The Bigger Picture SIMPLIFYING EXPRESSIONS A.1 AND SOLVING EQUATIONS. I. Simplifying Expressions A. Real Numbers 1. Add: (Sec. 1.

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The Bigger Picture

SIMPLIFYING EXPRESSIONS AND SOLVING EQUATIONS

A.1

A p p e n d i x

I. Simplifying Expressions A. Real Numbers

1. Add: (Sec. 1.4)

Adding like signs.

Add absolute values. Attach common sign.

Adding different signs.

Subtract absolute values. Attach the sign of the number with the larger absolute value.

2. Subtract: Add the first number to the opposite of the second number.

(Sec. 1.5)

3. Multiply or divide: Multiply or divide the two numbers as usual. If the signs are the same, the answer is positive. If the signs are different, the answer is negative. (Sec. 1.6)

B. Exponents (Sec. 3.2)

C. Polynomials

1. Add: Combine like terms. (Sec. 3.4)

= 12y² - 5y - 8

13y²+ 6y + 72 + 19y² - 11y - 152 = 3y² + 6y + 7 + 9y²- 11y - 15 x⁷

#

_x5

= x¹²; 1x⁷2⁵ = x³⁵; x⁷

x⁵ = x²; x⁰ = 1; 8^-² = 1 8² =

1 64 -10

#

_{3 = - 30,} _-_{81 ,} _{1-32 = 27}

17 - 25 = 17 + 1-252 = -8 - 7 + 3 = - 4 - 1.7 + 1-0.212 = -1.91

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4. Divide: (Sec. 3.7)

a. To divide by a monomial, divide each term of the polynomial by the monomial.

b. To divide by a polynomial other than a monomial, use long division.

D. Factoring Polynomials

See the Chapter 4 Integrated Review for steps. (Sec. 4.5)

Factor out GCF—always first step.

Factor trinomial.

Factor further—each difference of squares.

E. Rational Expressions

1. Simplify: Factor the numerator and denominator. Then divide out factors of 1 by dividing out common factors in the numerator and denominator.

(Sec. 5.1)

2. Multiply: Multiply numerators, then multiply denominators. (Sec. 5.2)

3. Divide: First fraction times the reciprocal of the second fraction. (Sec. 5.2)

4. Add or subtract: Must have same denominator. If not, find the LCD and write each fraction as an equivalent fraction with the LCD as denominator. (Sec. 5.4)

F. Radicals

1. Simplify square roots: If possible, factor the radicand so that one factor is a perfect square. Then use the product rule and simplify. (Sec. 8.2)

275 = 225

#

_{3 = 225}

#

_{23 = 523}

= 9x + 45 - 10x - 10 101x + 52 =

-x + 35 101x + 52 9

10 - x + 1 x + 5 =

91x + 52 101x + 52 -

101x + 12 101x + 52 14

x + 5 , x + 1

2 =

14 x + 5

#

²

x + 1 =

28 1x + 521x + 12 5z

2z² - 9z - 18

#

^{22z + 33}

10z =

5

#

_z

12z + 321z - 62

#

¹¹^{12z + 32}

2

#

₅

#

_z ⁼

11 21z - 62 x² - 9

7x² - 21x = 1x + 321x - 32 7x1x - 32 =

x + 3 7x

= 31x + 521x - 521x + 121x - 12 = 31x² - 2521x² - 12

3x⁴ - 78x² + 75 = 31x⁴ - 26x² + 252 x - 6 + 40

2x + 5 2x + 5^2x² ^- ^{7x + 10}

2x² +

-

5x - 12x + 10 -

+

12x -

+

30 40 8x² + 2x - 6

2x =

8x² 2x +

2x 2x -

6

2x = 4x + 1 - 3 x

A P P E N D I X A I THE BIGGER PICTURE

669

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2. Add or subtract: Only like radicals (same index and radicand) can be added or subtracted. (Sec. 8.3)

3. Multiply or divide: (Sec. 8.4)

4. Rationalizing the denominator: (Sec. 8.4) a. If denominator is one term,

b. If denominator is two terms, multiply by 1 in the form of

II. Solving Equations and Inequalities

A. Linear Equations: Power on variable is 1 and there are no variables in denominator. (Sec. 2.3)

Linear equation. (If fractions, multiply by LCD.) Use the distributive property.

Add 21 to both sides.

Subtract 4x from both sides.

Divide both sides by 3.

B. Linear Inequalities: Same as linear equation except if you multiply or divide by a negative number, then reverse direction of inequality. (Sec. 2.7)

Linear inequality.

Subtract 11 from both sides.

Simplify.

x Ú 3

Divide both sides by 4 and reverse the direction of the inequality symbol.

-4x -4 Ú

-12 -4 - 4x … - 12 - 4x + 11 … - 1

x = 9 3x = 27 7x = 4x + 27 7x - 21 = 4x + 6 71x - 32 = 4x + 6

13 3 + 22

= 13 3 + 22

#

^{3 - 22}

3 - 22

=

13

A

^{3 - 22}

B

9 - 2 =

13

A

^{3 - 22}

B

7 conjugate of denominator

conjugate of denominator. 5

211 =

5

#

₂₁₁

211

#

₂₁₁ ⁼

5 211 11 211

#

_{23 = 233;}²¹⁴⁰

27 = A

140

7 = 220 = 24

#

_{5 = 2 25}

1a

#

_{2b = 2ab;}^1a

2b = A

a b.

8 210 - 240 + 25 = 8 210 - 2 210 + 25 = 6 210 + 25

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A P P E N D I X A I THE BIGGER PICTURE

671

D. Equations with Rational Expressions: Make sure the proposed solution does not make the denominator 0. (Sec. 5.5)

Equation with rational expressions.

Multiply through by

Simplify.

Use the distributive property.

Simplify and move variable terms to right side.

Divide both sides by 2.

E. Proportions: An equation with two ratios equal. Set cross products equal, then solve. Make sure the proposed solution does not make the denominator 0.

(Sec. 5.6)

Set cross products equal.

Multiply.

Write equation with variable terms on one side and constants on the other.

F. Equations with Radicals: To solve, isolate a radical, then square both sides.

You may have to repeat this. Check possible solution in the original equation.

(Sec. 8.5)

Subtract 7 from both sides.

Square both sides.

Set terms equal to 0.

Factor.

Set each factor equal to 0 and solve.

x = 0 or x = 15

0 = x1x - 152 0 = x² - 15x x + 49 = x² - 14x + 49 2x + 49 = x - 7

2x + 49 + 7 = x x = 15 10x - 15 = 9x 512x - 32 = 9

#

_x

5 x =

9 2x - 3

-3 2 = x - 3 = 2x 3x - 3 - x = 4x 31x - 12 - x

#

_{1 = x}

#

₄

x1x - 12.

x1x - 12

#

³

x - x1x - 12

#

¹

x - 1 = x1x - 12

#

⁴

x - 1 3

x - 1 x - 1 =

4 x - 1

← ←

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A p p e n d i x

Although the sum of two squares usually does not factor, the sum or difference of two cubes can be factored and reveal factoring patterns. The pattern for the sum of cubes can be checked by multiplying the binomial and the trinomial The pattern for the difference of two cubes can be checked by multiplying the binomial x - yby the trinomial x² + xy + y².

x² - xy + y².

x + y

Factoring Sums and Differences of Cubes

EXAMPLE 1

Factor

Solution: First, write the binomial in the form

Write in the form

If we replace a with x and b with 2 in the formula above, we have

= 1x + 221x² - 2x + 42 x³ + 2³ = 1x + 22[x² - 1x2122 + 2²]

a³+ b³.

x³ + 8 = x³ + 2³

a³ + b³. x³ + 8.

Sum or Difference of Two Cubes a³ - b³ = 1a - b21a² + ab + b²2 a³ + b³ = 1a + b21a² - ab + b²2

When factoring sums or differences of cubes, notice the sign patterns.

x³+y³=(x+y)(x²-xy+y²) same sign

opposite sign always positive

Helpful

Hint

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A P P E N D I X B I FACTORING SUMS AND DIFFERENCES OF CUBES

673 EXAMPLE 2

^Factor

Solution:

Write in the form

= 1y - 321y² + 3y + 92 = 1y - 32[y² + 1y2132 + 3²]

a³- b³.

y³ - 27 = y³ - 3³

y³ - 27.

EXAMPLE 3

Factor Solution:

= 14x + 12116x² - 4x + 12 = 14x + 12[14x2² - 14x2112 + 1²] 64x³ + 1 = 14x2³ + 1³

64x³ + 1.

EXAMPLE 4

Factor

Solution: Remember to factor out common factors first before using other fac- toring methods.

Factor out the GCF 2.

Difference of two cubes.

= 213a - 2b219a² + 6ab + 4b²2

= 213a - 2b2[13a2² + 13a212b2 + 12b2²] = 2[13a2³ - 12b2³]

54a³ - 16b³ = 2127a³ - 8b³2 54a³ - 16b³.

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Student Solutions Manual PH Math/Tutor Center CD/Video for Review MathXL^® MyMathLab F O R E X T R A H E L P

B E X E R C I S E S E T

Factor the sum or difference of two cubes. See Examples 1 through 4.

1.

3.

5.

7.

9.

11.24x⁴ - 81xy³ 3x12x - 3y214x² + 6xy + 9y²2 1x + 521x² - 5x + 252

x³ + 125

1xy - 421x²y² + 4xy + 162 x³y³ - 64

51k + 221k² - 2k + 42 5k³ + 40

12a + 1214a² - 2a + 12 8a³ + 1

1a + 321a² - 3a + 92 a³ + 27

Factor the binomials completely.

13.

15.

17.

19.s³ - 64t³ 1s - 4t21s² + 4st + 16t²2 1t - 721t² + 7t + 492 t³ - 343

81r - 221r² + 2r + 42 8r³ - 64

13 - t219 + 3t + t²2 27 - t³

2.

4.

6.

8.

10.

12.375y⁶ - 24y³ 3y³15y - 22125y² + 10y + 42 1a - 621a² + 6a + 362

a³ - 216

12x - y214x² + 2xy + y²2 8x³ - y³

61r - 321r² + 3r + 92 6r³ - 162

14x - 12116x² + 4x + 12 64x³ - 1

1b - 221b² + 2b + 42 b³ - 8

14.

16.

18.

20.8t³ + s³ 12t + s214t² - 2ts + s²2 1s + 621s² - 6s + 362 s³ + 216

213r + 1219r² - 3r + 12 54r³ + 2

15 + r2125 - 5r + r²2 125 + r³

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Two other measures of central tendency are the median and the mode.

The median of an ordered set of numbers is the middle number. If the number of items is even, the median is the mean of the two middle numbers. The mode of a set of numbers is the number that occurs most often. It is possible for a data set to have no mode or more than one mode.

675

A p p e n d i x

Mean, Median, and Mode

It is sometimes desirable to be able to describe a set of data, or a set of numbers, by a single “middle” number. Three such measures of central tendency are the mean, the median, and the mode.

The most common measure of central tendency is the mean (sometimes called the arithmetic mean or the average). The mean of a set of data items, denoted by is the sum of the items divided by the number of items.

x,

EXAMPLE 1

Seven students in a psychology class conducted an experiment on mazes. Each student was given a pencil and asked to successfully complete the same maze. The timed results are below.

Student Ann Thanh Carlos Jesse Melinda Ramzi Dayni

Time (seconds) 13.2 11.8 10.7 16.2 15.9 13.8 18.5

a. Who completed the maze in the shortest time? Who completed the maze in the longest time?

b. Find the mean.

c. How many students took longer than the mean time? How many students took shorter than the mean time?

Solution:

a. Carlos completed the maze in 10.7 seconds, the shortest time. Dayni completed the maze in 18.5 seconds, the longest time.

b. To find the mean, find the sum of the data items and divide by 7, the number of items.

c. Three students, Jesse, Melinda, and Dayni, had times longer than the mean time. Four students, Ann, Thanh, Carlos, and Ramzi, had times shorter than the mean time.

x = 13.2 + 11.8 + 10.7 + 16.2 + 15.9 + 13.8 + 18.5

7 =

100.1 7 = 14.3 x,

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EXAMPLE 2

Find the median and the mode of the following set of numbers. These numbers were high temperatures for fourteen consecutive days in a city in Montana.

76, 80, 85, 86, 89, 87, 82, 77, 76, 79, 82, 89, 89, 92 Solution:

First, write the numbers in order.

76, 76, 77, 79, 80, 82, 82, 85, 86, 87, 89, 89, 89, 92

two mode

middle numbers

Since there are an even number of items, the median is the mean of the two middle numbers.

The mode is 89, since 89 occurs most often.

median = 82 + 85 2 = 83.5

 

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The eight tallest buildings in the United States are listed below. Use this table for Exercises 9 through 12.

Building Height (Feet)

Sears Tower, Chicago, IL 1454

Empire State, New York, NY 1250

Amoco, Chicago, IL 1136

John Hancock Center, Chicago, IL 1127

First Interstate World Center, Los Angeles, CA 1107

Chrysler, New York, NY 1046

NationsBank Tower, Atlanta, GA 1023

Texas Commerce Tower, Houston, TX 1002

9. Find the mean height for the five tallest buildings.

1214.8 ft

11. Find the median height for the eight tallest buildings.

1117 ft

677 C E X E R C I S E S E T

For each of the following data sets, find the mean, the median, and the mode. If necessary, round the mean to one decimal place.

1. 21, 28, 16, 42, 38

mean: 29, median: 28, no mode

3. 7.6, 8.2, 8.2, 9.6, 5.7, 9.1

mean: 8.1, median: 8.2, mode: 8.2

5. 0.2, 0.3, 0.5, 0.6, 0.6, 0.9, 0.2, 0.7, 1.1 mean: 0.6, median: 0.6, mode: 0.2 and 0.6

7. 231, 543, 601, 293, 588, 109, 334, 268 mean: 370.9, median: 313.5, no mode

2. 42, 35, 36, 40, 50

mean: 40.6, median: 40, no mode

4. 4.9, 7.1, 6.8, 6.8, 5.3, 4.9

mean: 6.0, median: 6.05, mode: 6.8 and 4.9

6. 0.6, 0.6, 0.8, 0.4, 0.5, 0.3, 0.7, 0.8, 0.1 mean: 0.5, median: 0.6, mode: 0.6 and 0.8

8. 451, 356, 478, 776, 892, 500, 467, 780 mean: 587.5, median: 489, no mode

10. Find the median height for the five tallest buildings.

1136 ft

12. Find the mean height for the eight tallest buildings.

Round to the nearest tenth. 1143.1 ft

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25. _____, _____ , 16, 18, _____

The mode is 21.

The median is 20.

21, 21, 20

26. _____, _____ , _____ , _____ , 40 The mode is 35.

The median is 37.

The mean is 38.

During an experiment, the following times (in seconds) were recorded: 7.8, 6.9, 7.5, 4.7, 6.9, 7.0.

In a mathematics class, the following test scores were recorded for a student: 86, 95, 91, 74, 77, 85.

The following pulse rates were recorded for a group of fifteen students: 78, 80, 66, 68, 71, 64, 82, 71, 70, 65, 70, 75, 77, 86, 72.

13. Find the mean.

6.8 sec

14. Find the median.

6.95 sec

15. Find the mode.

6.9 sec

16. Find the mean. Round to the nearest hundredth.

84.67

85.5

no mode

19. Find the mean.

73

71

70 and 71

22. How many rates were higher than the mean?

6

23. How many rates were lower than the mean?

9

Find the missing numbers in each set of numbers. (These numbers are not necessarily in numerical order.) 24. Have each student in your algebra class take his/her

pulse rate. Record the data and find the mean, the median, and the mode.

answers may vary

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679

A p p e n d i x

The set 56is not the empty set. It is a set with one element,.

EXAMPLE 1

Given the following sets, determine whether each statement is true or false.

a.

b.

c.

d.

Solution:

a. True, since 2 is a listed element of N

b. False, since 5 is not listed as an element of E c. True, since 5 is an odd number

d. False, since 10 is not an odd number 10 H O

5 H O 5 H E 2 H N

N = 50, 1, 2, 3, 4, 56 E = 50, 2, 4, 6, 8, 106 O = 5x ƒ x is an odd number6

Sets

Objective

Determining Whether an Object Is an Element of a Set

A set is a collection of objects, called elements or members. The elements of a set are listed or described between a pair of braces, Two common ways of representing a set are by roster form or by set builder notation. Examples of each are as follows:

Roster form—elements are listed.

Set builder notation

Set B is read as “the set of all x such that x is less than or equal to 3.”

The symbol means “is an element of.” For set A above, we can write

means the number 3 is an element of set A.

Also,

means the number 6 is not an element of set A.

A set that has no elements is called the empty set or the null set. The empty set (or null set) is symbolized by or For example, if set C is the set of all positive numbers less than 0, then

C = 5 6 or C = .

5 6 .

6 x A 3 H A

H B = 5x ƒ x … 36 A = 51, 2, 3, 4, 56

5 6.

A

O b j e c t i v e s

Determine Whether an Object Is an Element of a Set.

Determine Whether a Set Is a Subset of Another Set.

Find Unions and Intersection of Sets.

C B A

Helpful

Hint

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Set A is a subset of set B if every element of A is also an element of B. In symbols, we write and this is illustrated to the right.

A 8 B,

Objective

Determining Whether a Set Is a Subset of Another Set

B

From Example 2b, we see that every set is a subset of itself. For example, Why? Because every element of A is always an element of A.

A 8 A.

means set A is a subset of set B. The symbol means “is not a subset.”

Thus,

means set A is not a subset of set B.

EXAMPLE 2

Determine whether a.

b.

c.

d.

Solution:

a. since every element of A is also an element of B.

b. since every element of A is also an element of B.

c. because there are elements of A that are not in B. For example, 3 is an element of A but is not an element of B.

d. A h Bbecause but n HA, n xB.

A h B A 8 B A 8 B

A = 5~, ^, n6, B = 5~, ^6 A = 5x ƒ x 6 56, B = 5x ƒ x 6 16 A = 52, 4, 66, B = 52, 4, 66

A = 50, 76, B = 50, 1, 2, 3, 4, 5, 6, 76 A 8 B.

A h B A 8 B h

A B

Helpful Hint

If is true, then there must be at least one element of set A that is not in set B.

A h B

Helpful

Hint

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A P P E N D I X D I SETS

681

•

You may list the elements of a set in any order.

•

There is no need to list an element of a set more than once.

EXAMPLE 3

Let and Find

a.

b.

Solution:

a. all the numbers in set R or set S (or both) b. the only numbers in both sets

EXAMPLE 4

Let and Find

a.

b.

Solution:

a. Any order of the elements is fine.

b. or There are no elements common to both sets. The intersection

is the empty set.

M ¨ N = 5 6

M ´ N = 55, 10, 15, 20, 256 M ¨ N

M ´ N

N = 55, 15, 256.

M = 510, 206 R ¨ S = 50, 26,

R ´ S = 50, 1, 2, 3, 4, 56, R ¨ S

R ´ S

S = 50, 1, 2, 3, 46.

R = 50, 2, 56

Helpful

Hint

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For each set described, write the set in roster form.

5^, n, ~6 8 5^, n, ~6

D E X E R C I S E S E T

7.

or

8.

or

5 65x ƒ x is a number that is both even and odd6 5 6

5x ƒ x is an integer between 1 and 26

Objectives A B Mixed Practice Determine whether each statement is true or false. See Examples 1 and 2.

9.3 H51, 3, 5, 7, 96 true 10.6 H51, 3, 5, 7, 96 false 11.536 8 51, 3, 5, 7, 96 true

12. false 13.

true

14. true

5a, e, i, o, u6 8 5a, e, i, o, u6 566 8 51, 3, 5, 7, 96

15. set of days of the week false

16. set of months of

the year false

17.

true

9 x5xƒx is an even number6 5Sunday6 8 the

5May6 8 the

18.

true

19.5a6 h theset of vowels false 20. the set of polygons false 10 x5xƒx is an odd number6

Objective Given the sets, find each union or intersection. See Examples 3 and 4.

A = 51, 2, 3, 4, 5, 66 B = 52, 4, 66 C = 51, 3, 56 D = 576 C

21. 22. 23.A ¨ B 52, 4, 66 24.A ¨ C 51, 3, 56

51, 2, 3, 4, 5, 66 A ´ C

51, 2, 3, 4, 5, 66 A ´ B

25.C ´ D 51, 3, 5, 76 26.C ¨ D 27. B ¨ D 28.B ´ D 52, 4, 6, 76

1. The set of negative integers from to 2. The set of integers between and 1.

506

-1 5-9, -8, -7, -66

-5.

-10

3. The set of the days of the week starting with the letter T.

4. The set of the first five letters of the alphabet 5a, b, c, d, e6

5Tuesday, Thursday6

5. The set of whole numbers 50, 1, 2, 3, 4, Á 6 6. The set of natural numbers 51, 2, 3, 4, 5, Á 6

5^6 h

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683

A p p e n d i x

Angles

An angle whose measure is greater than 0° but less than 90° is called an acute angle.

A right angle is an angle whose measure is 90°. A right angle can be indi- cated by a square drawn at the vertex of the angle, as shown below.

An angle whose measure is greater than 90° but less than 180° is called an obtuse angle.

An angle whose measure is 180° is called a straight angle.

Two angles are said to be complementary if the sum of their measures is 90°.

Each angle is called the complement of the other.

Two angles are said to be supplementary if the sum of their measures is 180°.

Each angle is called the supplement of the other.

Acute angle

Complementary angles m1 m2 90

Supplementary angles m3 m4 180

Right angle Obtuse angle Straight angle

1 2

3

4

Review of Angles, Lines, and Special Triangles

The word geometry is formed from the Greek words, geo, meaning earth, and metron, meaning measure. Geometry literally means to measure the earth.

This appendix contains a review of some basic geometric ideas. It will be as- sumed that fundamental ideas of geometry such as point, line, ray, and angle are known. In this appendix, the notation is read “angle 1” and the notation is read “the measure of angle 1.”

We first review types of angles.

m∠1

∠1

EXAMPLE 1

If an angle measures 28°, find its complement.

Solution: Two angles are complementary if the sum of their measures is 90°. The complement of a 28° angle is an angle whose measure is To check, notice that

Plane is an undefined term that we will describe. A plane can be thought of as a flat surface with infinite length and width, but no thickness. A plane is two

28° + 62° = 90°.

90° - 28° = 62°.

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dimensional. The arrows in the following diagram indicate that a plane extends indefinitely and has no boundaries.

Lines

Two lines are parallel if they lie in the same plane but never meet. Intersecting lines meet or cross in one point.

Two lines that form right angles when they intersect are said to be perpendicular.

Parallel lines

Intersecting lines

Intersecting lines that are perpendicular

Figures that lie on a plane are called plane figures. Lines that lie in the same plane are called coplanar.

Two intersecting lines form vertical angles. Angles 1 and 3 are vertical angles.

Also angles 2 and 4 are vertical angles. It can be shown that vertical angles have equal measures.

4 2

3

1

m1 m3 m2 m4

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A P P E N D I X E I REVIEW OF ANGLES, LINES, AND SPECIAL TRIANGLES

685

Parallel Lines Cut by a Transversal

1. If two parallel lines are cut by a transversal, then a. corresponding angles are equal and

b. alternate interior angles are equal.

2. If corresponding angles formed by two lines and a transversal are equal, then the lines are parallel.

3. If alternate interior angles formed by two lines and a transversal are equal, then the lines are parallel.

Corresponding angles: and and and and and

Exterior angles: and

Interior angles: and

Alternate interior angles: and and

These angles and parallel lines are related in the following manner.

∠5.

∠6, ∠4

∠3

∠6.

∠3, ∠4, ∠5,

∠8.

∠1, ∠2, ∠7,

∠8.

∠4

∠6,

∠7, ∠2

∠5, ∠3

∠1

EXAMPLE 2

Given that lines m and n are parallel and that the measure of angle 1 is 100°, find the measures of angles 2, 3, and 4.

Solution:

since angles 1 and 2 are vertical angles.

since angles 1 and 4 are alternate interior angles.

since angles 4 and 3 are supplementary angles.

A polygon is the union of three or more coplanar line segments that intersect each other only at each end point, with each end point shared by exactly two segments.

A triangle is a polygon with three sides.The sum of the measures of the three an- gles of a triangle is 180°. In the following figure, m∠1 + m∠2 + m∠3 = 180°.

m∠3 = 180° - 100° = 80°

m∠4 = 100°

m∠2 = 100°

1 2 l

m

n

4 3

2 3

1

EXAMPLE 3

Find the measure of the third angle of the triangle shown.

95 45

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Solution: The sum of the measures of the angles of a triangle is 180°. Since one angle measures 45° and the other angle measures 95°, the third angle measures

Two triangles are congruent if they have the same size and the same shape. In congruent triangles, the measures of corresponding angles are equal and the lengths of corresponding sides are equal. The following triangles are congruent.

180° - 45° - 95° = 40°.

1 3

a 2 b

c

4 6

x 5 y

z

Corresponding angles are equal: and

Also, lengths of corresponding sides are equal:

and

Any one of the following may be used to determine whether two triangles are congruent.

c = z.

a = x, b = y, m∠3 = m∠6.

m∠1 = m∠4, m∠2 = m∠5,

Congruent Triangles

1. If the measures of two angles of a triangle equal the measures of two angles of another triangle and the lengths of the sides between each pair of angles are equal, the triangles are congruent.

and

2. If the lengths of the three sides of a triangle equal the lengths of corre- sponding sides of another triangle, the triangles are congruent.

and

3. If the lengths of two sides of a triangle equal the lengths of corresponding c = z b = y a = x a = x m∠2 = m∠4 m∠1 = m∠3

1

2

a 3

4 x

a b

c

x y

z

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A P P E N D I X E I REVIEW OF ANGLES, LINES, AND SPECIAL TRIANGLES

687

corresponding sides are in proportion. The following triangles are similar. (All similar triangles drawn in this appendix will be oriented the same.)

Corresponding angles are equal: and

Also, corresponding sides are proportional:

Any one of the following may be used to determine whether two triangles are similar.

a x = b

y = c z.

m∠3 = m∠6.

m∠1 = m∠4, m∠2 = m∠5,

a

x

b y

c

z 1

2 3

4

5 6

Similar Triangles

1. If the measures of two angles of a triangle equal the measures of two angles of another triangle, the triangles are similar.

and

2. If three sides of one triangle are proportional to three sides of another tri- angle, the triangles are similar.

3. If two sides of a triangle are proportional to two sides of another triangle and the measures of the included angles are equal, the triangles are similar.

and a x = b

y m∠1 = m∠2

a x =

b y =

c z m∠3 = m∠4 m∠1 = m∠2

a

x

b y

c

z 1

3

2 4

a

b

x

y 1 2

EXAMPLE 4

Given that the following triangles are similar, find the missing length x.

10 x

2 3

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Solution: Since the triangles are similar, corresponding sides are in proportion.

Thus, To solve this equation for x, we cross multiply.

The missing length is 15 units.

A right triangle contains a right angle. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. The Pythagorean theorem gives a formula that relates the lengths of the three sides of a right triangle.

x = 15 2x = 30

2 3 =

10 x 2 3 =

10 x .

The Pythagorean Theorem

If a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse, then a² + b² = c².

a c

b

hypotenuse

legs

EXAMPLE 5

Find the length of the hypotenuse of a right triangle whose legs have lengths of 3 centimeters and 4 centimeters.

Solution: Because we have a right triangle, we use the Pythagorean theorem.The legs are 3 centimeters and 4 centimeters, so let and in the formula.

3² + 4² = c² a² + b² = c²

b = 4 a = 3

4 centimeters

3 centimeters

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689

Find the complement of each angle. See Example 1.

E E X E R C I S E S E T

1. 19° 71° 2. 65° 25° 3. 70.8° 19.2°

Find the supplement of each angle.

4. 5. 783 6. 19.6° 70.4°

4° 111

4° 441

3° 452

3°

7. 150° 30° 8. 90° 90° 9. 30.2° 149.8°

10. 81.9° 98.1° 11. 12. 141

9° 1658

9° 1001

2° 791

2° 13. If lines m and n are parallel, find the measures of

angles 1 through 7. See Example 2.

14. If lines m and n are parallel, find the measures of angles 1 through 5. See Example 2.

m∠4 = 110°, m∠5 = 120°

m∠1 = 60°, m∠2 = 50°, m∠3 = 70°, m∠2 = m∠3 = m∠4 = m∠6 = 70°

m∠1 = m∠5 = m∠7 = 110°,

In each of the following, the measures of two angles of a triangle are given. Find the measure of the third angle.

See Example 3.

18. 44°, 19° 117° 19. 30°, 60° 90° 20. 67°, 23° 90°

15. 11°, 79° 90° 16. 8°, 102° 70° 17. 25°, 65° 90°

In each of the following, the measure of one angle of a right triangle is given. Find the measures of the other two angles.

21. 45° 45°, 90° 22. 60° 30°, 90° 23. 17° 73°, 90°

24. 30° 60°, 90° 25. 26. 72.6° 17.4°, 90°

501 4°, 90°

393 4°

Given that each of the following pairs of triangles is similar, find the missing length x. See Example 4.

27. x = 6 28. x = 8

m l

n

2 1

110 3

4 5

7 6

m

n 2

70 1

4 3 5

60

12 4

18 x

4 4 x x

7 14

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29. x = 4.5 30. x = 48

Use the Pythagorean theorem to find the missing lengths in the right triangles. See Example 5.

31. 10 32. 13

33. 12 34. 16

3

x 6

9

4

2 x

60

24 5

6

8

5

12

13 5

12

20

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691 Tables

TABLE OF SQUARES AND SQUARE ROOTS

F.1

A p p e n d i x

n n² n n²

1 1 1.000 51 2601 7.141

2 4 1.414 52 2704 7.211

3 9 1.732 53 2809 7.280

4 16 2.000 54 2916 7.348

5 25 2.236 55 3025 7.416

6 36 2.449 56 3136 7.483

7 49 2.646 57 3249 7.550

8 64 2.828 58 3364 7.616

9 81 3.000 59 3481 7.681

10 100 3.162 60 3600 7.746

11 121 3.317 61 3721 7.810

12 144 3.464 62 3844 7.874

13 169 3.606 63 3969 7.937

14 196 3.742 64 4096 8.000

15 225 3.873 65 4225 8.062

16 256 4.000 66 4356 8.124

17 289 4.123 67 4489 8.185

18 324 4.243 68 4624 8.246

19 361 4.359 69 4761 8.307

20 400 4.472 70 4900 8.367

21 441 4.583 71 5041 8.426

22 484 4.690 72 5184 8.485

23 529 4.796 73 5329 8.544

24 576 4.899 74 5476 8.602

25 625 5.000 75 5625 8.660

26 676 5.099 76 5776 8.718

27 729 5.196 77 5929 8.775

28 784 5.292 78 6084 8.832

29 841 5.385 79 6241 8.888

30 900 5.477 80 6400 8.944

31 961 5.568 81 6561 9.000

32 1024 5.657 82 6724 9.055

33 1089 5.745 83 6889 9.110

34 1156 5.831 84 7056 9.165

35 1225 5.916 85 7225 9.220

36 1296 6.000 86 7396 9.274

37 1369 6.083 87 7569 9.327

38 1444 6.164 88 7744 9.381

39 1521 6.245 89 7921 9.434

40 1600 6.325 90 8100 9.487

41 1681 6.403 91 8281 9.539

42 1764 6.481 92 8464 9.592

43 1849 6.557 93 8649 9.644

44 1936 6.633 94 8836 9.695

45 2025 6.708 95 9025 9.747

46 2116 6.782 96 9216 9.798

47 2209 6.856 97 9409 9.849

48 2304 6.928 98 9604 9.899

49 2401 7.000 99 9801 9.950

50 2500 7.071 100 10,000 10.000

2n 2n

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Percent, Decimal, and Fraction Equivalents

Percent Decimal Fraction

1% 0.01

5% 0.05

10% 0.1

12.5% or 0.125

or

20% 0.2

25% 0.25

30% 0.3

or

37.5% or 0.375

40% 0.4

50% 0.5

60% 0.6

62.5% or 0.625

or

70% 0.7

75% 0.75

80% 0.8

or

87.5% or 0.875

90% 0.9

100% 1.0 1

110% 1.1

125% 1.25

or

150% 1.5 1¹₂

1¹₃ 1.3

133¹₃% 133.3%

1¹₄ 1₁₀¹

9 10 7

87¹₂% 8

5

08.3 6

83¹₃% 83.3%

4 5 3 4 7 10 2

0.6 3

66²₃% 66.6%

5

62¹₂% 8

3 5 1 2 2 5 3

37¹₂% 8

1

0.3 3

33¹₃% 33.3%

3 10 1 4 1 5 1

0.16 6

16²₃% 16.6%

1

12¹₂% 8

1 10 1 20 1 100