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Holt McDougal Florida

Larson Geometry

Lesson Transparencies

LAH_GE_11_FL_LT_FM.indd iLAH_GE_11_FL_LT_FM.indd i 11/13/09 10:52:54 AM11/13/09 10:52:54 AM

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ISBN 13: 978-0-547-24235-4ISBN 10: 0-547-24235-2

1 2 3 4 5 6 7 8 9 XXX 15 14 13 12 11 10 09

LAH_GE_11_FL_LT_FM.indd iiLAH_GE_11_FL_LT_FM.indd ii 11/13/09 10:52:37 AM11/13/09 10:52:37 AM

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iiiGeometry

Lesson Transparencies

ContentsChapter 1 Essentials of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2 Reasoning and Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter 3 Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 4 Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter 5 Relationships Within Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Chapter 6 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 7 Right Triangles and Trigonometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 8 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 9 Properties of Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter 10 Properties of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Chapter 11 Measuring Length and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Chapter 12 Surface Area and Volume of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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LAH_GE_11_FL_LT_FM.indd ivLAH_GE_11_FL_LT_FM.indd iv 2/27/09 12:54:56 PM2/27/09 12:54:56 PM

Graph each inequality.

1. x ≤ 1 2. 22 ≤ x ≤ 3

3. Juan has more than 5 but fewer than 11 fi sh in his aquarium. Write an inequality to express the number of fi sh f Juan has.

Warm-Ups: 1. 0 2 42224

2. 0 2 42224

3. 5 < f < 11

ANSWERS

Warm-Up ExercisesLESSON

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1Geometry

Warm-Up Transparencies

LAH_GE_11_FL_LT_CH01.indd 1LAH_GE_11_FL_LT_CH01.indd 1 2/25/09 11:57:50 PM2/25/09 11:57:50 PM

1. Solve 3x 1 5 1 2x 2 4 5 36.

2. Find three cities on this map that appear to be collinear.

ChicagoRockford

BloomingtonQuincy

Springfield

Cairo

Lesson QuizFor use after Lesson 1.1

Use this fi gure for Exercises 1–4.

S

DB

F

A

C

E

1. Give two other names for ‹ ] › AE .

2. Give another name for plane S.

3. Name three collinear points.

4. Name the intersection of ‹ ] › AC

and plane S.

Warm-Ups: 1. 7 2. Chicago, Bloomington, Springfi eld

Lesson Quiz: 1. ‹ ] › EC , ‹ ] › AC 2. plane DEF or EBF

3. A, E, C or D, E, B 4. E

ANSWERS

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1.2

2 Geometry



1. Find a point between A(23, 5) and B(7, 5).

2. Find the average of 211 and 5.

3. Solve x 1 7

} 2 5 5.

4. Find Ï}

30 to the nearest hundredth.

5. Find Ï}

5 1 Ï}

20 to the nearest hundredth.


1. Measure } AB to the nearest 1 }

8 inch.

in. 1

A B

2. Find CE.

C D

7 15

E

3. Plot the points F(23, 5), G(2, 5), H(3, 1), and J(3, 23) in a coordinate plane. Are } FG and } HJ congruent? Explain.

4. Find KL.

0 2 42224

LK

Warm-Ups: 1. Sample: (2, 5) 2. 23 3. 3 4. 5.48 5. 6.71

Lesson Quiz: 1. 1 3 }

8 in. 2. 22

3. No, FG 5 5 and HJ 5 4. 4. 7

ANSWERS

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1.3

3Geometry



Solve each equation.

1. 5x 1 6 1 2x 2 14 5 90

2. 4x 2 30 1 x 1 25 5 180

3. x }

3 1 2 5 x 2 5


1. } AB bisects } CD at E. If CE 5 2 1 }

4 in., fi nd CD.

2. Point M is the midpoint of } XY . Find XM.

X M Y

2x 1 7 8x 2 23

3. Point M is the midpoint of } PQ with endpoints P(2, 26) and Q(28, 0). Find the coordinates of M.

4. The midpoint of } GH is M(4, 21). One endpoint is G(5, 3). Find the coordinates of H.

5. To fi nd the distance between the swing and the sandbox in his backyard, Darren made a graph and found the coordinates of the swing to be (7, 2) and the coordinates of the sandbox to be (23, 8). Find the distance between the swing and the sandbox to the nearest tenth of a unit.

Warm-Ups: 1. 14 2. 37 3. 10.5

Lesson Quiz: 1. 4 1 }

2 in. 2. 17 3. (23, 23)

4. (3, 25) 5. 11.7

ANSWERS

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1.4

4Geometry



1. The sum of two numbers is 90 and one number is 4 times the other. Write an equation and solve to fi nd the numbers.

2. Find m∠ABD. What kind of angle is it?

1688

128

D

C

A B


Classify each angle as acute, obtuse, right, or straight.

1. m∠A 5 908

2. m∠B 5 628

3. m∠C 5 1198

4. If m∠DEG 5 848, fi nd m∠FEG.

E

D

F

G

(x 2 6)8

(3x 1 10)8

5. If ]

› XY bisects ∠ZXW and m∠ZXY 5 368, fi nd m∠ZXW.

Warm-Ups: 1. x 1 4x 5 90; 18, 72 2. 1808, straight

Lesson Quiz: 1. right 2. acute 3. obtuse

4. 148 5. 728

ANSWERS

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1.5

5Geometry



1. Draw an acute angle and shade the interior.

2. Find the measure of the supplement of a 1308 angle.

3. Find the measure of the complement of an 868 angle.


1. ∠1 and ∠2 are supplementary. If m∠1 5 978, fi nd m∠2.

2. ∠3 and ∠4 are complementary angles. If m∠3 5 748, fi nd m∠4.

3. Find m∠EFH. 4. Find m∠ABC.

GE F

H

(3x 2 6)8 (2x 1 16)8

D

C

B

A

(x 1 9)8

(2x)8

5. Is it possible to draw a fi gure that contains exactly one pair of vertical angles? Explain.

Warm-Ups: 1. 2. 508 3. 48

Lesson Quiz: 1. 838 2. 168 3. 968 4. 3685. No; once you have drawn a pair of vertical angles, you have drawn two pairs of opposite rays. This automatically gives another pair of vertical angles.

ANSWERS

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1.6

6Warm-Up TransparenciesGeometry


1. Find the approximate distance between (2, 3) and (21, 4). Round to the nearest tenth.

2. Change 72 inches to feet.

3. Solve 1 }

2 (5x) 5 30.

4. Michelle will put braid around the edges of a rug. She calculates that she needs 14 yards. How many feet of braid does she need?


1. Draw a convex hexagon.

2. This fi gure shows the tiles on a kitchen fl oor. What type of polygon are the tiles? Are they regular polygons?

3. This fi gure is a regular polygon. Find the length of each side.

(x2 1 6x) cm

(x2 2 10x 1 32) cm

Warm-Ups: 1. 3.2 2. 6 ft 3. 12 4. 42 ft

Lesson Quiz: 1.

2. quadrilaterals; not regular 3. 16 cm

ANSWERS

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1.7

7Geometry


LAH_GE_11_FL_LT_CH01.indd 7LAH_GE_11_FL_LT_CH01.indd 7 11/11/09 11:33:12 AM11/11/09 11:33:12 AM

Find the perimeter and area of each fi gure.

1.

4 m

7 m

2. 13 in.

5 in.

12 in.

3. Find the approximate circumference and area of a circle with radius 5 cm.

4. The area of a square game board is 144 in.2. Find the length of a side.

Lesson Quiz: 1. 22 m, 28 m2 2. 30 in., 30 in.2

3. 31.4 cm, 78.5 cm2 4. 12 in.

ANSWERS

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LESSON

1.7 Lesson QuizFor use after Lesson 1.7

GeometryWarm-Up Transparencies 8


1. Find the length of a segment with endpoints A(1, �3) and B(�2, �7).

2. If M(4, �3) is the midpoint of � RS , and the coordinates of R are (8, �2), fi nd the coordinates of S.

3. ∠A and ∠B are supplementary. If the measure of ∠A is three times the measure of ∠B, fi nd the measure of ∠B.

Warm-Ups: 1. 5 2. (0, �4) 3. 45�

ANSWERS


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9Geometry



Classify each of the following angles as acute, right, or obtuse.

1. 102� 2. 37�

3. Find the complement and supplement of ∠XYZ if m∠XYZ � 80�.

4. If XY � YZ, is Y the midpoint of � XZ ? If not, give a counterexample.


1. Describe a pattern in the numbers. Write the next number in the pattern. 20, 22, 25, 29, 34, . . ..

2. Find a counterexample for the following conjecture: If the sum of two numbers is positive, then the two numbers must be positive.

3. The scatter plot shows the average

Ho

urs

of

ho

mew

ork

00 2 4 6 8

2

4

8

6

Week number

number of hours of homework done per week by a student during the fi rst 10 weeks of a school term. Make a conjecture that could be true. Explain your reasoning.

Warm-Ups: 1. obtuse 2. acute 3. 10�; 100� 4. No; X, Y, and Z need not be collinear.

Lesson Quiz: 1. Start by adding 2 to 22, then add numbers that suc-cessively increase by 1; 40. 2. Sample: 20 � (�10) � 10 3. Sample answer: The student will do about 11 hours of homework in week 11. The number of hours of homework per week increased steadily during the fi rst 10 weeks.

ANSWERS


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10Geometry



1. Write the converse, inverse, and contrapositive of “If the measure of an angle is less than 90�, then the angle is acute.”


1. For the given statement, write the if-then form, the converse, the inverse, and the contrapositive. Sophomores take driving lessons.

2. Decide if the statement about

CD

A

B

the diagram is true. Explain. �

› BD bisects ∠ABC.

Determine whether the statements are good defi nitions.

3. If two angles are supplementary, then the sum of their measures is 180�.

4. If an angle is obtuse, then its measure is greater than that of an acute angle.

Warm-Ups: 1. Converse: If an angle is acute, then the measure is less than 90�. Inverse: If the measure of an angle is not less than 90�, then the angle is not acute. Contrapositive: If an angle is not acute, then its measure is not less than 90�.

Lesson Quiz: 1. If-then: If you are a sophomore, then you take driv-ing lessons. Converse: If you take driving lessons, then you are a sophomore. Inverse: If you are not a sophomore, then you do not take driving lessons. Contrapositive: If you do not take driving lessons, then you are not a sophomore. 2. Yes; point D is in the interior of ∠ABC and m∠ABD � m∠CBD, so

�

› BD bisects ∠ABC.

3. yes 4. no

ANSWERS


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11Geometry



State which postulate justifi es each statement.

1. If D is in the interior of ∠ABC, then m∠ABD � m∠DBC � m∠ABC.

2. If M is between X and Y, then XM � MY � XY.

3. Find the length of � MN if N is between M and P, MP � 6x � 2, MN � 4x, and MP � 16.


1. Use the Law of Detachment to make a valid conclusion in the true situation. If two angles are complementary, then the sum of their measures is 90�. ∠C and ∠D are complementary.

2. Use the Law of Syllogism to make a valid conclusion in the true situation. If water is at room temperature, then it is a liquid. If water is a liquid, then it is not frozen.

Use the true statements below to determine whether the conclusion is true or false. Explain your reasoning. If Jeanine does her homework, then she goes to the movies. If Joaquin goes to the movies, then he buys popcorn. If Jeanine goes to the movies, then she buys popcorn. Jeanine does her homework.

3. Jeanine buys popcorn.

4. If Jeanine goes to the movies, Joaquin goes to the movies.

Warm-Ups: 1. Angle Addition Postulate 2. Segment Addition Postulate 3. 12

Lesson Quiz: 1. m∠C � m∠D � 90� 2. If water is at room temperature, then it is not frozen. 3. true; the Law of Syllogism 4. False; no laws of logic apply.

ANSWERS


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12 Geometry



Solve each equation.

1. 6x � 2 � �3x � 16 2. �4(x � 8) � �16

3. Solve P � 2l � 2w for w.

4. ∠S is the complement of ∠R. If m∠R � 38�, fi nd m∠S. If ∠T is also the complement of ∠R, what is true about m∠S and m∠T?


1. Use the diagram to write an example of Postulate 7.

RH

F

nG

mJE

2. Decide whether the statement “Two planes can intersect in exactly one point R” is true or false. If false, give a reason.

Use the diagram to determine

BE

D

FN

M

H

G

A

C

if the statement is true or false.

3. The intersection of planes M and N is points B, E, and H.

4. ‹ � › GH lies in plane M.

5. ∠ABE and ∠HED are vertical angles.

Warm-Ups: 1. �2 2. 12 3. w � P � 2 l

� 2 4. 52�; m∠S � m∠T

Lesson Quiz: 1. Line m intersects line n in point F and no other point. 2. False; Postulate 11 says that if two planes intersect, their intersection is a line. 3. false 4. true 5. false

ANSWERS


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13Geometry



Use a property of equality to complete the statement.

1. If m∠1 � m∠3, then m∠3 � ? .

2. If AB � CD and CD � TU, then ? .

3. If RS � WX, then ? � AB � ? � AB.

4. If m∠EFG � 28� and m∠GFH � 62�, then ? � 62� � m∠EFG � m∠GFH.


Solve. Give a reason for each step.

1. �5x � 18 � 3x � 38 2. �3(x � 5) � 2(x � 10)

Identify the property of equality.

3. If m∠3 � m∠5 and m∠5 � m∠8, then m∠3 � m∠8.

4. If CD � EF, then EF � CD.

Warm-Ups: 1. m∠1 2. AB � TU 3. RS; WX 4. 28�

Lesson Quiz:

1. Equation (Reason) 2. Equation (Reason)

�5x � 18 � 3x � 38 (Given) �3(x � 5) � 2(x � 10) (Given)�8x � 18 � �38 (Subtr. Prop. of Eq.) �3x � 15 � 2x � 20

(Distributive Prop.)�8x � �56 (Subtr. Prop. of Eq.) 15 � 5x � 20

(Addition Prop. of Eq.)x � 7 (Division Prop. of Eq.) �5 � 5x (Subtr. Prop. of Eq.) �1 � x (Division Prop. of Eq.)

3. Transitive Prop. of Eq. 4. Symmetric Prop. of Eq.

ANSWERS

GeometryWarm-Up Transparencies


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14


Give a reason for each statement.

1. If m∠1 � 90� and m∠2 � 90�, then m∠1 � m∠2.

2. If � AB ⊥ � BC , then ∠ABC is a right angle.

3. If � FG � � RS , then FG � RS.


1. Copy and complete the proof. 2. Use the given information

M HA T

to prove the statement.

GIVEN: MA � TH

21

PROVE: MT � AH

Statements (Reasons) GIVEN: m∠1 � m∠2 � 90�;

1. MA � TH ( ? ) m∠1 � 59�

2. ? (Refl exive Prop. of Eq.) PROVE: m∠2 � 31�

3. MA � AT � AT � TH ( ? )

4. MA � AT � MT; AT � TH � AH ( ? )

5. ? (Substitution Prop. of Eq.)

Warm-Ups: 1. (Transitive Prop. of Eq.) 2. (Def. of perpendicular) 3. (Def. of segment congruence) Lesson Quiz:

1. 1. Given 2. Statements (Reasons)

2. AT � AT 1. m∠1 � m∠2 � 90�; m∠1 � 59� (Given)

3. Addition Prop. of Eq. 2. m∠2 � 90� � m∠1 (Subtraction Prop. of Eq.)

4. Segment Add. Post. 3. m∠2 � 90� � 59� (Substitution Prop. of Eq.)

5. MT � AH 4. m∠2 � 31� (Simplify.)

ANSWERS


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15Geometry



1. Give the reason for each step.

GIVEN: ∠1 � ∠5 1 23

54

PROVE: ∠1 is supplementary to ∠4.

Statements (Reasons)

1. ∠1 � ∠5

2. m∠1 � m∠5

3. ∠4 and ∠5 are a linear pair.

4. ∠4 and ∠5 are supplementary.

5. m∠4 � m∠5 � 180°

6. m∠4 � m∠1 � 180°

7. ∠1 is supplementary to ∠4.

Lesson Quiz: 1. (Given) 2. (Def. of �) 3. (Def. of linear pair) 4. (Linear Pair Post.) 5. (Def. of supplementary) 6. (Substitution Prop. of Eq.) 7. (Def. of supplementary)

ANSWERS


LESSON

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1. Name the hypothesis and conclusion of the statement, “If an animal fl ies, then it is a bird.”

2. Write the converse, inverse, and contrapositive of the conditional, “If a polygon is regular, then it has congruent sides.”

Lesson QuizFor use after 2.3 Focus On Reasoning

1. Make a truth table for the logical statement ~p r ~q.

2. Let p be “the angle is an acute angle” and let q be “the measure of the angle is less than 90°.” Write the conditional statement p r q in words.

Warm-Ups: 1. hypothesis: an animal fl ies; conclusion: it is a bird 2. Converse: If a polygon has congruent sides, then it is regular. Inverse: If a polygon is not regular, then it does not have congruent sides. Contrapositive: If a polygon does not have congruent sides, then it is not regular.

Lesson Quiz: 1. p q ~p ~q ~p r ~qT T F F TT F F T TF T T F FF F T T T

2. If an angle is acute, then its measure is less than 90°.

ANSWERS

LESSON

2.3 Focus On ReasoningWarm-Up Exercises

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Geometry Warm-Up Transparencies 17


BA

CD

1. Name a line that does not intersect ‹ ] › AD .

2. What is the intersection of ‹ ] › AD and

‹ ] › DB ?

Warm-Ups: 1. ‹ ] › BC 2. D

ANSWERS

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3.1

18Geometry



Identify the type of angles.

1014

913

1216

1115

48

37

26

15

1. ∠5, ∠7 2. ∠3, ∠6 3. ∠1, ∠8


1. Name the pairs of corresponding angles.

78

34

56

12

2. Name the pairs of alternate interior angles.

3. Name the pairs of consecutive interior angles.

4. The fi gure shows an easel. A B

C D

Does ‹

] ›

AB appear to be parallel,

perpendicular, or skew to ‹ ] › CD ?

Warm-Ups: 1. corresponding 2. alternate interior 3. alternate exterior

Lesson Quiz: 1. ∠1 and ∠3; ∠5 and ∠7; ∠2 and ∠4; ∠6 and ∠8 2. ∠2 and ∠7; ∠3 and ∠6 3. ∠2 and ∠3; ∠6 and ∠7 4. parallel

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3.2

19Geometry



1. Find the value of x.

(x 1 20)8

528

2. Write the converse of the following statement. If it is raining, then Josh needs an umbrella.


What theorem justifi es each statement?

1. ∠3 > ∠6

2. ∠4 and ∠6 are supplementary. 8

76

5

432

1

3. If m∠2 5 1158, fi nd m∠7.

4. Find the values of x and y.

6x8

(y 1 10)8

308

(2x 1 10)8

5. The fi gure shows a plant trellis. 1

2

If m∠1 5 828, fi nd m∠2.

Warm-Ups: 1. 32 2. If Josh needs an umbrella, then it is raining.

Lesson Quiz: 1. Alt. Interior ? Thm. 2. Consec. Interior ? Thm. 3. 1158 4. 17.5, 35 5. 828

ANSWERS

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3.3

20Geometry



1. Evaluate a 2 b

} c 2 d

if a 5 5, b 5 2, c 5 1, and d 5 7.

2. Solve x 2 3

} 3 2 4

5 1 } 5 . 3. What is the reciprocal of

2 }

3 ?

4. Julie was thinking of a number. The product of her number and 6 is 21. What was Julie’s number?


1. Find the value of x that makes p i q.

(2x 1 43)8

p

q

(3x)8

2. Can you prove a i b? If so, what theorem would you use?

a

b

3. Which lines are parallel?

858 1008

808958

E F

D G

4. In the fi gure, if } HJ i } KL and } KL i } MN , what can you conclude?

M N

K L

H J

What theorem justifi es your conclusion?

Warm-Ups: 1. 2 1 } 2 2.

14 } 5 3.

3 }

2 4. 2

1 } 6

Lesson Quiz: 1. 43 2. Yes; Alternate Interior Angles Converse

3. ‹ ] › EF i

‹ ] › DG 4. } HJ i } MN by Transitive Property of Parallel Lines

ANSWERS

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3.4

21Geometry



1. If 2x 1 5y 5 220 and x 5 0, what is y?

2. What is the slope of the line containing the points (2, 7) and (3, 210)?

3. Orphelia bought x apples. If each apple costs $.59, write an equation for y, the total cost of x apples.


1. Find the slope of the line containing the points (4, 23) and (5, 2).

2. Line k passes through the points (21, 2) and (3, 5). Line n passes through the points (3, 7) and (6, 3). Are lines k and n parallel, perpendicular, or neither?

3. A highway has a grade of 7 percent. For each 200 feet it goes horizontally, how many feet does it rise?

Warm-Ups: 1. 24 2. 217 3. y 5 0.59x

Lesson Quiz: 1. 5 2. perpendicular 3. 14 ft

ANSWERS

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3.5

22Geometry



1. What is the distance between the points (2, 3) and (5, 7)? 2.

C

D

A B

If m∠DBC 5 908, what is m∠ABD?


1. Write an equation of the line that has slope 3 }

4 and y-intercept 21.

2. Write an equation of the line that passes through the point (4, 22) and has slope 3.

3. Write an equation of the line that passes through the point (7, 1) and is parallel to the line with equation x 5 4.

4. Write an equation of the line that passes through the point (7, 1) and is perpendicular to the line with equation 6x 2 3y 5 8.

5. Toni’s puppy weighed 10 pounds when it was 2 months old. It gained 2 pounds a month for 6 months. Write an equation for the puppy’s weight w after m months during this time.

Warm-Ups: 1. 5 2. 908

Lesson Quiz: 1. y 5 3 } 4 x 2 1 2. y 5 3x 2 14

3. x 5 7 4. y 5 2 1 } 2 x 1

9 } 2 5. w 5 2m 1 6

ANSWERS

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3.6

23Geometry



1. Find m∠3.

3

728

2. How do you know that a and b c

da b

are parallel?

3. Find the distance between the two parallel 2

y

22 x (24, 21)

(0, 4)

(5, 0)

(1, 25)

lines. Round to the nearest tenth.

Lesson Quiz: 1. 188 2. Both are perpendicular to c. 3. 6.4

ANSWERS

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LESSON

3.6



John walks 5 blocks north, 4 blocks east, 2 more blocks north, and 1 more block east.

1. How many blocks north of his starting point is he?

2. How many blocks east is he?

3. How many blocks did he walk?


1. Find the taxicab distance between the points (1, 1) and (–2, 4).

2. Draw the taxicab circle with radius 3 and center (2, –3).

Warm-Ups: 1. 7 2. 5 3. 12

Lesson Quiz: 1. 6 2. 1

1 x

y

ANSWERS

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LESSON


25Geometry



Classify each angle as acute, obtuse, or right.

1. 908 2. 728

3. 1168

4. How do you know that ∠1 > ∠2?

2

1

Warm-Ups: 1. right 2. acute 3. obtuse 4. Alt. Int. ? Thm.

ANSWERS

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4.1

26Geometry



1. When are two angles congruent?

2. In nABC, if m∠A 5 648 and m∠B 5 718, what is m∠C?

3. What property of angle congruence is illustrated by this statement? If ∠A > ∠B and ∠B > ∠C, then ∠A > ∠C.


1. Graph nABC with vertices A(0, 6),

1

x

y

1

A

CB

B(24, 21), and C(4, 21). Classify it by its sides. Then determine if it is a right triangle.

2. Find x. Then classify the triangle

(2x)8508

(3x 1 20)8

by its angles.

3. Find the measure of the exterior

(x 1 72)8

(2x)8

408

G

angle shown.

4. Find x and y.

508x8

y8

328

Warm-Ups: 1. when they have the same measure

2. 458 3. Transitive Property

Lesson Quiz: 1. isosceles; not a right triangle 2. 22; acute 3. 1048 4. 82, 58

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4.2

27Geometry



N O

M

R

P

1. Write a congruence statement.

2. How do you know that ∠N > ∠R?

3. Find x.

(7x 2 50)8

(3x)8

(2x 1 10)8


In the diagram, nABC > nDEF. Complete each statement.

1. m∠A 5 ?

A

E

F

CD

B

408

808

2. } FD > ?

3. nEDF > ?

4. Write a congruence statement for the two small triangles.

Z YW

X

Explain your reasoning.

Warm-Ups: 1. nMNO > nPRQ 2. Third ? Thm. 3. 30

Lesson Quiz: 1. 608 2. } CA 3. nBAC

4. nWXZ > nYXZ; The diagram tells us that ∠W > ∠Y and ∠WZX > ∠YZX. ∠WXZ > ∠YXZ by the Third ? Thm. From the diagram } WX > } YX and } WZ > } YZ , and } XZ > } XZ by Refl . Prop. of >

ANSWERS

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4.3

28Geometry



Given: } DF bisects } CE , } DC > } DE

Prove: nCDF > nEDF

F

E

C

D


1. The vertices of nGHI and nRST are G(22, 5), H(2, 5), I(22, 2), R(29, 8), S(25, 8), and T(29, 5). Is nGHI > nRST ? Explain.

2. Is nABC > nXYZ? Explain.

4

4

5

5

6

6

X

Y

A

BC

Z

Warm-Ups: It is given that } DC > } DE and } DF bisects } CE . } CF > } EF by the def. of bisector. } DF > } DF by the Refl . Prop. of > Segs. So nCDF > nEDF by the SSS > Post.

Lesson Quiz:

1. Yes. GH 5 RS 5 4, HI 5 ST 5 5, and IG 5 TR 5 3. By the SSS > Post., it follows that nGHI > nRST. 2. Yes. By the Seg. Add. Post., } AC > } XZ . Also, } AB > } XY and } BC > } YZ . So nABC > nXYZ by the SSS > Post.

ANSWERS

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4.4

29Geometry



Warm-Ups: Yes; HL > Thm.

Lesson Quiz: 1. SAS > Post. 2. HL > Thm.

3. ∠S > ∠Y 4. } ST > } YZ

ANSWERS

Tell whether the pair of triangles is congruent or not and why.


Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem.

1. nABE, nCBD 2. nFGH, nHJK

D E CA

B

J

H

G F

K

State a third congruence that would

R T

S

X Z

Y

allow you to prove nRST > nXYZ by the SAS Congruence Postulate.

3. } ST > } YZ , } RS > } XY

4. ∠T > ∠Z, } RT > } XZ

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4.5

30Geometry



Suppose that nXYZ > nRST. Complete each statement.

1. } XY > ? 2. ∠Z > ?

3. m∠S 5 m∠ ?

4. If ∠A > ∠B, m∠A 5 (2x 1 40)8, and m∠B 5 (3x 2 10)8, fi nd x.


Tell whether each pair of triangles is congruent by SAS, ASA, SSS, AAS, or HL. If it is not possible to prove the triangles congruent, write not necessarily congruent.

1. 2.

3. Write a fl ow proof.

A D C

B

Given: } BD bisects ∠ABC, ∠A > ∠CProve: nABD > nCBD

Warm-Ups: 1. } RS 2. ∠T 3. Y 4. 50

Lesson Quiz:

1. ASA 2. not necessarily congruent 3.

∠ABD > ∠CBD

Given

Def. of ∠ Bisector

Given

∠A > ∠C

AAS > Thm.

Reflex. Prop.of > Segs.

nABD > nCBD

BD > BD

BD bisects ∠ABC.

ANSWERS

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4.6

31Geometry



Warm-Ups: 1. equilateral 2. isosceles 3. scalene4. 608 5. 778, 778

Lesson Quiz: 1. nAEC > nDEB by the AAS Cong. Thm. or by the ASA Cong. Post. 2. Show } LM > } LM by the Refl . Prop. of > Segs. Hence nOLM > nNML by the SAS Cong. Post. This gives ∠NLM > ∠OML, since Corr. Parts of > ns are >. So∠1 > ∠2 by the Vert. ? Thm. and properties of > ?.

ANSWERS

Classify each triangle by its sides.

1. 2 cm, 2 cm, 2 cm 2. 7 ft, 11 ft, 7 ft

3. 9 m, 8 m, 10 m

4. In nABC, if m∠A 5 708 and m∠B 5 508, what is m∠C ?

5. In nDEF, if m∠D 5 m∠E and m∠F 5 268, what are the

measures of ∠D and ∠E?


1. Tell which triangles you can show are congruent in order to prove AE 5 DE. What postulate or theorem would you use?

B D

E

A C

2. Write a plan to prove ∠1 > ∠2. L

NO

2M1

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4.7

32Geometry



Warm-Ups: 1. 13 2. (9, 5) 3. Yes, by the SSS Cong. Post. or by the SAS Cong. Post.

Lesson Quiz: 1. 8 2. 3 3. 348, 348 4. 66 cm

ANSWERS

1. Find the length of } AB for A(2, 7) and B(7, 25).

2. What point is 6 units to the right of (3, 5)?

3. Are these triangles congruent?

1

x

y

21


Find the value of x.

1.

x

8

2.

7x 1 3

24

19

3. If the measure of the vertex angle of an isosceles triangle is 1128,

what are the measures of the base angles?

4. Find the perimeter of the triangle.

(3x 2 5) cm

(2x 1 4) cm(x 1 13) cm

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4.8

33Geometry



1. Use coordinate notation to describe the translation 3 units to the

left and 1 unit up.

2. Use a refl ection in the x-axis to

1

x

y

21

draw the other half of the fi gure.

3. Tell whether } XY is a rotation of } GH about the origin if the points are X(�6, 2), Y(�4, 3), G(6, �2), and H(4, �3). If so, give the angle and direction of rotation.

Lesson Quiz: 1. (x, y) r (x � 3, y � 1)

2.

1

x

y

21

3. Yes; 180� clockwise orcounterclockwise

ANSWERS

LESSON


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34Geometry



In Exercises 1–4, use A(0, 10), B(24, 0), and C(0, 0).

1. Find AB. 2. Find the midpoint of } CA .

3. Find the midpoint of } AB . 4. Find the slope of } AB .

Warm-Ups: 1. 26 2. (0, 5) 3. (12, 5) 4. 2 5 } 12

ANSWERS

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5.1

35Geometry



1. Solve 3x 5 8x 2 15. 2. Solve 6x 1 3 5 8x 2 14.

3. If M is the midpoint of } AB , AM 5 5x 2 2, and MB 5 3x 1 6, fi nd AB.


Use the fi gure below for Exercises 1–4.

1. If UV 5 13, fi nd RT.

2. If ST 5 20, fi nd UW.

R W T

VU

S

3. If the perimeter of nRST 5 68 inches, fi nd the perimeter of nUVW.

4. If VW 5 2x 2 4 and RS 5 3x 2 3, what is VW?

5. Place a rectangle in a coordinate plane so its vertical side has length a and its horizontal side has width 2a. Label the coordinates of each vertex.

Warm-Ups: 1. 3 2. 8.5 3. 36

Lesson Quiz: 1. 26 2. 10 3. 34 in. 4. 65.

x

y

(0, 0)

(0, a)

(2a, 0)

(2a, a)

ANSWERS

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5.2

36Geometry



1. ]

› AD bisects ∠BAC. Find x.

C

D

B

A

(5x 2 32.5)8

(3x 1 7.5)8

2. Solve x2 1 122 5 169.

3. The legs of a right triangle measure 15 feet and 20 feet. Find the length of the hypotenuse.


In Exercises 1 and 2, fi nd AB.

1.

A C

B

4x 2 73x 1 1

2. C

AB 7x

5x 1 7

3. In this diagram, the perpendicular

C

G

E

B

D

FA

7

5bisectors of nABC meet at point G. Find EC and GC.

Warm-Ups: 1. 20 2. 65 3. 25 ft

Lesson Quiz: 1. 25 2. 24.5 3. 5; 7

ANSWERS

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5.3

37Geometry



1. For A(24, 8) and B(5, 8), fi nd the midpoint of } AB .

2. For A(23, 2) and B(4, 21), fi nd the length of } AB .

3. For A(0, 4) and C(18, 4), fi nd the length of } AB , where B is a

point 2 }

3 the distance from A to C.



1.

(3x 1 4)8

(4x 2 8)8

2. (4x 1 10)8

3. Point D is the incenter of nXYZ. Find DB.

BD

2624

Z

C

X

A

Y

Warm-Ups: 1. 1 1 } 2 , 8 2 2. Ï

}

58 3. 12

Lesson Quiz: 1. 12 2. 20 3. 10

ANSWERS

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5.4

38Geometry



1. Solve 3x 1 8 < 29.

2. Solve 15 > 22x 2 9.

3. Solve 2x 2 2 < (3x 2 4) 1 (x 2 8).

4. Find all integer solutions for 4 < x < 11.

5. A triangle has angle measures 828 and 348. Find the measure of the third angle.


In Exercises 1–3, use the diagram. G is the centroid of nABC.

E

B

CF

G

A

D 1. If BG 5 9, fi nd BF.

2. If BD 5 12, fi nd AD.

3. If CD 5 27, fi nd GC.

4. Find the centroid of nABC.

1

x

y B

C

A

21

5. Which type of triangle has its orthocenter on the triangle?

Warm-Ups: 1. x < 7 2. x > 212 3. x > 5 4. 5, 6, 7, 8, 9, 10 5. 648

Lesson Quiz: 1. 13.5 2. 12 3. 18 4. (1, 1)5. a right triangle

ANSWERS

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5.5

39Geometry



1. Write the if-then form, converse, inverse, and contrapositive of the given statement. 3x 2 8 5 22 because x 5 10.

2. In nABC, BC 5 18, AB 5 13, and AC 5 16. List the angles of the triangle from least to greatest.


In Exercises 1 and 2, list the sides or angles in order from least to greatest.

1.

508608

708

A C

B 2.

R 30

2546

S

T

Tell whether the side lengths can form a triangle.

3. 37 m, 35 m, 18 m

4. 3 ft, 3 ft, 6 ft

5. Jeremy wants to build a triangular toy using sticks. He has one stick that is 12 inches and another that is 10 inches. What are the possible lengths of the third side of the triangle?

Warm-Ups: 1. If x 5 10, then 3x 2 8 5 22. Con: If 3x 2 8 5 22, then x 5 10. Inv: If x Þ 10, then 3x 2 8 Þ 22. Contra: If 3x 2 8 Þ 22, then x Þ 10. 2. ∠C, ∠B, ∠A

Lesson Quiz: 1. } AB , } BC , } AC 2. ∠R, ∠T, ∠S

3. yes 4. no 5. greater than 2 in. and less than 22 in.

ANSWERS

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5.6

40Geometry



Complete each statement with < , > , or �.

1. KL ? NP 2. m∠1 ? m∠2

LJ

M

N

P

K

758

808

Y

ZW

X

1222

28

2

128

3. Suppose you want to write an indirect proof of this statement: “In �ABC, if m∠A > 90� then �ABC is not a right triangle.” What temporary assumption should start your proof?

Lesson Quiz: 1. � 2. �

3. Assume �ABC is a right triangle.

ANSWERS


LESSON

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41Algebra 1



1. Simplify 3x2

_ 9x

. 2. Solve 2n � 18 p 32.

3. Solve 4x � 2x � 12. 4. Solve x � ��

36 p 16 .

5. A triangle has angle measures 18� and 82�. Find the measure of the third angle.

Warm-Ups: 1. x �

3 2. 288 3. 6 4. 24 5. 80�

ANSWERS


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42Geometry



Solve each proportion.

1. 3 �

8 �

6 � x � 1

2. 8 �

3 �

x � 1 � 6

3. 3 � 6 �

8 � x � 1

4. The area of a rectangle is 750 square meters. The ratio of the width to the length is 5 : 6. Find the width and the length.


1. Simplify the ratio 1200 cm : 1.8 m.

2. The perimeter of a rectangle is 528 millimeters. The ratio of the length to the width is 8 : 3. Find the length and the width.

3. Solve 1 �

s � 8 �

3 � 36 .

4. Find the geometric mean of 42 and 12.

5. The extended ratio of the angles of a triangle is 5 : 12 : 13. Find the angle measures of the triangle.

6. The area of a rectangle is 720 square inches. If the ratio of the length to width is 5 : 4, fi nd the perimeter of the rectangle.

Warm-Ups: 1. 15 2. 15 3. 15 4. width: 25 m; length: 30 m

Lesson Quiz: 1. 2 : 3 2. 192 mm, 72 mm 3. 4 4. 6 ��

14 5. 30�, 72�, 78� 6. 108 in.

ANSWERS


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43Geometry



1. Solve 12

� 10

� x � 60 .

2. The scale of a map is 1 cm : 10 mi. The actual distance between two towns is 4.3 miles. Find the length on the map.

3. A model train engine is 9 centimeters long. The actual engine is 18 meters long. What is the scale of the model?


1. In the diagram, RS

� XY

� ST

� YZ . Write four true proportions.

R T

S

4524

X Z

Y30

x

2. In the diagram, MR

� RN

� PS

� SN .

M

P

S

RN

20

12

18x

Find MN and MR.

3. On a map, a street is 2 inches long. The actual length is 1 mile. Find the scale.

4. A map is scaled so that 1 cm represents 20 km. If two towns are 8.2 centimeters apart on the map, what is the actual distance between the towns?

Warm-Ups: 1. 72 2. 0.43 cm 3. 1 cm : 2 m

Lesson Quiz: 1. Sample answer: 45

� 30

� 24

� x ; 30

� 45

� x � 24 ;

45 �

24 �

30 � x ;

45 � 30

� 30

� 24 � x

� x 2. 48; 30 3. 1 in. : 0.5 mi 4. 164 km

ANSWERS


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44Geometry



1. In nABC and nXZW, m∠A � m∠X and m∠B � m∠Z. What can you conclude about m∠C and m∠W?

2. Solve x �

18 �

54 � 9 .

3. nABC � nDEF. Find x.

CA

B

8

12 FD

E

x

15


1. Determine whether the polygons

E G

H

F

30

26 22

16

K M

N

L 15

13 11

8 are similar. If they are, writea similarity statement and fi nd the scale factor of EFGH to KLMN.

2. In the diagram, nDEF � nHJK. Find the value of x.

D F

E

1012

H K

J

8

9x

3. Two similar triangles have the scale factor 5 : 4. Find the ratio of their corresponding altitudes and medians.

4. Two similar triangles have the scale factor 3 : 7. Find the ratio of their corresponding perimeters and areas.

Warm-Ups: 1. They are the same. 2. 108 3. 10

Lesson Quiz: 1. yes; EFGH � KLMN; the scale factor is 2 : 1. 2. 13.5 3. 5 : 4; 5 : 4 4. 3 : 7; 9 : 49

ANSWERS


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45Geometry



Determine whether the two triangles are similar.

1. nABC: m∠A � 90�, m∠B � 44�; nDEF: m∠D � 90�, m∠E � 46�

2. nABC: m∠A � 132�, m∠B � 24�; nDEF: m∠D � 90�, m∠F � 24�

3. Solve 6 �

12 �

x � 1 � 8 .


Determine if the two triangles are similar. If they are, write a similarity statement.

1.

D C

BE

A

558

558

2. NM

Y

P

X

558 708

458 3. Find the length of � BC .

BA

D

8

E

12

5C

x

4. A tree casts a shadow that is 30 feet long. At the same time a per-son standing nearby, who is fi ve feet two inches tall, casts a shadow that is 50 inches long. How tall is the tree to the nearest foot?

Warm-Ups: 1. similar 2. not similar 3. 5

Lesson Quiz: 1. yes; nABE � nACD 2. no 3. 7.5 4. 37 ft

ANSWERS


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46Geometry



1. Give the postulate or theorem that justifi es why the triangles are similar.

808

808

2. Solve 16

� 32

� 12 � x

� x .


1. Verify that nABC � nDEF for the given information. nABC: AC � 6, AB � 9, BC � 12; nDEF: DF � 2, DE � 3, EF � 4

2. Show that the triangles are similar and write a similarity statement. Explain your reasoning.

60

45

40

30

1508

1508

CA

ZX

B

Y

Warm-Ups: 1. AA Similarity Postulate 2. 8

Lesson Quiz: 1. AC

� DF � AB

� DE � BC

� EF � 3 � 1 . The ratios are

equal, so nABC � nDEF by the SSS Similarity Theorem.

2. XY

� AB

� YZ

� BC � 3 � 4 and ∠Y � ∠B. So nXYZ � nABC by the SAS

Similarity Theorem.

ANSWERS


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47Geometry



Give the coordinates of a point twice as far from the origin along a ray from (0, 0).

1. (3, 5) 2. (�2, 0) 3. � 1 __ 2 , � 1 __ 2 � Give the coordinates of a point one-half as far from the origin along a ray from (0, 0).

4. (6, �4) 5. (0, �7)


Find the value of the variable.

1. 20

106

x

2.

x

12 16

18

3.

1410

y17

1 2

∠1 > ∠2

4. This diagram represents a tract

Webster Street

Center Street371.9 yd

80 yd

85 yd

120 yd 150 yd

Lot CLot A

Lot B40 yd

80 yd

110 yd

130 yd

of land being developed for homes. Which lot has the greatest perimeter?

Warm-Ups: 1. (6, 10) 2. (�4, 0) 3. (1, �1) 4. (3, �2) 5. (0, �3.5)

Lesson Quiz: 1. 12 2. 13.5 3. 23.8 4. Lot B

ANSWERS


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48Geometry



1. Draw a dilation with scale factor 2 of nABC with vertices A(1, 0), B(0, 3), and C(3, 2). Label the image nDEF.

2. nABC has vertices A(0, 0), B(0, 8), and C(6, 0). Draw a dilation

of nABC using a scale factor of 1 �

2 . Label the image nDEF.

Lesson Quiz:

1.

F

E

C

A

B

D

1

x5

y 2.

FCA

B

D

E

1

x5

y

ANSWERS


LESSON

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1. Determine whether the polygons are similar. If they are, write a similarity statement and fi nd the scale factor of nABC to nDEF.

42 B

A

28

C

52.5

95°

D E12

8

F

15

2. In the fi gure below, if nJKL ~ nRST, fi nd the value of x.

18

x

12

6

159

J L

KS

R T

Lesson QuizFor use after 6.3 Focus On Patterns

1. Find the fi fth, sixth, seventh, and eighth terms of the Fibonacci sequence.

2. Find the ratios of the consecutive terms you found in Question 1.

Warm-Ups: 1. nABC ~ nDEF, 7 �

2 2. 10

Lesson Quiz: 1. 5, 8, 13, 21 2. 8 � 5 � 1.6,

13 �

8 � 1.625,

21 �

13 " 1.6154

ANSWERS

LESSON

6.3 Focus On PatternsWarm-Up Exercises

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1. Solve x2 5 100. 2. Solve x2 1 9 5 25.

3. Simplify Ï}

20 . 4. Find x.

x

10 cm10 cm

12 cm

Warm-Ups: 1. 10, 210 2. 4, 24 3. 2 Ï}

5 4. 6 cm

ANSWERS

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7.1

51Geometry



1. Find x.

5

x2

2. Simplify (5 Ï}

3 )2.

3. Find x. 12x

15


1. Find the length of the 2. Find the area of the hypotenuse of the right triangle. isosceles triangle.

36

x15

h 51 cm51 cm

48 cm

3. Find the unknown side length x.

126

x10Write your answer in simplest radical form.

Warm-Ups: 1. Ï}

29 2. 75 3. 9

Lesson Quiz: 1. 39 2. 1080 cm2 3. 4 Ï}

13

ANSWERS

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7.2

52Geometry



1. Are these triangles similar? If so, give the reason.

708

808

308

808

2. Find x. 14

x

48


1. Tell whether the triangle is a 51

4524

right triangle.

Decide if the segment lengths can form a triangle. If so, tell whether the triangle is acute, right, or obtuse.

2. 7, 11, 3 Ï}

17 3. 4.1, 9.2, 5.6

4. A playground has a slide, a swing, 50 ftSlide Swing

32 ft

Sandbox

48 ft

and a sandbox. The slide and the swing are 50 feet apart, the swing and sandbox are 32 feet apart, and the slide and sandbox are 48 feet apart. Do the three pieces of apparatus form a right triangle?

Warm-Ups: 1. Yes; the AA Similarity Postulate 2. 50

Lesson Quiz: 1. yes 2. yes; acute because 153 < 170 3. yes; obtuse because 84.64 > 48.17 4. no

ANSWERS

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7.3

53Geometry



In Exercises 1–3, simplify.

1. 6 Ï}

2 p Ï}

2 2. 6 }

Ï}

3

3. 5 } Ï

}

2

4. Find m∠DBC in square ABCD.

D

A

C

B


1. Identify the three similar right triangles in the diagram.

X W Z

Y

Find the value of the variable.

2. 3.

a

6

4

x

5

9

Warm-Ups: 1. 12 2. 2 Ï}

3 3. 5 Ï

}

2 }

2 4. 458

Lesson Quiz: 1. nXYZ , nXWY , nYWZ 2. 3 Ï}

5 3. 2 Ï}

15

ANSWERS

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7.4

54Geometry



Find the values of the variables.

1.

y

x6

4 2.

30 x

x


Use these triangles for Exercises 1–4.

a

ba

c

de

608

1. Find a if b 5 10 Ï}

2 . 2. Find b if a 5 19.

3. Find d and e if c 5 4. 4. Find c and d if e 5 50 Ï}

3 .

5. Find x, y, and z.

y

xz

458 308

6

Warm-Ups: 1. x 5 2 Ï}

6 , y 5 2 Ï}

15 2. 15 Ï}

2

Lesson Quiz: 1. 10 2. 19 Ï}

2 3. d 5 4 Ï}

3 , e 5 8

4. c 5 25 Ï}

3 , d 5 75 5. x 5 3 Ï}

2 , y 5 3 Ï}

6 , z 5 6 Ï}

2

ANSWERS

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7.5

55Geometry



Use this diagram for Exercises 1–4.

Y

X Z

1. Name the hypotenuse. 2. Name the leg opposite ∠X.

3. Name the leg adjacent to ∠X. 4. If XY 5 17 and m∠X 5 418, fi nd YZ.



C A

B

b

ca

1. If a 5 18, b 5 80, and c 5 82, fi nd tan B and tan A. Write each answer as a fraction and a decimal rounded to 4 places.

2. If a 5 17 and m∠A 5 318, fi nd b to the nearest tenth.

3. If a 5 9 and m∠B 5 748, fi nd b to the nearest tenth.

4. If a 5 5 and m∠B 5 798, fi nd the number of square units in the area of nABC to the nearest tenth.

Warm-Ups: 1. } XZ 2. } YZ 3. } XY 4. 14.78

Lesson Quiz: 1. tan B 5 40

} 9 5 4.4444;

tan A 5 9 } 40 5 0.2250 2. 28.3 3. 31.4 4. 64.3

ANSWERS

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7.6

56Geometry




1. If PR 5 12 and m∠R 5 198, fi nd p.

2. If m∠P 5 588 and r 5 5, fi nd p.

P

R

r

p

q

3. If m∠P 5 608, and p 5 9, fi nd q.

4. If r 5 8 and p 5 12, fi nd q.



1. If x 5 4 Ï}

5 , y 5 4, and z 5 4 Ï}

6 , fi nd sin X, sin Y, cos X, and cos Y.

Z

X

Y

y

x

z

2. If y 5 10 and m∠Y 5 158, fi nd z to the nearest tenth.

3. If z 5 12 and m∠X 5 848, fi nd y to the nearest tenth.

4. A lamppost is 11 feet tall. If the

498

11 ftx

angle of elevation through the top of the lamppost to the sun is 498, approximately how far is the top of the lamppost from the tip of its shadow?

Warm-Ups: 1. 11.3 2. 8.0 3. 10.4 4. 14.4

Lesson Quiz: 1. sin X 5 Ï

}

30 } 6 ø 0.9129, sin Y 5

Ï}

6 } 6 ø 0.4082,

cos X 5 Ï

}

6 } 6 ø 0.4082, cos Y 5

Ï}

30 } 6 ø 0.9129 2. 38.6

3. 1.3 4. 14.6 ft

ANSWERS

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7.7

57Geometry




Y Z

yz

x

X

1. If x 5 9 and z 5 11, fi nd m∠X to the nearest tenth of a degree.

2. If y 5 5 and z 5 12, fi nd m∠X to the nearest tenth of a degree.

3. If m∠Y 5 17.48 and z 5 12, solve nXYZ.

Lesson Quiz: 1. 54.98 2. 65.4° 3. The angles are 17.48, 72.68, and 908; the sides are 12, about 3.6, and about 11.5.

ANSWERS

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LESSON

7.7



1. Let nABC be a right triangle with acute /A. Defi ne sin A, cos A, and tan A.

AC

B

2. Find sin B, cos B, and tan B.

A C

B

25

15

20

Lesson QuizFor use after 7.6 Focus On Trigonometry

1. Find cot A, sec A, and csc A. Write each answer as a fraction and as a decimal to four places, if necessary.

C

A

B40

419

Warm-Ups: 1. sin A = BC

} AB

, cos A = AC

} AB

, tan A = BC

} AC

2. sin B = 3 } 5 , cos B =

4 } 5 , tan B =

3 }

4

Lesson Quiz: 1. cot A = 9 }

40 = 0.225, sec A =

41 }

9 ø 4.5556, csc A =

41 }

40 = 1.025

ANSWERS

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LESSON

7.6 Focus On TrigonometryWarm-Up Exercises



1. If the measures of two angles of a triangle are 19� and 80�, fi nd the measure of the third angle.

2. Solve (x � 2)180 � 1980.


548 x 8

Warm-Ups: 1. 81� 2. 13 3. 126

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60Geometry




988

1208

828x 8

2. Find the coordinates of the midpoint of � AB with endpoints A(�2, 6) and B(10, 4).

3. Find the distance between (�2, 6) and (10, 4).


1. Find the sum of the measures of the interior angles of a convex 24-gon.

2. The sum of the measures of the interior angles of a convex polygon is 5040�. Classify the polygon by the number of sides.


1108

468

728

x 8

(3x)8

4. Find the measure of an interior angle and an exterior angle of a regular 36-gon.

Warm-Ups: 1. 60 2. (4, 5) 3. 2 ��

37

Lesson Quiz: 1. 3960� 2. 30-gon 3. 33 4. 170�, 10�

ANSWERS


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61Geometry



1. What congruence postulate shows that nABE � nCDE?

D C

A B

E

2. If ∠E � ∠G, fi nd m∠E.

G H

628

F E628


1. Find the values of a and b in the parallelogram.

36830

a 2 12(b 1 10)8

2. Find the measure of ∠K in the parallelogram.

648

J M

K L

Find the indicated measures in parallelogram ABCD.

3. m∠BCD

4. If BE � 9, fi nd BD. 378

1008

B CE

A D

Warm-Ups: 1. SAS 2. 118�

Lesson Quiz: 1. 42, 26 2. 116� 3. 43� 4. 18

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62Geometry



1. Give fi ve ways to prove that a quadrilateral is a parallelogram.

2. Find x in the parallelogram.

2x 1 105x 2 32


Tell how you know that the quadrilateral is a parallelogram.

1. 2.

808

808

1008

3. For what value of x is the 4. In a coordinate plane, draw the quadrilateral a parallelogram? quadrilateral ABCD with vertices

(x 1 30)8

2x 8 A(�1, 3), B(7, �1), C(6, �4), and D(�2, 0). Show that ABCD is a parallelogram.

Warm-Ups: 1. opp. sides �, opp. sides �, diags. bisects each other, opp. angles �, a pair of opp. sides � and � 2. 14

Lesson Quiz: 1. The diagonals bisect each other.2. Both pairs of opposite angles are congruent. You can also conclude that opposite sides are parallel and use the defi nition of a parallelogram. 3. 504. AB � CD � 4 �

� 5 and BC � AD � �

� 10 ,

so two pairs of opposite sides are congruent.

ANSWERS

1x

y

C

D

A

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63Geometry



Use the fi gure to answer the questions.

A B

X Y

558 558

x 8 y 8

1. What are the values of x and y?

2. If �

› AX and

�

› BY intersect at point P, what kind of triangle is nXPY?


Name each type of quadrilateral (parallelogram, rectangle, rhombus, and square) for which the statement is true.

1. Both pairs of opposite angles are congruent.

2. The quadrilateral is equilateral.

3. Given rhombus MNOP, 4. Given rectangle QRST, fi nd m∠NPO. fi nd m∠RUS.

M P

N O488

T

U

R S

648

Warm-Ups: 1. 125, 125 2. isosceles

Lesson Quiz: 1. parallelogram, rectangle, rhombus, square 2. rhombus, square 3. 48� 4. 128�

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64Geometry



Find the value of x. 1. 1158

x 8

2.

x 8

308

3. x 8

208


1. Find m∠A, m∠C, and m∠D. 2. Find the length of the midsegment of the trapezoid.

17

33

A

C

D

B568

Use the fi gure to fi nd the indicated measures.

3. If m∠XYZ � 80� and m∠XWZ � 48�, fi nd m∠YZW.

X

YW

Z

O 4. If XO � 2, OZ � 2, YO � 6, and OW � 8, fi nd the lengths of the sides of the kite.

Warm-Ups: 1. 65 2. 120 3. 70

Lesson Quiz: 1. 124�, 56�, 124� 2. 25 3. 116�

4. 2 ��

10 , 2 ��

17

ANSWERS


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65Geometry



1. Find the distance between the points A(1, �3) and B(�2, 4).

2. Determine if the triangles with the given vertices are similar.

A(�3, 3), B(�4, 1), C(�2, �1)

1

y

21 x

B

C

D

E

F

A D(3, 5), E(2, 1), F(4, �3)


For Exercises 1–3, write true or false.

1. The diagonals of a rectangle are always perpendicular.

2. The diagonals of a rhombus are always congruent.

3. One pair of opposite angles of a kite are congruent.

4. Give the most specifi c name for the quadrilateral. Explain.

5. Points A(1, 4), B(6, 21), C(1, 26), and D(24, 21) are the vertices of a quadrilateral. Give the most specifi c name for ABCD. Explain.

Warm-Ups: 1. ��

58 2. nABC and nDEF are not similar.

Lesson Quiz: 1. false 2. false 3. true 4. Rhombus; it is a � since two pairs of opp. � are �. Since two consec. sides are �, all sides are �. 5. Square; slope of } AB � slope of } CD � �1, slope of } BC � slope of } AD � 1. Opp. sides are �, so ABCD is a �. Two consec. sides are � and AB 5 BC 5 5 �

� 2 , so ABCD is a square.

ANSWERS

LESSON

8.7 Warm-Up ExercisesCo

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1. Find the side lengths of the quadrilaterals with the given vertices. Then determine if the quadrilaterals are congruent.

J(1, 1), K(2, 4), L(5, 4), M(4, 1); N(�1, �3), O(0, 0), P(4, 0), Q(3, �3)

2. Find the side lengths of the quadrilaterals with the given vertices. Then determine if the quadrilaterals are similar.

R(�2, 3), S(�2, 1), T(�4, 1), U(�4, 3); V(5, 2), W(5, �2), X(1, �2), Y(1, 2)

3. Without introducing any new variables, supply the missing coordinates for G so that EFGH is a rectangle.

y

x

G(?, ?)F(a, c)

E(a, b) H(d, b)

Lesson Quiz: 1. JK � LM � NO � PQ � ��

10 , KL � JM � 3 but OP � NQ � 4 ; not congruent 2. RS � ST � TU � RU � 2, VW � WX � XY � VY � 4; ratios of corresponding side lengths are equal; corresponding angles are congruent; similar 3. (d, c)

ANSWERS


LESSON

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67Geometry



1. What is a translation?

2. Translate A(3, 5) 4 units right and 2 units down. What are the coordinates of the image?

3. Find the length of } BC with endpoints B(23, 5) and C(1, 2).

4. If you translate the points M(4, 7) and N(21, 5) by using (x, y) → (x 2 3, y 1 6), what is the distance from M9 to N9?

Warm-Ups: 1. a transformation that moves every point of a fi gure the same distance in the same direction 2. (7, 3) 3. 5 4. Ï

}

29

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68Geometry



1. Write the component form of the

1

x

y

5

B9

A9

C9

B

AC

vector that translates nABC to nA9B9C9.

2. If P(28, 4) is translated by (x, y) → (x 1 6, y 1 1), what is the image of P?


1. The vertices of nXYZ are X(23, 2), Y(1, 1), and Z(21, 23). Graph the image of the triangle using prime notation after the translation (x, y) → (x 1 1, y 2 2).

2. Translate A(28, 6) using D

C

(x, y) → (x 2 7, y 1 10).

3. Name the vector and write it in component form.

4. Use the point M(8, 22). Find the component form of the vector that describes the translation to M9(7, 5).

Warm-Ups: 1. ⟨26, 21⟩ 2. (22, 5)

Lesson Quiz: 1.

1

x

y

24

YX

Z

Y9X9

Z9

2. (215, 16)

3. ### Y CD , ⟨4, 6⟩

4.⟨21, 7⟩

ANSWERS


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69Geometry



1. The y-axis is the perpendicular bisector of } AB . If point A is at (23, 5), what is the location of point B?

2. What is the slope of the segment with endpoints at (27, 4) and (4, 27)?

3. Multiply F 21 0 0 1GF 6 22 1

3 4 7G.


Add or subtract

1. [2.8 29.2] 1 [3.5 6.1]

2. F 8 22 23 5 G 2 F 4 3

1 27G 3. Triangle ABC has vertices A(2, 21), B(1, 3), and C(22, 22).

Write a matrix equation that shows how to fi nd the image matrix that represents the translation of nABC 4 units left and 5 units up.

4. Multiply F 2 25 1 0GF 6 21

3 22G.

Warm-Ups: 1. (3, 5) 2. 21 3. F 26 2 21 3 4 7G

Lesson Quiz: 1. [6.3 23.1] 2. F 4 25 24 12G

3. F 24 24 24 5 5 5G 1 F 2 1 22

21 3 22 G 5 F 22 23 26 4 8 3G 4. F 23 8

6 21G

ANSWERS


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70Geometry



1. Use a protractor to draw an angle with measure 208.

2. Given the points A(3, 5), B(25, 3), and the origin O(0, 0), fi nd OA, OB, and m∠BOA.

3. Multiply F 0 1 21 0GF 2 3

5 27G.


Graph the polygon and its refl ection in the given line.

1. Quadrilateral with vertices A(22, 1), B(1, 4), C(3, 1), D(2, 21) over y 5 1

2. Triangle P(22, 2), Q(3, 4), R(4, 1) over y 5 x

3. Use matrix multiplication to fi nd the image A B C

matrix when nABC is refl ected in the y-axis. F 2 21 0 3 4 21G

4. Find point C on the x-axis so AC 1 BC is a minimum for A(2, 5) and B(7, 3).

Warm-Ups: 1. 208

2. Ï}

34 , Ï}

34 , 908 3.

F 5 27 22 23G

Lesson Quiz: 1.

4

x

y

21

CC9

B

B9

AA9

D9

D

2.

1

x

y

21

9

R

R9

P

P9

3. A9 B9 C9 4. 1 41 }

8 , 0 2 F 22 1 0

3 4 21G

ANSWERS


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71Geometry



Find the image of (2, 3) under each transformation.

1. translation (x, y) → (x 2 6, y 2 2)

2. refl ection in the y-axis

3. refl ection in the line y 5 6

4. rotation 908 about the origin


1. Construct an equilateral triangle DEF. Then draw the image of nDEF for a 308 rotation about the center C of the triangle.

In Exercises 2 and 3, use the quadrilateral HIJK with vertices H (3, 0), I(1, 24), J(0, 24), and K(21, 23).

2. Use the coordinate rules to name the vertices of the image of HIJK for a rotation of 1808 about the origin.

3. Use matrix multiplication to fi nd the image matrix that represents a 2708 rotation of HIJK about the origin. Then graph the polygon and its image.

Warm-Ups: 1. (24, 1) 2. (22, 3) 3. (2, 9) 4. (23, 2)

Lesson Quiz:

1. E

FD

E9

F 9

D 9

C

2. H9(23, 0), I9(21, 4), J9(0, 4), K9(1, 3)

3. H9 I9 J9 K9

F 0 24 24 23 23 21 0 1G

1

x

y

21

I

H

KJ

I9

J9

K9

H9

ANSWERS


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72Geometry



Find the coordinates of the image of A(23, 8) for each transformation.

1. Refl ection in the line y 5 x

2. Rotation of 908 about the origin

3. Translation 4 units down


1. The endpoints of } AB are A(3, 2) and B(1, 4). Graph the image of } AB after the glide refl ection. Translation: (x, y) → (x, y 2 2) Refl ection: in the line x 5 1

2. The vertices of nMNK are M(1, 1), N(2, 3), and K(0, 2). Graph the image of nMNK after the composition of the refl ection followed by the rotation. Refl ection: in the y-axis Rotation: 1808 about the origin

Warm-Ups: 1. (8, 23) 2. (28, 23) 3. (23, 4)

Lesson Quiz:

1.

2

x

y

21

B

AB 0B9

A9

x 5 1

A0

2.

1

x

y

21 M 0

N 0

K 0

N

M

K9N9

M9

K

ANSWERS


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73Geometry



If nABC ~ nDEF, fi nd each value.

20

18

10

A

C

B

8D

F

E

1. EF 2. AC 3. scale factor


How many lines of symmetry does each fi gure have?

1. a rhombus that is not a square 2. equilateral triangle

Does the fi gure have rotational symmetry? If so, describe the rotations that map the fi gure onto itself.

3. 4.

Warm-Ups: 1. 36

} 5 2. 25 3. 5 }

2

Lesson Quiz: 1. 2 2. 3 3. yes; 728, 1448 4. no

ANSWERS


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74Geometry



1. Find the scale factor. Tell whether the dilation is a reduction or an enlargement. Find the value of x.

3 2

2x

C

P 9

P

2. Find the image matrix that represents a dilation of nXYZ with vertices X(3, 2), Y(8, 21), and Z(21, 21) centered at the origin

with scale factor 1 }

2 . Then graph the polygon and its image.

Lesson Quiz: 1. 5 }

3 ; enlargement; 1.2

2. X' Y' Z'

F 1.5 4 20.5

1 20.5 20.5 G

ANSWERS

1

x

y

6

Y

X

Y 9

ZZ9

X 9

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75Geometry



What is the measure of each interior angle of each regular polygon?

1. equilateral triangle

2. pentagon

3. hexagon

4. octagon

5. Draw a trapezoid and its image for a refl ection in one base.

Lesson QuizFor use after 9.5 Focus On Patterns

1. Make a tessellation using a rectangle. Change both pairs of opposite sides.

2. Describe the transformation(s) used to make the tessellation.

Warm-Ups: 1. 60° 2. 108° 3. 120° 4. 135° 5.

Lesson Quiz:

1. Sample: 2. translation and refl ection, or translation and rotation

ANSWERS

LESSON

9.5 Focus On PatternsWarm-Up Exercises

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76Geometry



1. What measure is needed to fi nd the circumference or area of a circle?

2. Find the radius of a circle with diameter 8 centimeters.

3. A right triangle has legs with lengths 5 inches and 12 inches. Find the length of the hypotenuse.

4. Solve 6x 1 15 5 33.

5. Solve (x 1 18)2 5 x2 1 242.

Warm-Ups: 1. radius or diameter 2. 4 cm 3. 13 in. 4. 3 5. 7

ANSWERS


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77Geometry



Find x and y.

1. x 8

y 8

1208 2.

x 8y 8

408

708


1. Give the name that best 2. Tell how many common describes the fi gure. tangents the circles have.

B

E

C

P FD

A

a. ‹ ]

› CD b.

‹ ] › AB c. } FD d. } EP

3. Is } AB tangent to (C? Explain. 4. Find x.

B C

A

34

30 16

A

D

B

2x 1 11

3x 2 1

C

Warm-Ups: 1. x 5 60; y 5 60 2. x 5 35; y 5 35

Lesson Quiz: 1. a. secant, b. tangent, c. chord, d. radius 2. One tangent; it is a vertical line through the point of tangency. 3. Yes; 162 1 302 5 1156 5 342 so } AB ⊥ } AC , and a line ⊥ to a radius at its endpoint is tangent to the circle. 4. 12

ANSWERS


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78Geometry



Tell whether the segment is best described as a radius, chord, or diameter of (C.

BA

DE

C

1. } DC 5. Solve 4x 5 8x 2 12.

2. } BD 6. Solve 3x 1 2 5 6x 2 4.

3. } DE

4. } AE


Use this diagram of (F for Exercises 1–4.

Describe each arc as a minor arc, major arc,

CE

AB

F328

or semicircle. Find the arc measure.

1. C BC

2. C CBE

3. C BCE

4. Explain why C AE > C BC .

5. Two diameters of (P are } AB and } CD . If m C AD 5 508, fi nd m C ACD and m C AC .

Warm-Ups: 1. radius 2. diameter 3. chord 4. chord 5. 3 6. 2

Lesson Quiz: 1. minor arc, 328 2. major arc, 2128 3. semi-circle, 1808 4. m∠AFE 5 m∠BFC because the angles are

vertical angles, so ∠AFE > ∠BFC. Then arcs C AE and C BC are arcs that

have the same measure in the same circle. By defi nition C AE > C BC . 5. 3108; 1308

ANSWERS


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79Geometry



1. The measure of the interior angles of a quadrilateral are 808, 1008, 558, and 5x8. Find the value of x.

2. Two supplementary angles have measures 6x8 and 12x8. Find each angle measure.

3. Solve 3x 5 1 } 2 (4x 1 12). 4. Solve 80 5

1 } 2 (360 2 2x).


Find the value of x in (C. Explain.

1.

C

3x 5x 2 12

2.

18C

4x 1 2

3. Determine whether } RS R

S

T UV

is a diameter. Explain.

Warm-Ups: 1. 25 2. 608; 1208 3. 6 4. 100

Lesson Quiz: 1. 6; If a diameter of a circle is ⊥ to a chord, then the diameter bisects the chord and its arc. 2. 4; In the same circle, if two chords are equidistant from the center, then they are >. 3. Yes. Sample answer: } RS is the ⊥ bisector of } TU by Theorem 5.3. Then } RS is a diameter of the circle by Theorem 10.4.

ANSWERS


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80Geometry



Solve for x.

1. 2x 5 84 1 32 2. x 5 1 } 2 (360 2 120)

3. 180 2 x 5 1 } 2 ((2x 1 4) 1 28)

4. One-half of the measure of an angle plus the angle’s supplement is equal to the measure of the angle. Find the measure of the angle.



1.

768

x 8

2.

288

x 8 3.

448

x 8

Find the value of each variable.

4. 1008

4y 8728

2x 8

5.

15x 89.5y 8

8.5y 821x 8

Warm-Ups: 1. 58 2. 120 3. 82 4. 1208

Lesson Quiz: 1. 38 2. 56 3. 44 4. x 5 54; y 5 20 5. x 5 5; y 5 10

ANSWERS


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81Geometry



Solve the equation.

1. x2 1 4x 1 3 5 0 2. 6x 2 9 5 x2

3. 3(x 1 4) 5 (2 1 4)(4) 4. 21(x 2 4) 5 (2x 2 7)(x 1 2)

5. If 10 is multiplied by 10 more than a number, the product is the square of 24. Find the number.


Line t is tangent to the circle. Find the value of x and y. 1.

t

2308

x 8

2.

688

368

x 8

3.

228 728

y 8

4.

t278

1358x 8

y 8

Warm-Ups: 1. 21, 23 2. 3 3. 4 4. 5, 7 5. 47.6

Lesson Quiz: 1. 115 2. 52 3. 25 4. x 5 81, y 5 40.5

ANSWERS


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82Geometry



Evaluate each expression.

1. Ï}}

(10 2 2)2 1 (8 1 7)2

2. Ï}}

(23 2 2)2 1 (28 2 4)2

3. In a map on a coordinate grid, two towns are at coordinates (2, 3) and (21, 4). If each unit on the grid is 1 kilometer, whatis the distance between the towns?


Find the value of x. Round to the nearest hundredth, if necessary.

1.

10

4

20

x

2.

18 4

x

x 1 1

3.

16

9

x

4.

45

40

18

x

5.

8 16x

x

Warm-Ups: 1. 17 2. 13 3. Ï}

10 km or about 3.16 km

Lesson Quiz: 1. 2 2. 8 3. 15 4. 20.25 5. 11.31

ANSWERS


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83Geometry



1. Write the standard equation of a circle with center (23, 5) and radius 2.

2. Graph the circle (x 1 1)2 1 ( y 2 3)2 5 9.

3. The line x 5 2 is tangent to the circle in Exercise 2. Find the coordinates of the point of tangency.

4. A circle has equation x2 1 y2 5 4. Determine whether line y 5 2x 1 2 is a secant, tangent, or neither.

Lesson Quiz: 1. (x 1 3)2 1 ( y 2 5)2 5 4

2.

1

x

y

21

3. (2, 3) 4. secant

ANSWERS

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1. List the properties that make a quadrilateral a parallelogram.

2. Solve x2 1 x2 5 102.

3. Solve 28 5 1 } 2 x(8).

4. Solve 12x 5 84.

Warm-Ups: 1. Opposite sides are congruent and parallel; opposite angles are congruent; consecutive angles are supplementary;

diagonals bisect each other 2. Ï}

51 3. 7 4. 7

ANSWERS


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85Geometry



1. Find the area of a parallelogram with height 12 feet and base 8 feet.

2. Find the area of an isosceles triangle with sides 20, 20, and 24 units.

3. What is the length of the midsegment of a trapezoid with bases 10 inches and 15 inches?

4. Solve 200 5 1 } 2 x(18 1 22).


Find the area of the polygon.

1.

20

24

2.

24

40

3. ABCE 4.

33 ft

19 ft20 ft

25 30

15

18

A

F

B

D E

C

5. Find the area of a triangle with side lengths 13 inches, 13 inches, and 10 inches.

Warm-Ups: 1. 96 ft2 2. 192 square units 3. 12.5 in. 4. 10

Lesson Quiz: 1. 480 square units 2. 480 square units 3. 375.4 square units 4. 973.5 ft2 5. 60 in.2

ANSWERS


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86Geometry



1. Two polygons are similar and the ratio of corresponding sides is 3 : 4. What is the ratio of the perimeters?

2. Solve 12

} x 5

18 } 6 .

3. A rectangle has area 108 square feet and the length is three times the width. What are the dimensions of the rectangle?


Find the area of each fi gure.

1. BA

CD1

x

y

21

2. 8 cm

36 cm

3.

13 ft

13 ft

15 ft

5 ft

4. 20 ft

38 ft

16 ft

5. The height of a trapezoid is 8 feet. One base is twice as long as the other, and the area is 72 square feet. How long is each base?

Warm-Ups: 1. 3 : 4 2. 4 3. 18 ft by 6 ft

Lesson Quiz: 1. 20 square units 2. 144 cm2 3. 135 ft2 4. 1072 ft2

5. 6 ft; 12 ft

ANSWERS


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87Geometry



Simplify the expression.

1. 45

} 360

2. 120

} 360

(9π) 3. 15π

} 3 4.

60 }

360 (15π)

5. A central angle of a circle measures 828. Find the measure of the intercepted arc.

6. Solve 3.8

} x 5

120 } 360 .


Figure I~Figure II. Find the ratio of the perimeters and the ratio of the areas. Then fi nd the unknown area.

1.

24 ft

18 ftI

II

A 5 60 ft2

2.

18 cm

12 cmI

II

A 5 780 cm2

3. Rectangle I ~ Rectangle II. In Rectangle I, the length is 20 feet and the perimeter is 64 feet. In Rectangle II, the width is 8 yards. Find the ratio of the area of Rectangle I to the area of Rectangle II.

Warm-Ups: 1. 1 }

8 2. 3π 3. 5π 4.

5π }

2 5. 828 6. 11.4

Lesson Quiz:

1. ratio of perimeters: 4 : 3; ratio of areas: 16 : 9; 33.75 ft2

2. ratio of perimeters: 2 : 3; ratio of areas: 4 : 9; 346 2 }

3 cm2 3. 1: 4

ANSWERS


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88Geometry



Find the area of a circle with the given measure.

1. Radius: 8 cm 2. Radius: 1 1 }

3 mi 3. Circumference: 4π ft

4. Solve x2 5 100

} π .


Find the indicated measure.

1. Circumference 2. Radius 3. length of C AB

13 in.

r

C 5 48 ft

9 cm

A

B558

4. Find the total circumference 32 cm

16 cm

of the circles.

Find the indicated measure.

5. Radius 6. Circumference

P r

42 cm

1108

92.8 in.

1508

Warm-Ups: 1. about 201.06 cm2 2. about 5.59 mi2 3. 4π ft2 4. about 65.64

Lesson Quiz: 1. about 81.68 in. 2. about 7.64 ft 3. 8.64 cm 4. 100.53 cm 5. about 21.88 cm 6. about 222.72 in.

ANSWERS


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89Geometry



1. An isosceles triangle has side lengths 20 meters, 26 meters, and 26 meters. Find the length of the altitude to the base.

2. Solve 18 5 1 } 2 p x.

3. Evaluate (4 cos 108)(10 sin 208).

4. Evaluate 10 }

tan 888 .


1. Find the area of the 2. Use the area of the sector to fi nd sectors formed by ∠DEF. the area of (R.

9 in.

D

EF

1508

R

1108

A 5 64.5 cm2

3. Find the area of the shaded region.

6 ft

18 ft

Warm-Ups: 1. 24 m 2. 36 3. 13.47 4. 0.3492

Lesson Quiz: 1. 106.03 in.2, 148.44 in.2 2. 211.09 cm2 3. 23.18 ft2

ANSWERS


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90Geometry



1. Find the area of a circle with radius 5.2 cm.

2. Find the area of a right triangle with hypotenuse 18 in. and side length 8 in.

3. A rectangle with sides 4 inches and 5 inches is cut out of a square with sides 8 inches. Find the area that remains.


1. Find the measure of a central angle of a regular polygon with 24 sides.

Find the area of each regular polygon.

2.

5.5 cm

8.0 cm 3.

12 cm6 3 cm

Find the perimeter and area of each regular polygon.

4.

15 in.

5.

4 m

Warm-Ups: 1. 84.9 cm2 2. 64.5 in.2 3. 44 in.2

Lesson Quiz: 1. 158 2. 110 cm2 3. 374.1 cm2 4. 99.4 in.; 745.6 in.2 5. 22.6 m; 32 m2

ANSWERS


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91Geometry



1. Find the probability that a point chosen at random on } JN is on } KN .

0 42

L M NJ K

222426

2. Find the probability that a randomly chosen point

4

in the fi gure will lie in the shaded region.

3. A bird is equally likely to be anywherein the garden shown. What is the

flowers

vegetables

probability that the bird will be in the vegetables?

Lesson Quiz: 1. 5 } 7 or 71.4% 2. 36.3% 3. about

24 }

51 or 47%

ANSWERS

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1. Find the length of a 60° arc in a circle with radius 8 meters.

2. Find the exact value of x if 12 }

2p • 8 5

x }

360 .

Find the exact length of the arc.

3. C ACB 4. C AB

P

C

20 m

A

B

P

A

B

608

Circumference 5 80 cm


Use the diagram and the given arc measure to fi nd thedistance between points A and B. Leave your answers in terms of p.

1. m C AB 5 80º 2. m C AB 5 100º

A P

B

C9

A P

B

C18

Warm-Ups: 1. about 8.38 m 2. 270

} p 3. 30p m 4. 13 1 } 3 cm

Lesson Quiz: 1. 2p 2. 5p

ANSWERS

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Name the polygon by the number of sides.

1. 6 2. 10 3. 5

4. What is a regular polygon?

5. What is the length of the diagonal of a square with side length 6?

Warm-Ups: 1. hexagon 2. decagon 3. pentagon 4. a polygon with all congruent sides and all congruent angles 5. 6 Ï

}

2

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94Geometry



1. Evaluate 2xy 1 2yz 1 2xz for x 5 9, y 5 6, and z 5 4.

2. Evaluate 2πx2 1 2πxy for x 5 4 and y 5 7. Use 3.14 for π.

3. Find the area of a circle with radius 8 cm. Use 3.14 for π.

4. Find the area of a right triangle with hypotenuse 13 in. and a side length of 5 in.

5. Find the area of a regular pentagon with apothem 4.13 ft and side length 6 ft.


Determine whether the solid is a polyhedron. If it is, name it. 1. 2. 3.

4. Find the number of faces, vertices, and edges of each polyhedron in Exercises 2 and 3.

5. A plane intersects a cone, but does not intersect the base of the cone. Describe the possible cross sections.

Warm-Ups: 1. 228 2. 276.32 3. 200.96 cm2 4. 30 in.2 5. 61.95 ft2

Lesson Quiz: 1. no 2. yes; rectangular pyramid 3. yes; pentagonal prism 4. pyramid: 5 faces, 5 vertices, 8 edges; prism: 7 faces, 10 vertices, 15 edges 5. a point, a circle, an ellipse

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95Geometry



Find the area of each regular polygon or circle.

1. Hexagon, radius 8 cm 2. Circle, radius 4 in.

3. Inscribed pentagon, radius 1 yd

4. Find the perimeter of an inscribed square with radius 1 ft.

5. Find the circumference of a circle with diameter 12.8 in.


1. Find the surface area of the

16 cm

4 cm solid formed by the net.

Find the surface area and lateral area of the right prism or right cylinder.

2.

25 m

15 m 38 m

3.

26 yd

20 yd

4. A rectangular prism has surface

8 ft

30 ft

x ft

area 1848 ft2. Solve for x.

Warm-Ups: 1. 166.28 cm2 2. 50.27 in.2 3. 2.38 yd2 4. 5.66 ft 5. 40.21 in.

Lesson Quiz: 1. 288 cm2 2. 2580 m2, 2280 m2 3. 2261.95 yd2, 1633.63 yd2 4. 18

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96Geometry



Find the area of each polygon or circle.

1. Trapezoid, bases 12 ft and 18 ft, height 3 ft

2. Circle, diameter 8.2 in.

3. Regular hexagon, side length 14 cm


1. Find the lateral area of the 2. Find the surface area of the cone. regular pyramid. Round to the nearest hundredth.

10 cm

15 cm

12 in.

36 in.

3. A right cone has radius 8 m and surface area 224π m2. Find its slant height.

Warm-Ups: 1. 45 ft2 2. 52.8 in.2 3. 509.22 cm2

Lesson Quiz: 1. 300 cm2 2. 1882.97 in.2 3. 20 m

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97Geometry



Find the area of each polygon or circle.

1. Regular hexagon, side length 9 in.

2. Circle, radius 15 m

3. Right triangle, hypotenuse 12 cm, side length 6 cm

4. Solve (2 Ï

}

3 )2 Ï}

3 }

12 p x 5 15 Ï

}

3 .


1. Find the volume of the solid with regular pentagonal bases.

14.5 in.

15 in.

10 in. 2. Find the volume of a right triangular prism

with height 32 in., base height 12 in., and base length 18 in.

3. Find the volume of a right cylinder with height 30 ft and diameter 14 ft.

4. A cylindrical beaker with diameter 10 in. and height 12 in. is fi lled with water that is then poured into a rectangular pan that is 14 in. by 9 in. by 3 in. What is the volume of each solid? Would the water overfl ow the pan? If so, what is the height of the water in the beaker after the pan is fi lled?

Warm-Ups: 1. 210.44 in.2 2. 706.86 m2 3. 31.18 cm2 4. 15

Lesson Quiz: 1. 5437.5 in.3 2. 3456 in.3 3. 4618.14 ft3 4. cylinder: 942.48 in.3; pan: 378 in.3; Yes, it would overfl ow the pan. After the pan is fi lled, the height of the water in the beaker will be about 7.2 in.

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98Geometry



Find the area of each circle.

1. Radius 17 m 2. Diameter 7.8 in.

Find the volume of each solid.

3. Cylinder, radius 6 ft, height 10 ft

4. Cone, radius 3.1 cm and height 11.8 cm

5. Find the circumference of a circle with radius 48 m.


Find the volume of each solid.

1.

8 in.

6 in.

2.

12 mm

24 mm

3. Find the volume of the cube afterthe cone is removed.

6.8 mm

6.8 mm

6.8 mm 4. The volume of a right cone with

height 32 in. and radius r in. is 2144.66 in.3. Find r.

Warm-Ups: 1. 907.92 m2 2. 47.78 in.2 3. 1130.97 ft3 4. 118.75 cm3 5. 301.59 m

Lesson Quiz: 1. 96 in.3 2. 3619.11 mm3 3. 232.11 mm3 4. 8.0 in.

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Find the surface area and volume of each solid.

1. Right rectangular prism, side lengths 8 in., 5 in., and 10 in.

2. Right cone, radius 3 m, height 4 m

3. Sphere, radius 7.3 ft


Find the surface area and volume of each sphere.

1.

r 5 6 ft

2.

d 5 12.4 m

3. A great circle of a sphere has circumference 16π in. Find the volume of the sphere.

4. Find the radius of a sphere with volume 4445.18 ft3.

5. Find the surface area and volume of the solid. 4 m

4 m

Warm-Ups: 1. 340 in.2; 400 in.3 2. 75.40 m2; 37.70 m3 3. 669.66 ft2; 1629.51 ft3

Lesson Quiz: 1. 452.39 ft2; 904.78 ft3 2. 483.05 m2; 998.31 m3 3. 2144.66 in.3 4. 10.2 ft 5. 251.33 m2; 335.10 m3

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100Geometry



1. Two triangular prisms are similar, with surface areas 9 ft2 and 25 ft2. What is the scale factor of the prisms?

2. Two spheres have volume 125π m3 and 216π m3. What is the scale factor of the spheres? What is the ratio of the surface areas?

3. The volumes of two similar square pyramids are 8 ft3 and 27 ft3. What is the ratio of the surface area of the smaller pyramid to the larger?

4. The two solids are similar. Find the surface area and volume of Solid II.

III

2.5 cm

5 cm

6 cm

Lesson Quiz: 1. 3 : 5 2. 5 : 6; 25 : 36 3. 4 : 9 4. S 5 361.28 cm2; V 5 575.96 cm3

ANSWERS


LESSON

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101Geometry



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