College Textbook sample 2 by Terri Wright

Elementary Geometry for College Students

David C. Alexander Parkland College

Geralyn N. Koeberlein Mahomet-Seymour High School

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

Contents

Preface ix Foreword xvii Index of Applications xviii

1 Line and Angle Relationships 1.1 Sets, Statements, and Reasoning 00 1.2 Informal Geometry and Measurement Calculations 00 1.3 Early Definitions and Postulates 00 1.4 Angles and Their Relationships 00 1.5 Introduction to Geometric Proof 00 1.6 Relationships: Perpendicular Lines 00

2 Parallel Lines

23 1.7 The Formal Proof of a Theorem 00 ■■ P ERSPECTIVE ON HISTORY: The Development of

Geometry 00 ■■ PERSPECTIVE ON APPLICATIONS: Patterns 00 ■■ SUMMARY 00 ■■ REVIEW EXERCISES 00 ■■ CHAPTER 1 TEST 00

2.1 The Parallel Postulate and Special Angles 00 2.2 Indirect Proof 00 3.3 Proving Lines Parallel 00 2.4 The Angles of a Triangle 00 2.5 Convex Polygons 00 2.6 Symmetry and Transformations 00

■ PERSPECTIVE ON HISTORY: Sketch of Euclid 00 ■ PERSPECTIVE ON APPLICATIONS: Non-Euclidian 00 ■ SUMMARY 00 ■ REVIEW EXERCISES 00 ■ CHAPTER 2 TEST 00

3 Triangles 127 3.1 The Parallel Postulate and Special Angles 00 3.2 Indirect Proof 00 3.3 Proving Lines Parallel 00 3.4 The Angles of a Triangle 00 3.5 Convex Polygons 00 3.6 Symmetry and Transformations 00

■■ PERSPECTIVE ON HISTORY: Sketch of Euclid 00 ■■ PERSPECTIVE ON APPLICATIONS: Non-Euclidian 00 ■■ SUMMARY 00 ■■ REVIEW EXERCISES 00 ■■ CHAPTER 3 TEST 00

4 Quadrilaterals 171 4.1 The Parallel Postulate and Special Angles 00 4.2 Indirect Proof 00 4.3 Proving Lines Parallel 00 4.4 The Angles of a Triangle 00

■■ PERSPECTIVE ON APPLICATIONS: Square Numbers

■■ PERSPECTIVE ON HISTORY: Sketch of Thales 00

■■ CHAPTER 4 TEST 00

as Sums 00 ■■ SUMMARY 00 ■■ REVIEW EXERCISES 00

iii

iv CONTENTS

8 Areas of Polygons and Circles 8.1 Area and Initial Postulates 00 8.2 Perimeter and Area of Polygons 00 8.3 REgular Polygons and Area 00 8.4 Circumference and Area of a Circle 00 8.5 More Area Relationships in the Circle 00

Preface 299

■■ PERSPECTIVE ON APPLICATIONS: Another Look at the

Pythagorean Theorem 00 ■■ SUMMARY 00 ■■ REVIEW EXERCISES 00 ■■ CHAPTER 8 TEST 00

■■ PERSPECTIVE ON HISTORY: Sketch of Pythagoras 00

9 Surfaces and Solids

389

9.1 The Parallel Postulate and Special Angles 00 9.2 Indirect Proof 00 9.3 Proving Lines Parallel 00 9.4 The Angles of a Triangle 00

■ PERSPECTIVE ON APPLICATIONS: Birds in Flight 00 ■ SUMMARY 00 ■ REVIEW EXERCISES 00 ■ CHAPTER 9 TEST 00

■■ PERSPECTIVE ON HISTORY: Sketch of Reneé

Descartes 00

10 Analytic Geometry

Elementary Geometry for College Students, Fifth Edition was written in a style that would teach students to explore principles of geometry, reason deductively, and perform geometric applications in the real world. Moloborting erostrud tatem velenit pratin henim do ercin ullut at nit am quipit augiat wis nit ut adion ut iriliqu atisim erat laorpero dipsum alit vullutat, quatincillan ulput lor aliquis dolobor suscili quisit luptat. Iliquamcon ut volortinisim voloreetue ea faccum il etuerillan verit lobore feu feugait autpatueros dolorem incidunt inis erosto dionsectem vel ulputpat. Ut nonse molobore feum veros accum dolortion vulputat veraestio odiatem adipit pratis exerosto consequat.

OUTCOMES FOR THE STUDENT Na con venim ipsummolore tatue cortie molorem digna faciduis ad eugiam augiamet ipit, venibh ex eum numsan ut ad mod te molortio consed tincip eu feuisi. ■

171

10.1 The Rectangular Coordinate System 00 10.2 Graphs of Linear Equations and Slope 00 10.3 Preparing to Do Analytic Proofs 00 10.4 Analytic Proofs 00 10.5 Equations of Lines 00 ■■ PERSPECTIVE ON HISTORY: The Banach-Tarski

Paradox 00

APPENDICES APPENDIX A: Algebra Review A1 APPENDIX B: Summary of Constructions, Postulates, and Theorems and Corollaraies A26

ANSWERS

AUTHOR'S PHILOSOPHY AND STYLE

■ ■ ■■ PERSPECTIVE ON APPLICATIONS: The Point-of-Division

Formulas 00 ■■ SUMMARY 00 ■■ REVIEW EXERCISES 00 ■■ CHAPTER 10 TEST 00

Some statements contain one or more variables. A variable is a letter that represents a number. A set is any collection of objects.

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FEATURES MAINTAINED FROM PREVIOUS EDITIONS Tue vel il ulla facil dio core dip etum zzrit utatue enisl elestie consent ut veliquat landipis nis et, quat. Ilit luptat. Ut am volenit diat. Dolobore min henibh etum veliquam, quat, senit wisci tat, quis at ulluptatio cor il ut ero delit luptat ipit luptat. Giate vullaor adipit ea consent iustincipsum er suscilla autat atis nim verci eu feu facilis nonsequ issenibh eu feum dolorpe rostie tinisl dolobor am zzrit utpat. Dolobore min henibh etum veliquam, quat, senit wisci tat, quis at ulluptatio cor il ut ero delit luptat ipit luptat. Giate vullaor adipit ea consent iustincipsum er suscilla autat atis nim verci eu feu facilis nonsequ issenibh eu feum dolorpe rostie tinisl dolobor am zzrit utpat. Dolobore min henibh etum veliquam, quat, senit wisci tat, quis at ulluptatio cor il ut ero delit luptat ipit luptat. Giate vullaor adipit ea consent iustincipsum er suscilla autat atis nim verci eu feu facilis nonsequ issenibh eu feum dolorpe rostie tinisl dolobor am zzrit utpat.

Selected Answers and Proofs A33

◆ Intuition

Glossary 503

We wish to thank Marc Bove, . . . as well as the members of the team at Cengae Learning. We express our gratitude to reviewers of previous editions, including

Index 527

Paul Allen, University of Alabama Jane C. Beatie, University of South Carolina at Aiken Steven Blasberg, West Valley College

Line and Angle Relationships

©John Peter Photography / Alamy Stock Photo.

CHAPTER OUTLINE

1.1 Sets, Statements, and Reasoning

1.2 Informal Geometry and Measurement Calculations

1.3 Early Definitions and Postulates

1.4 Angles and Their Relationships

1.5 Introduction to Geometric Proof

1.6 Relationships: Perpendicular Lines

1.7 The Formal Proof of a Theorem

Magical ! Geometry, figures can be drawn to create an illusion. Careful inspection of this Una Stanza di Poliphilo, 1994 print reveals that the artist created numerous perceptual flaws: compare the staircases, ladders, and windows in the print in order to discover the wrong-doing. This chapter opens with a discussion of statements and a type of reasoning used in geometry. In geometry, figures can be drawn that create an illusion. Careful inspection of this Escher print reveals that the artist created numerous perceptual flaws: compare the staircases, ladders, and windows in the print in order to discover the wrong-doing. This chapter opens with a discussion of statements and a type of reasoning used in geometry. In geometry, figures can be drawn that create an illusion. This chapter opens with a discussion of statements.

n PERSPECTIVE ON

HISTORY: The Development of Geometry n PERSPECTIVE ON

APPLICATIONS: Patterns n SUMMARY

101

102 CHAPTER 1 n LINE AND ANGLE RELATIONSHIPS

n Introduction to Geometric Proof 1.1

Warning

1.1 Sets, Statements, and Reasoning Statement Conjunction Disjunction Disjunction Negation

EXAMPLE 2

In the box, the argument on the left and pattern after Example:

Give the negation of each statement.

1. The argument on the right is

KEY CONCEPTS Implication (Conditional) Hypothesis Conclusion Intuition

Induction Deduction Law of Detachment Sets Subset

Intersection Union Union Venn Diagram

103

a) 4 + 3 = 7

invalid; ths form was given in Example 10.∠1 ≅ ∠2).

b) All fish can swim

b) one fish can swim.

SOLUTION a) 4 + 3 ≠ 7 (≠ means “is not equal to.”) b) Some fish cannot swim. (To negate “All fish can swim,” we say that at least

one fish cannot swim.)

EXAMPLE 3

A set is any collection of objects; in particular, the objects are known as the elements of the set. A = (1, 2, 3) is read, “A is the set of elements 1, 2, and 3.” In geometry, geometric figures such as lines and angles are actually sets of points. Where A = (1, 2, 3) and B = {A is a subset of B because each element in A is also in B; in symbols, A = B. In Chapter 2, we will discover that T = {all triangles} is a subset of P = {all polygons}. Some statements contain one or more variables; a variable is a letter that represents a number.

Assume that statements P and Q are true.

P: 4 + 3 = 7 Q: An angle has two sides.

STATEMENTS

Classify the following statements as true or false. 1. 4 + 3 ≠ 7 and an angle has two sides.

(from AB = 9)

2. 4 + 3 ≠ 7 or an angle has two sides.

(from BC = 8)

DEFINITION

Sid

A statement is a set of words and symbols that collectively make a claim that can be classified as true or false

Side 2 Figure 1.1

Figure 1.2 Alternate sides of an angle

EXAMPLE 1 SOLUTION Statement 1 is false because the conjunction has the form “F and T.”

Classify each of the following as a true statement, a false statement, or neither.

Statement 2 is true because the disjunction has the form “F or T.”

1. 4 + 3 = 7 2. An Angle has two sides, (See Figure 1.1)

Some statements contain one or more variables; a variable is a letter that represents a number. The claim “x + 5 = 6” is called an open sentence or open statement.

3. Robert E. Lee played shortstop for the Yankees. 4. 7 < 3 (This is read "7 is less than 3.") 5. Look out! Reminder Numbers that measure may be equal (AB = CD or m∠2) whereas geometric figures may be congruent (AB ≅ CD or ∠1 ≅ ∠2).

REASONING

SOLUTION 1 and 2 are true statements; 3 and 4 are false statements; 5 is not a false

For instance, x + 5 = 6 is true if x = 1; for x not equal to 1, x + 5 = 6 is false. The negation of a given statement P makes a claim opposite that of the statement.

statement. A set is any collection of objects; in particular, the objects are known as the elements of the set. A = (1, 2, 3) is read, “A is the set of elements 1, 2, and 3.” In geometry, geometric figures such as lines and angles are actually sets of points. 1. Some statements contain one or more variables. 2. A variable is a letter that represents a number. 3. A set is any collection of objects.

The claim “ x + 5 = 6” is called an open sentence or open statement because it can be classified as true or false, depending on the replacement value of x. For instance, x + 5 = 6 is true if x = 1; for x not equal to 1, x + 5 = 6 is false. For instance, x + 5 = 6 is true if x = 1; for x not equal to 1, x + 5 = 6 is false.

Technology Exploration Use computer software if available. 1. Construct AC and DB to intersect

at point 0. (See Figuire 1.62) 2. Measure <1, <2, <3, and <4. 3. Show that M<1 = m<3 and

M,2 = m<4.

◆ Intuition If the given statement is true, its negation is false, and vica versa. If P is a statement, we use ~ P (which is read “not P”) to indicate its negation. If P is a statement, we use ~ P (which is read “not P”) to indicate its negation. Some statements contain one or more variables; a variable is a letter that represents a number. ■■ Some

statements contain one or more variables. variable is a letter that represents a number. ■■ A set is any collection of objects. ■■ A

If P is a statement, we use ~ P (which is read “not P”) to indicate its negation. Some statements contain one or more variables; a variable is a letter that represents a number. For instance, x + 5 = 6 is true if x = 1; for x not equal to 1, x + 5 = 6 is false.

104 CHAPTER 1 n LINE AND ANGLE RELATIONSHIPS

In the St. Louis area, an interview of 100 sports enthusiass shows that 74 support the Cardinals basball team and 58 support the Rams football team. All of those interviewed support one team or the other or both. How many support both teams? ANSWER

Geometry in Nature

POSTULATE 16 ■ Central Angle Postulate

A statement is a set of words and symbols that collectively make a claim that can be classified as true or false

An icycle formed from freezing water assumes a vertical path.

Discover

n Introduction to Geometric Proof 1.1

32 (74 + 58 - 100)

Reflective property: aRa (5 = 5; equality of numbers has a reflective property) Symmetric property: If aRb, then bRa. (If 𝓵 ⫠ m, then m ⫠ 𝓵; perpendicularity of lines has a symmetric property)

}

premises

C. ∴ Q

}

conclusion

If P is a statement, we use ~ P (which is read “not P”) to indicate its negation. If the given statement is true, its negation is false, and vica versa. If P is a statement, we use ~ P (which is read “not P”) to indicate its negation. Where A = (1, 2, 3) and B = {A is a subset of B because each element in A is also in B; in symbols, A = B. In Chapter 2, we will discover that T = {all triangles} is a subset of P = {all polygons}. Corollary 6.2.4

The line whose slope is m and whose y intercept is b has the equation y = 487

Where A = (1, 2, 3) and B = {A is a subset of B because each element in A is also in B; in symbols, A = B. In Chapter 2, we will discover that T = {all triangles} is a subset of P = {all polygons}. Some statements contain one or more variables; a variable is a letter that represents a number.

TYPES OF PROPERTIES

1. If P, then Q f premises 2. P

105

NOTE: In geometry, the reference numbers used with potulates (as in Postulatae 1 need not be memorized).

EXAMPLE 1

NOTE: The symbol ∴ means “therefore.”

If you accept the following statements 1 and 2 as true, what must you conclude? 1. If a student plays on the Rockville High School boys’ varsity basketball

Some statements contain one or more variables; a variable is a letter that represents a number. The claim “ x + 5 = 6” is called an open sentence or open statement because it can be classified as true or false, depending on the replacement value of letter x. TABLE 1.1

Theorem 10.6.8 ■ Slope Intercept Form of a Line

The Conjunction

The line whose slope is m and whose y intercept is b has the equation y = 243

P and Q

PROOF

If the given statement is true, its negation is false, and vica versa. If P is a statement, we use ~ P (which is read “not P”) to indicate its negation. .

SSG

The Disjunction

P and Q

A ruler can be used to verify that this claim is true. Note 2: Using methods found in Chapter 3, we could use deduction to prove. Note 1:

CONSTRUCTIONS

Geometry in the Real World

Another tool used in geometry is the compass. The ancient Greeks insisted that only two tools (a compass and a straightedge) be used for geometric constructions, which were idealized drawings assuming poerfection in the use of these tools. .

Hypothesis: Two sides of a triangle are equal in length. Conclusion: Two angles of the triangle are equal in measure.

TABLE 1.2

CONCLUSION Todd is a talented athlete.

EXS. 1–7

For the true statemnt, "If P, then Q," the hypothetical situation described in P implies the conclusion described in Q. This type of statement is often used in reasoning so, we turn our attention to this matter.

Parts AB and CD of the chain represent common external tangents to the circular gears.

team, then he is a talented athlete. 2. Todd plays on the Rockville High School boys’ varsity basketball team.

CONSTRUCTION 1 To construct a segment congruent to a given segment. GIVEN: AB in Figure 1.26a, as sow on page 16. CONSTRUCT: CD on line m so that CD ≅ AB (or CD = AB) CONSTRUCTION: With your compass open to the length of AB, place the stationary point of the compass at C and mark off a length equal to AB as shown in Figures 1.23 and 1.24. Then CD = AB.

Figure 1.3

This type of statement is often used in reasoning so: 1. The definition is reversible

i) A line segment is the part of a line between and including two points. ii) The part of a line between and including two points.

Figure 1.4

Figure 1.5

SOURCE: Mathematics in the 21st Century, Putnam, 2009.

In geometry, geometric figures such as lines and angles are actually sets of points. Some statements contain variables; a variable is a letter that represents a number.

106 CHAPTER 1 n LINE AND ANGLE RELATIONSHIPS

n Introduction to Geometric Proof 1.1

If P is a statement, we use ~ P (which is read “not P”) to indicate its negation. Where A = (1, 2, 3) and B = {A is a subset of B because each element in A is also in B; in symbols, A = B. In Chapter 2, we will discover that T = {all triangles} is a subset of P = {all polygons}. If P is a statement, we use ~ P (which is read “not P”) to indicate its negation. Where A = (1, 2, 3) and B = {A is a subset of B because each element in A is also in B; in symbols, A = B. In Chapter 2, we will discover that T = {all triangles} is a subset of P = {all polygons}

STRATEGY FOR PROOF ■ The First Line of Proof

The measure of an angle formed by a tangent and a chord drawn to the point of tangency is one-half the measure of the intercepted arc (See Figure 6.29).

Figure 1.6

107

Discover

EXAMPLE 3 GIVEN: MN > PQ (Figure 1.57) PROVE: MP > NQ PROOF 1. Given

2. ∠ s AOC and DOB are straight ∠ s, with m ∠ AOC = 180 and m ∠ DOB = 180

2. The measure of a straight angle is 180˚

3. m ∠ AOC = m∠ DOB

3. Substitution

NOTE: The last proportion is the inverted form of the given proportion. In any of the equivalent proportions, a ∙ d = b ∙ c (product of means = product of extremes).

Some statements may contain one or more variables; a variable is a letter that represents a number. The statement or claim “ x + 5 = 6” is called an open sentence or open statement.

SAMPLE PROOFS For instance, x + 5 = 6 is true if x = 1; for x not equal to 1, x + 5 = 6 is false. The negation of a given statement P makes a claim opposite that of the original statement. If the given statement is true, its negation is false, and vica versa. The negation of a given statement P makes a claim opposite that of the original statement. If the given statement is true, its negation is false, and vica versa.

One edge of the index card coincides with both of the angle's sides Straight Angle

1. AC intersects BD at O

Figure 1.8

In Exercises 5 - 10, which sentences are settlements? If a sentence is a statement, classify it as true or false..

1. a) Where do you live?

5. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5. b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

3. a) Where do you live?

Sides

Center of the Circle

The measure of the intercepted arc.

Angles

Angles in the interior of the circle

One half the sum of the measures of the intercepted arcs

Now, squaring each side and simplifying, we get 12(x - 0)2 + ( y - p)222 = ( y + p)2 x2 + y2 - 2py + p2 = y2 + 2py + p2 x2 = 4py

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

2. a) Chicago is located in the state of Illinois.

Methods for Measuring Angles Related to a Circle

Location of vertex and angles for measuring and centering

Card exposes the second side of the angle.

In Exercises 1 and 2, which sentences are settlements? If a sentence is a statement, classify it as true or false.

TABLE 1.3 Rules for Measuring the Angle

Card hides the second side of the angle

Exercises 1.1

In Exercises 3 and 4, which sentences are settlements? If a sentence is a statement, classify it as true or false.

Location of the Vertex

Sides of the angle coincide with two edges of the card Right Angle

Reasons

Acute angle

Statements

Obtuse angle

Figure 1.7

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

e Sid

Side 2

4. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. (HINT: Use Exercise 28 of Section 1.6 as a guide.)

Exercises 5-9 6. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) x + 3 = 7 when x = 5.

POSTULATES 7. a) Where do you live?

b) 4 + 7 ≠ 5. 8. a) Chicago is located in the state of Illinois.

b) Get out of here! 9. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president.

108 CHAPTER 1 n LINE AND ANGLE RELATIONSHIPS 15. a) Where do you live?

n Introduction to Geometric Proof 1.1

18. a) Chicago is located in the state of Illinois.

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

16. a) Chicago is located in the state of Illinois.

19. a) Where do you live?

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

17. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president.

20. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

10.4 Some Constructions and Inequalities for the Circle

PERSPECTIVE ON HISTORY THE DEVELOPMENT OF GEOMETRY One of the first written accounts of geometric knowledge appears in the Rhind papyrus, a collection of documents that date back to more than 1000 years before Christ. In this document, Ahmes (an Egyptian scribe) describes how north-south and east-west lines were redrawn following the overflow of the Nile River. Astronomy was used to lay out the north-south line. The rest was done by people known as “rope-fasteners.” By tying knots in a rope, it was possible to separate the rope into segments with lengths that were in the ratio 3 to 4 to 5.

KEY CONCEPTS Construction of Tangents to a Circle

SSG

EXS. 1–7

Inequalities in the Circle

Constructions and Inequalities

109

Figure 10.69

In Figure 1.69, the right angle is formed so that one side (of length 4, as shown) lies in the north-south line, and the sec-

ond side (of length 3, as shown) lies in the east-west line. The principle that was used by the rope-fasteners is known as the Pythagorean Theorem. However, we also know that the ancient Chinese were aware of this relationship. That is, the Pythagorean Theorem was known and applied many centuries before the time of Pythagoras (the Greek mathematician for whom the theorem is named). Ahmes describes other facts of geometry that were known to the Egyptians. Perhaps the most impressive of these facts was that their approximation of [&|pi.3.Opt5|&] was 3.1604. To four decimal places of accuracy, we know today that the correct value of [&|pi.3.Opt5|&] is 3.1416. Like the Egyptians, the Chinese treated geometry in a very pr actical way. In their constructions and designs, the Chinese used the rule (ruler), the square, the compass, and the level. Unlike the Egyptians and the Chinese, the Greeks formalized and expanded the knowledge base of geometry by pursuing geometry as an intellectual endeavor. According to the Greek scribe Proclus (about 50 b.c.), Thales (625–547 b.c.) first established deductive proofs for several of the known theorems of geometry. Proclus also notes that it was Euclid (330–275 b.c.) who collected, summarized, ordered, and verified the vast quantity of knowledge of geometry in his time.

THE CONVERSE OF THE PYTHAGOREAN THEOREM PERSPECTIVE ON APPLICATIONS

DEFINITION

A Pythagorean triple is a set of three natural numbers (a,b,c) for which a2 + b2 = c2.

PATTERNS

EXAMPLE 3

In much of the study of mathematics, we seek patterns related to the set of counting numbers N = 51,2,3,4,5, . . . 6. Some of these patterns are geometric and are given special names that

EXAMPLE 3 Study the picture proof of Theorem 10.23. See Figure 10.15 (a). PICTURE PROOF OF THE THEOREM 10.23

GIVEN: AB in Figure 1.26a, as sow on page 16. PROVE: CD on line m so that CD ≅ AB (or CD = AB) PROOF: We reflect △ABV across AC to form an equiangular and therefore equilateral △ABD. As shown in Figures 5.35 (b) and (c), we have AB = 2a. To find b in Figure 5.35(c) we apply the Pythagorean Theorem.

Figure 10.23

c2 = a2 + b2 (4a)2 = a2 + b2 b2 = 3a2 (2a)2 = a2 + b2

That is, AC = a√3 a) 4 + 3 = 7 d) 4 + 8 = 12

b) All fish can swim e) Three fish can swim

c) one fish can swim. f ) all fish can swim.

Find the fourth number in the pattern of triangular numbers shown in Figure 1.8 (a).

reflect the configuration of sets of points. For instance, the set of square numbers are shown geometrically in Figure 1.70 and, of course, correspond to the numbers 1, 4, 9, 16, . . . .i.

• • • • • • • Figure 1.8

• • • • • • • • • • • •

In much of the study of mathematics, we seek patterns related to the set of counting numbers N = 51,2,3,4,5, . . . 6. Some of these patterns are geometric and are given special names that reflect the configuration of sets of points. For instance, the set of square numbers are shown geometrically in Figure 1.70. For instance, the set of square numbers are shown geometrically in Figure 1.70

Figure 1.8 (a)

SOLUTION By continuing to add and join midpoints in the third figure, we form a figure like the one shown in Figure 1.74(b).

Figure 1.8 (b)

110 CHAPTER 1 n LINE AND ANGLE RELATIONSHIPS

SOLUTION By continuing to add and join midpoints in the third figure, we form a figure like the one shown in Figure 1.8 (b).

• • • • • • •

Some patterns of geometry lead to principles known as postulates and theorems. One of the principles that we will explore in the next example is based on the total number of diagonals found in a polygon with a given number of sides. A diagonal of a polygon (many-sided figure) joins two onconsecutive vertices of the polygon together.

Figure 1.8

• • • • • • • • • • • •

Summary

Overview ◆ Chapter 1 Line and Line Segment Relationships Figure A B C D

Summary

Relationship

Symbols

Parallel line

ℓ ∙ m or AB ∙ CD;

(and segments)

AB ∙ CD

Intersecting lines

ℓ ∙ m or AB ∙ CD;

(and segments)

AB ∙ CD

Perpendicular lines

t⊥v

(t shown vertical, v shown horizontal)

A Look Back at Chapter 1 Geometry, figures can be drawn to create an illusion. Careful inspection of this Una Stanza di Poliphilo, 1994 print reveals that the artist created numerous perceptual flaws: compare the staircases, ladders, and windows in the print in order to discover the wrong-doing. This chapter opens with a discussion of statements and atype of reasoning used in geometry. In geometry, figures can be drawn that create an illusion. Careful inspection of this Escher print reveals that the artist created numerous perceptual flaws: compare the staircases, ladders, and windows in the print in order to discover the wrong-doing. This chapter opens with a discussion of statements and atype of reasoning used in geometry. .

A Look Ahead to Chapter 2 Geometry, figures can be drawn to create an illusion. Careful inspection of this Una Stanza di Poliphilo, 1994 print reveals that the artist created perceptual flaws: compare the staircases, ladders, and windows iin order to discover the wrong-doing.

Key Concepts 1.1 Statement • Variable • Conjunction • Disjunction • Negation • Implication (Conditional) • Hypothesis • Conclusion • Intuition • Induction • Deduction • Argument (Valid and Invalid) • Law of Detachment • Set • Subets • Venn Diagram • Intersection • Union

1.2 Point • Line • Plane • Collinear Points • Vertex • Line Segment • Betweenness of Points • Midpoint •

Congruent • Protractor • Parallel Lines • Bisect • Straight Angle • Right Angle • Intersect • Perpendicular • Compass • Constructions • Circle

1.3 Mathematical System • Axiom or Postulate • Assumption • Theorem • Ruler Postulate • Distance • Segment-Addition • Postulate • Congruent Segments • Midpoint of a Line Segment • Bisector of a Line Segment • Union • Ray • Opposite Rays • Intersection of Two Geometric Figures • Parallel Lines • Plane • Coplanar Points • Space • Parallel, Vertical, Horizontal Planes

Perpendicular lines

ℓ ∙ m or AB ∙ CD;

(t shown vertical, v

AB ∙ CD

shown horizontal)

1.4 Angle • Sides of an Angle • Vertex of an Angle • Protractor Postulate • Acute, Right, Obtuse, Straight, and Reflex Angles • Angle-Addition Postulate • Adjacent Angles • Congruent Angles • Bisector of an Angle • Complementary Angles • Supplementary Angles • Vertical Angles

1.5 Algebraic Properties • Proof

1.6 Vertical Lines and Horizontal Lines • Perpendicular Lines • Relations • Reflexive, Symmetric, and Transitive Properties of Congurence • Equivalence Relation • Perpendicular Bisector of a Line Segment

1.7 Formal Proof of a Theorem • Converse of a Postulate, Theorem • Reflexive, Symmetric Properties.

Line and Line Segment Relationships Figure A B C D

Relationship

Symbols

Parallel line

ℓ ∙ m or AB ∙ CD;

(and segments)

AB ∙ CD

Intersecting lines

ℓ ∙ m or AB ∙ CD;

(and segments)

AB ∙ CD

111

112 CHAPTER 1 n LINE AND ANGLE RELATIONSHIPS

Chapter 1 Review Exercises In Exercises 1 and 2, which sentences are settlements? If a sentence is a statement, classify it as true or false. 1. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

2. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

In Exercises 3 and 4, which sentences are settlements? If a sentence is a statement, classify it as true or false. 3. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

Sid

9. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president.

10. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

(HINT: Use Exercise 28 of Section 1.6 as a guide.)

In Exercises 13and 14 which sentences are settlements? If a sentence is a statement, classify it as true or false.. PROOF Statements

Reasons

1. AC intersects BD at O

1. Given

2. ∠ s AOC and DOB are straight ∠ s, with m ∠ AOC = 180 and m ∠ DOB = 180

2. The measure of a straight angle is 180˚

3. m ∠ AOC = m∠ DOB 3. Substitution

5. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

In Exercise 22 which sentences is a settlement? If a sentence is a statement, classify it as true or false.

20. a) Where do you live? b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

22. a) Where do you live? b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5. e) Babe Ruth is remmbered as a great football player.

21. a) Chicago is located in the state of Illinois. b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10.

Chapter 1 Test

12. a) Chicago is located in the state of Illinois.

In Exercises 5 - 10, which sentences are settlements? If a sentence is a statement, classify it as true or false..

In Exercises 20 and 21, which sentences are settlements? If a sentence is a statement, classify it as true or false..

11. a) Where do you live?

4. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10.

13. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

In Exercises 1 and 2, which sentences are settlements? If a sentence is a statement, classify it as true or false. 1. a) Where do you live?

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10.

In Exercises 15 to 19, find the equation of the conic section that satisfies the given conditions. 15. Ellipse with vertices at (7, 3) and (-3, 3); length of the

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) x + 3 = 7 when x = 5.

7. a) Where do you live?

b) 4 + 7 ≠ 5.

8. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

minor axis is 8. 16. Hyperbola with vertices at (4, 1) and (-2, 1);

eccentricity is ¾. 17. Hyperbola with foci at (-5, 2) and (1, 2); length of

transverse axis is 4. 18. Parabola with focus at (2, -3) and directrix x = 6\ (y + 3)2

= -8(x - 4) [5.1] 19. Parabola with vertex at (0, -2) and passing through the

point

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

6. a) Chicago is located in the state of Illinois.

Statements

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

In Exercises 3 and 4, which sentences are settlements? If a sentence is a statement, classify it as true or false. b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5. A B C D 4. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10.

(HINT: Use Exercise 28 of Section 1.6 as a guide.)

In Exercises 5 - 10, which sentences are settlements? If a sentence is a statement, classify it as true or false.. 5. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

Reasons

1. AC intersects BD at O

1. Given

2. ∠ s AOC and DOB are straight ∠ s, with m ∠ AOC = 180 and m ∠ DOB = 180

2. The measure of a straight angle is 180˚

3. m ∠ AOC = m∠ DOB 3. Substitution

3. a) Where do you live?

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) x + 3 = 7 when x = 5. PROOF

2. a) Chicago is located in the state of Illinois.

14. a) Chicago is located in the state of Illinois.

6. a) Chicago is located in the state of Illinois.

113

In Exercises 11 and 12 which sentences are settlements? If a sentence is a statement, classify it as true or false..

Side 2

Chapter Test

7. a) Where do you live?

b) 4 + 7 ≠ 5.

8. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

9. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president.

10. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

In Exercises 11 and 12 which sentences are settlements? If a sentence is a statement, classify it as true or false.. 11. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

12. a) Chicago is located in the state of Illinois.

b) Get out of here!

Appendix A Answers Algebra Review A.1 ALGEBRAIC EXPRESSIONS A set collection of objects; in particular, the objects are known as the elements of the set. A = (1, 2, 3) is read, “A is the set of elements 1, 2, and 3.” In geometry, geometric figures such as lines and angles are actually sets of points. EXAMPLE 1 Classify each of the following as a true statement, a false statement, or neither. a) 4 + 3 = 7 b) An Angle has two sides, (See Figure 1.1) c) Robert E. Lee played shortstop for the Yankees.

SOLUTION a) Reflective

b) Symmetric

c) Substitution

FORMS OF THE DISTRIBUTIVE AXIOM

a(b + c) = a • b ≠ a • c a • b + c = a(b + c) a(b - c) = a • b - a • c A set is any collection of objects; in particular, the objects are known as the elements of the set. A = (1, 2, 3) is read, “A is the set of elements 1, 2, and 3.”A set is any collection of objects; in particular, the objects are known as the elements of the set. A = (1, 2, 3) is read, “A is the set of elements 1, 2, and 3.”

Exercises 1.1 In Exercises 1 and 2, which sentences are settlements? If a sentence is a statement, classify it as true or false. 1. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5.

2. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

In Exercises 3 and 4, which sentences are settlements? If a sentence is a statement, classify it as true or false. 3. a) Where do you live?

b) 4 + 7 ≠ 5. c) Washington was the first U.S. president. d) x + 3 = 7 when x = 5. c) 7 – (–3) d) 14 – (–2)

6. a) Chicago is located in the state of Illinois. b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. 7. a) Where do you live?

b) 4 + 7 ≠ 5. c) x < 6 (read as "x is less than 6") when x = 10.

8. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player.

9. a) Where do you live?

b) 4 + 7 ≠ 5. c) x < 6 (read as "x is less than 6") when x = 10.

10. a) Chicago is located in the state of Illinois.

b) Get out of here! c) x < 6 (read as "x is less than 6") when x = 10. d) Babe Ruth is remmbered as a great football player. 115

Appendix B Summary of Constructions, Postulates, and Theorems and Corollaries ◆ Constructions Section 1.2 1. To construct a segment congruent to a given segment. 2. To construct the midpoint M of a given line segment

AB. 3. To construct an angle congruent to a given angle. 4. construct the midpoint wM of a given line segment. 5. To construct the line perpendicular to a given line ata a

specified point on the given line. 6. To construct the line that is perpendicular to a given line

from a point not on the given line. 7. To construct the line that is perpendicular to a given line

from a point not on the given line. 8. To construct a segment congruent to a given segment. 9. To construct the midpoint M of a given line segment. Section 4.1

6. If two lines intersect, they intersect at a point. 7. Through two distinct point, there is exactly one line. 8. (Ruler Postulate) The measure of any line segment is a

unique positive number. . If two lines intersct, they intersect at a point. 9 10. If two lines intersect, they intersect at a point. 11. Through two distinct point, there is exactly one line. 12. (Ruler Postulate) The measure of any line segment is a

unique positive number. 13. If two lines intersct, they intersect at a point. 14. If two lines intersect, they intersect at a point. 15. Through two distinct point, there is exactly one line. 16. (Ruler Postulate) The measure of any line segment is a

unique positive number. 17. If two lines intersct, they intersect at a point.

◆ Theorems and Corollaries

1. To construct a segment congruent to a given segment. 2. To construct the midpoint M of a given line segment AB.

1.3.1 The midpoint of a line segment is unique. 1.4.1 There is one and only one angle bisector for a given

Section 5.2

1.6.1 If two lines are perpendicular, then they meet to form

1. To construct a segment congruent to a given segment.

angle. right angles. 1.6.2 If two lines are perpendicular, then they meet to form

◆ Postulates Section 1.2 1. Through two distinct point, there is exactly one line. 2. (Ruler Postulate) The measure of any line segment is a unique positive number. 3. If two lines intersct, they intersect at a point. 4. Through two distinct point, there is exactly one line. To construct the midpoint M of a given line segment. To the midpoint M of a given line segment. 5. (Ruler Postulate) The measure of any line segment is a unique positive number.

V = 𝓵wh

where 𝓵 measures the length, w the width, and h the altitude of the prism. 6. If two lines intersect, they intersect at a point. 7. Through two distinct point, there is exactly one line. 8. (Ruler Postulate) The measure of any line segment is a unique positive number. 9. If two lines intersct, they intersect at a point.

right angles. 1.6.3 The midpoint of a line segment is unique. 1.6.4 There is one and only one angle bisector for a given

angle. 1.7.1 If two lines are perpendicular, then they meet to form

right angles. 1.7.2 If two lines are perpendicular, then they meet to form

right angles. 1.7.3 The midpoint of a line segment is unique. 1.7.4 There is one and only one angle bisector for a given

angle. 1.7.5 If two lines are perpendicular, then they meet to form

right angles. 1.7.6 If two lines are perpendicular, then they meet to form

right angles. 2.1.1 The midpoint of a line segment is unique. 2.1.2 There is one and only one angle bisector for a given

angle. 2.1.3 If two lines are perpendicular, then they meet to form

right angles. 2.1.4 If two lines are perpendicular, then they meet to form

right angles AB.

Section 1.3

2.6.1 If two lines are perpendicular, then they meet to form

1. Through two distinct point, there is exactly one line. 2. (Ruler Postulate) The measure of any line segment is a

2.7.1 If two lines are perpendicular, then they meet to form

unique positive number. 3. If two lines inersct, they intersect at a point. 4. (Ruler Postulate) The measure of any line segment is a

unique positive number. 5. (Ruler Postulate) The measure of any line segment is a unique positive number.

right angles. right angles. 2.8.1 If two lines are perpendicular, then they meet to form

right angles. 2.9.1 If two lines are perpendicular, then they meet to form

right angles.

117

Chapter 1 Answers Selected Exercises and Proofs

Chapter 1

13. H: The diagonals of a parallelogram are perpendicular. C: The parallelogram is a rhombus.

1.1 Exercises 1. (a) Not a statement (b) Statement; true

(c) Statement; true (d) Statement; false 3. (a) Christopher Columbus did not cross the Atlantic Ocean. (b) Some jokes are not funny. 5. Conditional 7. Simple 9. Simple 11. H. You go to the game. You will have a great time. 13. H: The diagonals of a parallelogram are perpendicular. C: The parallelogram is a rhombus. 1. (a) Not a statement (b) Statement; true (c) Statement; true (d) Statement; false 3. (a) Christopher Columbus did not cross the Atlantic Ocean. (b) Some jokes are not funny. 5. Conditional 7. Simple 9. Simple 11. H. You go to the game. You will have a great time. 13. H: The diagonals of a parallelogram are perpendicular. C: The parallelogram is a rhombus. 1. (a) Not a statement (b) Statement; true 1.2 Exercises

Chapter 1 Review Exercises Selected Proofs 1. (a) Not a statement (b) Statement; true (c) Statement;

true (d) Statement; false PROOF Statements

Reasons

1. AC intersects BD at O

1. Given

2. ∠ s AOC and DOB are straight ∠ s, with m ∠ AOC = 180 and m ∠ DOB = 180

2. The measure of a straight angle is 180˚

3. m ∠ AOC = m∠ DOB 3. Substitution

PROOF Statements

Reasons

3. (a) Christopher Columbus did not cross the Atlantic

1. AC intersects BD at O

1. Given

Ocean. (b) Some jokes are not funny. 5. Conditional 7. Simple 9. Simple 11. H. You go to the game. You will have a great time. 13. H: The diagonals of a parallelogram are perpendicular. C: The parallelogram is a rhombus. 1. (a) Not a statement (b) Statement; true (c) Statement; true (d) Statement; false

2. ∠ s AOC and DOB are straight ∠ s, with m ∠ AOC = 180 and m ∠ DOB = 180

2. The measure of a straight angle is 180˚

1.3 Exercises 3. (a) Christopher Columbus did not cross the Atlantic

Ocean. (b) Some jokes are not funny. 5. Conditional 7. Simple 9. Simple 11. H. You go to the game. You will have a great time. 13. H: The diagonals of a parallelogram are perpendicular.

C: The parallelogram is a rhombus. 1. (a) Not a statement (b) Statement; true (c) Statement; true (d) Statement; false 3. (a) Christopher Columbus did not cross the Atlantic Ocean. (b) Some jokes are not funny. 5. Conditional 7. Simple 9. Simple 11. H. You go to the game. You will have a great time. 13. H: The diagonals of a parallelogram are perpendicular. C: The parallelogram is a rhombus. 1. (a) Not a statement (b) Statement; true C: The parallelogram is a rhombus. 1. (a) Not a statement (b) Statement; true 1.4 Exercises 3. (a) Christopher Columbus did not cross the Atlantic

Ocean. (b) Some jokes are not funny.

3. m ∠ AOC = m∠ DOB 3. Substitution 3. (a) Christopher Columbus did not cross the Atlantic

(c) Statement; true (d) Statement; false 11. H. You go to the game. You will have a great time. 13. H: The diagonals of a parallelogram are perpendicular.

C: The parallelogram is a rhombus.

Chapter 2 2.1 Exercises 1. (a) Not a statement (b) Statement; true

Ocean. (b) Some jokes are not funny. (c) Statement: true (d) Statement False (e) Ambiguous 5. Conditional 7. Simple

5. Conditional 7. Simple 9. Simple 11. H. You go to the game. You will have a great time.

119

Glossary

acute angle The y-intercept of a line is the piont.

diagonal of polygon The y-intercept of a line is the piont (0,

altitude of cylinder (prism) The y-intercept of a line is the pi-

b). where the line intersects the y-axis. . diameter The y-intercept of a line is the piont dodecagon The y-intercept of a line is the piont (0, b). The yintercept of a line is the piont (0, b). dodecahedron (regular) The y-intercept of a line is the piont (0, b). where the line intersects the y-axis.

altitude of parallelogram The y-intercept of a line is the piont

edge of polyhedron The y-intercept of a line is the piont (0,

acute triangle They-intercept of a line is the piont (0, b). where

the line intersects the y-axis. .

adjacent angles The y-intercept of a line is the piont (0, b).

where the line intersects the y-axis. .

altitude of cone (pryamid) The y-intercept of a line is the piont

ont (0, b).

(0, b). where the line intersects the y-axis. . altitude of trapezoid The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. altitude of triangle The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. angle The y-intercept of a line is the piont. The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. angle bisector The y-intercept of a line is the piont. arc The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. base The y-intercept of a line is the piont (0, b).

base angles of isosoceles triangle The y-intercept of a line is

the piont (0, b). where the line intersects the y-axis. . bases of trapezoid The y-intercept of a line is the piont bisector of angle The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve center of circle The y-intercept of a line is the piont (0, b).

where the line intersects the y-axis.

central angle of circle The y-intercept of a line is the piont

b).ewhere the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line. equilangular polygon The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve equivalent equations The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. equilangular polygon The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve equivalent equations The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. exterior To find b, substitute 0 for x in the equaion of the line and solve for y. extremes of a proportion The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. union the joining together of any two sets, such as geometric

figures

Valid argument an argument in which the conclusion follows-

vlogically from previusly stated (and accepted) premises or assumptions

(0, b).

zero The y-intercept of a line is the piont (0, b).

the piont.

zero equation The y-intercept of a line is the piont (0, b). where

central angle of regular polygon The y-intercept of a line is chord of circle The y-intercept of a line is the piont (0, b).

where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. decagon The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. deduction The y-intercept of a line is the piont. degree The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. .

zero and negative-integers The y-intercept of a line is the piont.

the line intersects the y-axis. .

zero exponent The y-intercept of a line is the piont (0, b).

where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve zero quotient The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve zero variable The y-intercept of a line is the piont (0, b). where the line intersects the y-axis. To find b, substitute 0 for x in the equaion of the line and solve for y. 0 for x in the equaion. 121

Index

A Acute angle, 9 Acute triangle, 209 Additional Property of Equality, 579 of Inequality, 23 Alternate interior angles, 209 Altitude of cones, 18 of cylinder, 259 of parallelogram, 260 of prism, 371, 374 of triangle, 39-40 B Base, 9 Base of angles of an isosceles triangle, 209, 240 Binomials complex numbers, 579 decimals, 23 fractions, 18 nonomials, 259 polynomials, 260 rational expressions and the use of, 371, 374 real numbers, 39-40 C Calculus value, 9 Careers and Mathematics Athletes, Coaches, and Related Workers, 579 Carpenters, 23 Construction Manager, 18 Fodd-processing Occupations baker, 259 Medical Scientist, 260 Psychologist, 371, 374 Retail Salesperson, 39-40 Circle value, 9 Coeffiient value functions, 209 Complex Numbers complex numbers, 579 decimals, 23 fractions, 18 nonomials, 259 polynomials, 260 rational expressions, 371, 374 real numbers, 39-40 Cone value, 9 Contraint value functions, 209 Coordinates complex numbers, 579 decimals, 23 fractions, 18

nonomials, 259 polynomials, 260 rational expressions, 371, 374 real numbers, 39-40 Cubing value, 9 Cube roots value functions, 209 Cylinder complex numbers, 579 decimals, 23 fractions, 18 nonomials, 259 polynomials, 260 rational expressions, 371, 374 real numbers, 39-40 D Absolute value, 9 Absolute value functions, 209 Adding complex numbers, 579 decimals, 23 fractions, 18 nonomials, 259 polynomials, 260 rational expressions, 371, 374 real numbers, 39-40 E Einstein, Albert, 9 Elements of a set, 209 Ellipses complex numbers, 579 decimals, 23 fractions, 18 nonomials, 259 polynomials, 260 rational expressions, 371, 374 real numbers, 39-40 F Factor value, 9 Functions, 209 Formulas complex numbers, 579 decimals, 23 fractions, 18 nonomials, 259 polynomials, 260 rational expressions, 371, 374 real numbers, 39-40 Functional Notations, 9 Fundamentals Athletes, Coaches, and Related Workers, 579 Carpenters, 23 Construction Manager, 18

Foodd-processing Occupations baker, 259 Medical Scientist, 260 Psychologist, 371, 374 Retail Salesperson, 39-40 Fundamental theorem value, 9 G Galileo, 9 Graping value functions, 209 Graphing Numbers complex numbers, 579 decimals, 23 fractions, 18 nonomials, 259 polynomials, 260 rational expressions, 371, 374 Graphing value, 9 Graphing value functions, 209 Graphings Coordinates complex numbers, 579 decimals, 23 fractions, 18 nonomials, 259 polynomials, 260 rational expressions, 371, 374 real numbers, 39-40 Graphing value, 9 Graphing roots value functions, 209 Graphing complex numbers, 579 decimals, 23 fractions, 18 nonomials, 259 polynomials, 260 rational expressions, 371, 374 real numbers, 39-40 Gravity, 336 Great Circle, 436 Great Pyramids, 403 H Heptagon, 100, 596 Hexagram, 105 Heptagon, 100, 596 Hexagram, 105 Heptagon, 100, 596 Hexagram, 105 Heptagon, 100, 596 Hexagram, 105 Heptagon, 100, 596 Hexagram, 105 Heptagon, 100, 596 Hexagram, 105 Heptagon, 100, 596 Hexagram, 105 123

Index of Applications

A Agriculture 9 Aluminum 209 Air duct 376 Allocation of supplies 209 Angle of depression (elevation) 487, 496, 500, 505 B Ball 9 Balloon 209, 240 Barn 579 Baseball 45 C Calculus value 9 Campsite 428 Circle value, 9 Coeffiient value functions, 209 Complex numbers Cone value, 9 Contraint value functions, 209 Coordinates Cubing value, 9 Cube roots value functions, 209 Cylinder D Deck construction 9 Detour 209 Dials Dice 421, 426, 433, 444 498, 538, 609, 742 E Electrician 224, 371 Excavation 339, 409 Exhause chute 420 F Factor value 9 Functions 209 Formulas Functional ntations 9 Fundamentals 579 Food-processing 259 Flagpole, 260 Floor pan 9 Fold-down bed 192 Foundaiton 1886 G Galileo, 9 Garden, 209 Gasket Graphing value, 9

Graphing value functions, 209 Graphings coordinates Graphing value, 9 Graphing roots value functions, 209 Graphing Galileo, 9 Garden, 209 Gasket Graphing value, 9 Graphing value functions, 209 Graphings coordinates Graphing value, 9 Graphing roots value functions, 209 Graphing Galileo, 9 Garden, 209 Gasket Graphing value, 9 Graphing value functions, 209 Graphings coordinates Graphing value, 9 Graphing roots value functions, 209 Graphing Tablet H Hanging sgn 9 Hinge 209 Home plate 106 Hot air balloon 9 House 209, 212, 218 Hinge 209 Home plate 106 Hot air balloon 9 I Ice cream cone, 9 Icicle 209 Ironing board 193 Island 9 J Jogging 259 Joing savings 209 Joy stick 193 K Jogging 259 Joing savings 20 L Ladder 23, 210, 244, 489, 497, 559 669, 708 Lamppost 20, 98 Lawn roller 94 Level 82 Light fixture 99

Logo 110, 114, 171 Ladder 23, 210, 244, 489, 497, 559 Lamppost 20, 98 Lawn roller 94 Level 82 Light fixture 99 Logo 110, 114, 171 Ladder 23, 210, 244, 489, 497, 559 Lamppost 20, 98 Lawn roller 94 Level 82 Light fixture 99 Logo 110, 114, 171 Ladder 23, 210, 244, 489, 497, 559 Lamppost 20, 98 Lawn roller 94 Level 82 Light fixture 99 Logo 110, 114, 171 Ladder 23, 210, 244, 489, 497, 559 Lamppost 20, 98 Lawn roller 94 Level 82 Logo 110, 114, 171 Ladder 23, 210, 244, 489, 497, 559 Lamppost 20, 98 Lawn roller 94 Level 82 Light fixture 99 Logo 110, 114, 171 M Manufacturing, 141 Maps, 98 Margarine tub, 432 Measuring wheel, 382 Mirrors, 85 N Negation, 2, 80 Negative inference, 81-82 Negative numbers, 552 Negative reciprocals, 552 Nine-point circle, 346 Nonagon, 100 Noncollinear points, 597 Noncoplanar points, 597 Non-Euclidean geometry, 118-120 0 Oblique, 2, 80 Oblique prism 81-82 Octagon, 100, 597 Open sentence, 2 Open statement, 4 ix