, 3).

5. The midpoint is (25, 1.5).

3. The midpoint is (22, 4).

IV. B.1. The point that divides directed segment AB

into a 1 : 3 ratio is (4, 3.5).

5. The point that divides directed segment ABinto a 2 : 1 ratio is (22, 26 1 __ 3 ).

3. The point that divides directed segment ABinto a 3 : 1 ratio is (3, 7).


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CIRCLES AND VOLUME: Skills Practice Answers • 243

Topic 1

CIRCLES AND VOLUME

MODULE 4Topic 1

Circles and VolumeI. A.

II. A.

II. B.

I. B.

1. The arc is 1 __ 9 of the circle’s circumference. 3. The arc is 1 __ 3 of the circle’s circumference.

1. 60° 3. 45° 5. 18°

1. 5π

___ 9 radians 3. π __ 6 radians 5. π

____ 180 radians

5. The arc is 7 ___ 24 of the circle’s circumference.

1. The arc length of ⏜ AB is 3π meters.

5. The arc length of ⏜ IJ is 20π

____ 9 centimeters.

3. The arc length of ⏜ EF is 8 __ 3 π inches.

7. The arc length of ⏜

MN is 25 ___ 6 π millimeters.

I. C.

1. The arc length of ⏜ AB is 12π centimeters.

5. The arc length of ⏜ IJ is 32 ___ 9 π centimeters.

3. The arc length of ⏜ EF is 15π inches.

7. The arc length of ⏜

MN is 55 ___ 9 π centimeters.


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Topic 1

CIRCLES AND VOLUME

9. The length of the arc is 25π

____ 3 millimeters. 11. The length of the arc is 47.2π

______ 9 centimeters.

5. The length of the arc is 2π

___ 3 meters. 7. The length of the arc is 4π yards.

II. C.1. The length of the arc is π centimeters. 3. The length of the arc is 15π millimeters.

III. A.

III. B.

III. C.

1. The area of sector AOB is 27π centimeters2. 3. The area of sector EOF is 75 ___ 4 π feet2.

1. The area of the shaded segment isapproximately 10.3 centimeters2.

3. The area of the shaded segment isapproximately 82.4 feet2.

1. The length of the radius is 8 feet. 3. The length of the radius is 2 meters.

5. The area of sector IOJ is 256 ____ 5 π centimeters2. 7. The area of sector MON is 216π centimeters2.

5. The area of the shaded segment isapproximately 178.1 meters2.

5. The length of the radius is 22 feet.


IV. A.

IV. B.

1. sphere 3. cylinder 5. cone

1. The base of the rectangle is equal to the radius of the cylinder’s base.

3. The radius of the circle is equal to the radius of the sphere.

5. Half the width of the square is equal to the radius of the cylinder’s base.


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CIRCLES AND VOLUME: Skills Practice Answers • 245

Topic 1

CIRCLES AND VOLUME

right triangle; right triangular prism square; square prism

IV. C.

1. 3.

parallelogram; rectangular prism

5.

IV. E.

V. A.

1. cone 3. triangular pyramid 5. rectangular pyramid

1. V ≈ 664.9 cubic meters 3. V ≈ 1570 cubic meters

5. V ≈ 301.4 cubic millimeters 7. V ≈ 763.0 cubic meters

IV. D.

1. cylinder 3. triangular prism 5. rectangular prism


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CIRCLES AND VOLUME

V. B.

V. C.

1. V ≈ 88.73 cubic centimeters 3. V ≈ 113.04 cubic inches

1. V 5 300 cubic inches 3. V ≈ 179.67 cubic centimeters

5. V ≈ 2355 cubic meters 7. V ≈ 170.08 cubic centimeters

5. V 5 6600 cubic feet 7. V 5 20,286 cubic inches

VI. A.

V. D.

1. SA 5 122 square units 3. SA 5 264 square inches

1. V ≈ 1436.0 cubic meters 3. V ≈ 4186.7 cubic inches

5. V ≈ 65.4 cubic centimeters

VI. B.

1. SA 5 146 square inches 3. SA 5 161 square meters

VI. C.

VI. D.

VI. E.

1. SA ≈ 408.41 square centimeters 3. SA ≈ 636.17 square inches

1. SA ≈ 267.04 square feet 3. SA ≈ 549.78 square inches

1. SA ≈ 706.86 square meters 3. SA ≈ 1520.53 square inches



high school math solution€¦ · trapezoid. is magda correct? explain. 4. sylvia drew what she...

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