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Key Concepts

ReOexive Property Symmetric Property Transitive Property

Properties of Congruence AB ~ AB LA~LA If AB ~ CD, then CD ~ AB. If LA ~ LB, then LB ~ LA. If AB ~ CD and CD ~ EF, then AB ~ EF. If LA ~ LB and LB ~ LC, then LA ~ LC.

Using Properties of Equality and Congruence

Name the property of equality or congruence that justifies each statement. a. LK ~ LK Reflexive Property of Congruence

b. If2x - 8

= 10, then 2x = 18. Addition Property of Equality

c. If RS ~ TW and TW ~ PQ, then RS ~ PQ. Transitive Property of Congruence

d. If mLA = mLB, then mLB = mLA.

Symmetric Property of Equality

ri! Quick Check e

EXERCISES

Name the property of equality or congruence illustrated. a.XY~XY b. If mLA = 45 and 45 = mLB , then mLA = mLB.

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

o

Practice by Example Examples 1 and 2

~

Algebra

Fill in the reason that justifies each step.

1. Solve for x. mLCDE + mLEDF

for Help

(page 104)

= 180

a.

.L:

x + (3x + 20) = 180 4x + 20 = 180 4x

x

b. ~

c.~

= 160 = 40

d.~

e.~

3(n

2. Solve for n. Given: XY

3(n

+ 4)

= 42

XY

a.~

x b.~

c.~

z

·

3n

y

XZ.+ ZY=

+ 4) + 3n = 42 3n + 12 + 3n = 42 6n + 12 = 42

6n

d.~

e._~ f.~

lesson 2-4 Reasoning in Algebra

=

30

n =5'"

105

~

Algebra 3.

Give a reason for each step.

1x - 5 = 10

z(!x x -

Given

4. 5(x

= 20 10 = 20 x = 30

5)

a· .-:L b.~ c.~

+ 3) = -4 5x + 15 = -4

5x = -19 x = _19

5

Given

a.~

b.~

c. ....L

Example 3

(page 105)

Name the property

that jnstifies each statement.

5. LZ

==

LZ

6. 2(3x

+ 5) = 6x +

10

7. If 12x

=

84, then x

=

=

7.

8. If ST

== QR,

then QR

== ST.

9. If mLA 11. If 3x

15, then 3mLA

= =

45.

10. XY=XY

12. If KL

+

14

=

80, then 3x y,

66.

= MN, then MN = KL.

13. If 2x

then

+ Y = 5 and x = 2x + x = 5.

14. IfAB - BC = 12, then AB = 12 + Be.

o

15. If Ll Apply Your Skills

== L2

and L2

==

L3, then L1

== L3.

17. Subtraction Property of Equality If 5x + 6 = 21, then Z. = 15.

19. Symmetric Property of Congruence If LH == LK, then ....L == LH. 21. Distributive Property 3(x - 1) = 3x - ....L 23. Transitive Property of Congruence If LXYZ == LAO Band LAOB == L WfT, then....L. is equivalent 180 to the left side of this

Use the given property

to complete

each statement.

16. Addition Property of Equality If 2x - 5 = 10, then 2x = ....L.

18. Symmetric Property of Equality If AB = YU, then ....L. 20. Reflexive Property

LPQR

==....L

of Congruence

22. Substitution Property If LM = 7 and EF + LM then....L = NP.

= NP,

24. Multiple Choice Which expression equation?

-4x

+ 7y + t(12x - 3y) = CD 6y + 8

® 8x + 4y

~ ~

CO 6y

® 8x

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25. Writing Jero claims that the statements LR == RL and LCBA == LABC are both true by the Reflexive Property of Congruence. Explain why Jero is correct. 26. Use what you know about transitive properties The Transitive Property of Falling Dominoes: to complete the following:

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If domino A causes domino B to fall, and domino B causes domino C to fall, then domino A causes domino ....L to fall.

106

Chapter 2

Reasoning and Proof

~

27. Algebra

Fill in the reason that justifies each step.

4x

Given: C is the midpoint of AD.

C is the midpoint of AD. AC= CD 4x = Ix + 12 2x = 12 x=6 ~

For a guide to solving Exercise 28, see p. 109.

a. ?

A

· C

2x

+ 12

· D

b.~

c.~

d.~

e.~

28. Algebra In the figure at the right, KM a. Solve for x. Justify each step. b. Find the length of KL.

= 35.

K

.2x- 5.

L

2x

M

bd 29. Algebra

~

In the figure at the right, mLGFI a. Solve for x. Justify each step. b. Find mLEFI.

=

128.

G

F I

30. ---=+ Algebra Point C is on the crease when you fold ~ BD onto BA. Give the reason that justifies each step. (Hint: See page 102, Exercises 4 and 5.)

--

BC bisects LABD. mLABC = mLCBD 6n + 1 2n

a. -.L b.-.L

(6" +

1)~".C

(4" + 19)0

D

= 4n + 19 = 18

«.s:

d.~

a

n=9

Challenge 31. Error Analysis Given: a

-:»:

The steps below "show" that 1

= 2. Find the error.

=

b

a=b ab = b2 ab - a2 = b2 - a2 a(b - a) = (b + a)(b - a) a=b+a a=a+a

a 1

=

Given Multiplication Property of Equality Subtraction Property of Equality Distributive Property Division Property of Equality Substitution Property Simplify. Division Property of Equality

2a 2

=

Relationships The relationships "is equal to" and "is congruent to" are reflexive, symmetric, and transitive. In a later chapter, you will see that this is' also trne for the relationship "is similar to." Consider the foUowing relationships among people. State whether each relationship is reflexive, symmetric, transitive, or none of these. Sample: The relationship "is younger than" is transitive. If Sue is younger than Fred and Fred is younger than Alana, then Sue is younger than Alana. The relationship "is younger than" is not reflexive because Sue is not younger than herself. It is also not symmetric because if Sue is younger than Fred, Fred is not younger than Sue.

Real-World

8Connection

President Calvin Coolidge, advice columnist Ann Landers, and musician BillWithers · were all born on the Fourth of July. Each one of them "has the same birthday as" ,~·either one of the others.

32. has the same birthday as 34. lives in the same state as 36. is the same height as

33. is taller than 35. lives in a different state than 37. is a descendant of

~ nline

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Lesson 2-4

Reasoning in Algebra

107

Multiple Choice

38. Which property justifies this statement? If 4x = 16, then 16 = 4x. A. Multiplication Property of Equality B. Transitive Property of Equality C Reflexive Property of Equality D. Symmetric Property of Equality 39. The Multiplication 3 F. If Lfx H.lf

=

Property of Equality justifies which statement below?

ix

6, then

""4

3x

=

6.

3 G. If Lfx

+ 5 - 6, then Lfx - 1.

_

3

_

= 6, then 3x = 24.

J. If ix - 18 = 6, then ix = 24.

40. A transitive property justifies which statement below? A.lfy17 = g,theny= 9 + 17. B. If AM = RS, then RS = AM. Clf 5(3a - 4) = 120, then 15a - 20 = 120. D. If LJ = LR and LR = LH, then LJ = LH. 41. Which equation follows from ~m of Equality? F. m Short Response

+ 1 = 10 by the Multiplication Property

H. 1 3m - 9 = 0 J. m - 27

+3

=

30

G.

1m

=

9

= 0

42. In the diagram, x = 2y + 15 and x + y = 120. a. Use a Property of Equality to explain why 3y + 15 = 120. b. Solve for y. Justify each step. Then find the value of x.

14. for

~Help

Lesson 2-3

Reasoning Use logical reasoning to draw a conclusion.

43. If a student is having difficulty in class, then that student's Elena is having difficulty in history class. teacher is concerned.

44. If a person has a job, then that person is earning money. If a person is earning money, then that person can save money each week.

Lesson 1-6

Use the diagram at the right and find each measure.

45. mLAOC 47. mLDOB

46.mLAOD 48.mLBOE

A

49. In the diagram, name an obtuse angle and a right angle.

lesson 1-1

Find the next two terms in each sequence.

50. 19,21.5,24,26.5 52. -2,6, -18,54 51. 3.4,3.45,3.456,3.4567 53.8, -4,2,-1

108

Chapter 2

Reasoning and Proof

Proof ~

~~

Proving Theorem 2.,2

. Study what you are given, what you are to prove, and the diagram. Write a paragraph proof.

Given: Prove: Ll and L2 are supplementary.

L3 and L2 are supplementary.

L1

==

L3

~

Proof: By the definition of supplementary angles, mL1 + mL2 = 180 and mL3 + mL2 = 180. By substitution, mLl + mL2 = mL3 + mL2. Subtract mL2 from each side. You get mLl = mL3, or Ll == L3.

@Quick Check

e

In the proof above, which Property of Equality allows you to subtract mL2 from each side of the equation?

Theorem 2-3 is like the Congruent Supplements Theorem. You can demonstrate its proof in Exercises 7 and 28.

Key Concepts

Congruent Complements Theorem

If two angles are congruent and supplementary, then each is a right angle.

You can complete proofs of Theorems 2-4 and 2-5 in Exercises 14 and 21, respectively.

EXERCISES

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

e

Practice by Example Example 1

(page 111)

Find the value of each variable.

1.

(80 - x)?

2.

3.

I~for

~Help

Find the measures of the labeled angles in each exercise. 4. Exercise 1 5. Exercise 2 6. Exercise 3

112

Chantor 7

R.,,,,cnninn ""rl Dr",,-/'

Example 2

(page 112)

7. Oeveloping

Proof

Complete this proof of one form of Theorem 2-3 by filling in

the blanks. If two angles are complements of the same angle, then the two angles are congruent.

Given: L1 and L2 are complementary.

L3 and L2 are complementary.

Prove: L1

== L3

Proof: By the definition of complementary angles,

o

mL1 + mL2 = a.~andmL3 + mL2 = b.~. Then mL1 + mL2 = mL3 + mL2 by c.~. Subtract mL2 from each side. You get mL1 = d.~,

Apply Your Skills ~

L/ ~L

or L1

== L3.

8. Writing

How is a theorem different from a postulate? Give an example of vertical angles in your home.

9. Open-Ended

10. Reasoning Explain why this statement is true: If mL1 + mL2 = 180 and mL3 + mL2 = 180, then L1

== L3.

11. Design The two back legs of the director's chair pictured at the left meet in a 72° angle. Find the measure of each angle formed by the two back legs. ~ Algebra Find the value of each variable and the measure of each labeled angle.

tz,

13.

(3x

+ Sr

(5x

(5x - 20)°

+ 4yr

14. Developing Proof Complete this proof of Theorem 2-4 by filling in the blanks. All right angles are congruent.

Given: LX and L Yare right angles. Prove: LX

Exercise 11

== L Y

vL

X Y

Park St.

By the definition of a. ~, mLX = 90 and mL Y = 90.. By the Substitution Property, mLX = b.~, or LX == L Y

Proof:

15. Multiple Choice What is the measure of the angle formed by Park St. and Oak St.?

® 35°

CD

55°

® 45° ® 90°

Name two pairs of congruent angles in each figure. Justify your answers.

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16.

B 17'~E

I

18.

K

F

G H

19. Coordinate Geometry LDOE contains points D(2, 3),0(0,0), and E(5, 1). ~ _fmJ:Lthe coordinates of a pointFso that OF is a sideofanangle that is 'adjacent and supplementary to LDOE.

lesson 2·5 Proving Angles Congruent

113

20. Coordinate Geometry LAOX contains points A(l, 3),0(0,0), and X( 4,0). a. Find the coordinates of a-pointB so that LBOA and LAOX are adjacent complementary angles. ----'> b. Find the coordinates of a point C so that OC is a side of a different angle that is adjacent and complementary to LAOX. 21. Developing Proof Complete this proof of Theorem 2-5 by filling in the blanks. If two angles are congruent and supplementary, then each is a right angle. Given: L Wand LV are congruent and supplementary. Prove: L Wand L V are right angles.

L\ ~

\:>/'

V

=

W

Proof: L Wand L V are congruent, so mL W = mL a. ...L.. L Wand L V are supplementary so mL W + mL V = b· ...L.. Substituting mL W for mL V, you get mL W + mL W By the c· ...L. Property of Equality, mL W = 90. Since L W 180, or 2mL W

=

180.

== LV, mL V = 90, too. Then both angles are d. ...L. angles.

@\

Exercise 22

22. Sports In the photograph, the wheels of the racing wheelchair are tilted so that L1 == L2. What theorem can you use to justify the statement L3 == L4? ~ Algebra Find the measure of each angle.

23. LA is twice as large as its complement, LB.

24. LA is half as large as its complement, LB.

25. LA is twice as large as its supplement, LB. 26. LA is half as large as twice its supplement, LB.

j!QQf. 27. Write a proof for this form of Theorem 2-2.

If two angles are supplements of congruent angles, then the two angles are congruent. Given: L1 and L2 are supplementary. L3 and L4 are supplementary. L2 == L4

Prove: Ll == L3 j!QQf. 28. Write a proof for this form of Theorem 2-3.

If two angles are complements of congruent angles, then the two angles are congruent. Given: L1 and L2 are complementary. L3 and L4 are complementary. L2 == L4

a

114

Prove: Ll == L3

Challenge 29. Paper Folding After you've done the Activity on page 110, answer these questions. a. How is the first fold line you make related to angles 3 and 4? b. How is the second fold line you make related to angles 1 and 2? c. How are the two fold lines related to each other? Give a convincing argument to support your answer.

Chapter 2

Reasoning and Proof

~

Algebra Find the value of each variable and the measure of each labeled angle. 30. 31. (x

+ Y + 5t

32.

Gridded Response

Find the measure of each angle. 33. an angle with measure 8 lessthan the measure of its complement 34. one angle of a pair of complementary vertical angles 35. an angle with measure three times the measure of its supplement Use the diagram at the right to find the measure of each of the following angles. 36. L 1 38. L3 37. L2 39. L4 4 3 2

70° 1

lesson 2~4

Use the given property

to complete

each statement.

40. Subtraction Property of Equality If 3x + 7 = 19, then 3x = ~. 41. Reflexive Property AB=~ 42. Substitution Property If MN = 3 and MN lesson 2-3 Use deductive reasoning of Congruence

+ NP = 15, then~.

to draw a conclusion. If not possible, write not possible.

43. If two lines intersect, then they are coplanar. Lines m and n are coplanar. 44. If two angles are vertical angles, then they are congruent. L1 and L2 are vertical angles. Lesson 2-2 Each conditional statement below is true. Write its converse. If the converse is also true, combine the statements as a biconditional. 45. Ify

+7

=

32, theny

=

25.

46. If you live in Australia, 47. Ifn

then you live south of the equator.

> 0, thenn2 > 0.

"inline

lesson quiz, PHSchool.com, Web Code: aua-0205

lesson 2-5

Proving Angles Congruent

115

Chapter Review

.,5Jj

biconditional (p. 87) conclusion (p. 80) conditional (p. 80) converse (p. 81) deductive reasoning (p. 94) hypothesis (p. 80) law of Detachment (p. 94) law of Syllogism (p. 95) paragraph proof (p. 111) Reflexive Property (p. 105) Symmetric Property (p. 105) theorem (p. 110) Transitive Property (p. 105) truth value (p. 81)

Choose the correct vocabulary term to complete each seutence. 1. The statement" LA ~ LA" is an example of the ~ Property of Congruence.

2. In a conditional statement, the part that directly follows if is the ~. 3. "If LA ~ LB and LB ~ LC, then LA ~ LC" is an example of the ~ Property of Congruence. 4. When a conditional and its converse are true, they may be written as a single true statement called a ~. 5. The ~ of a conditional switches the hypothesis and the conclusion. Property

6. "If LA ~ LB, then LB ~ LA" is an example of the ~ of Congruence.

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7. The part of a conditional statement that follows "then" is the ~. 8. A conditional has a ~ of true or false.

For: Vocabulary quiz Web Code: auj-02S1

9. Reasoning logically from given statements to a conclusion is ~. 10. A statement that you prove true is a .L.

Ski lis and Conce~ts

2·1 and 2·2 Objectives

" To recognize conditional statements To write converses of conditional statements

An if-then statement is a conditional. The part following ifis the hypothesis. The part following then is the conclusion. You find the truth value of a conditional by determining whether it is true or false. The symbolic form of a conditional is p --+ q. The converse of a conditional switches the hypothesis and the conclusion. The symbolic form of the converse of p --+ q is q --+ p. When a conditional and its converse are true, you can combine them as a true biconditional. To write a biconditional, you join the two parts of each conditional with the phrase if and only if The symbolic form of a biconditional is p -H q. For Exercises 11-13, (a) write the converse and (b) determine the truth value of the conditional aad its converse. (c) If both statements are true, write a biconditional. 11. If you are a teenager, then you are younger than 20. U. If an angle is obtuse, then its measure is greater than 90 and less than 180. 13. If a figure is a square, then it has four sides.

V

Y To write biconditionals " To recognize good definitions

Chapter 2

Chapter Review

117

14. Write the following sentence as a conditional: All flowers are beautiful. A good definition is precise. A good definition uses terms that have been previously defined or are commonly accepted. 15. Rico defines a book as something you read. Explain why this is not a good definition. 16. Write this definition as a biconditional: An oxymoron is a phrase that contains contradictory terms. 17. Write this biconditional as two statements, a conditional and its converse: Two angles are complementary if and only if the sum of their measures is 90.

2-3 Objectives

V To use the Law of Detachment V To use the Law of Syllogism

Deductive reasoning is the process of reasoning logically from given statements to a conclusion. If the given statements are true, deductive reasoning produces a true conclusion. The following are two important laws of deductive reasoning: Law of Detachment: If p ~ q is a true statement and p is true, then q is true. Law of Syllogism: If p ~ q and q ~ r are true statements, then p ~ r is true. Use the Law of Detachment to make a conclusion. 18. If you practice table tennis every day, you will become a better player. Lucy practices table tennis every day. 19. Line e and line m are perpendicular. If two lines are perpendicular, they intersect to form right angles. 20. If two angles are supplementary, then the sum of their measures is 180. L1 and L2 are supplementary. Use the Law of Syllogism to make a conclusion. 21. If Kate studies, she will get good grades. If Kate gets good grades, she will graduate. 22. If a, then b. If b, then c. 23. If the weather is wet, the Huskies will not play soccer. If the Huskies do not play soccer, Nathan can stop at the ice cream shop.

2-4 Objective

". To connect reasoning in algebra and geometry

In algebra, you use deductive reasoning and properties to solve equations. In geometry, each statement in a deductive argument is justified by a property, definition, or postulate. Some of the properties you need are listed below. Properties of Equality Addition Property Subtraction Property Multiplication Property Division Property Substitution Property Distributive Property If a If a If a If a If a

= b, then a + c = b + c.

=

=

b, then a - c

b, then a . c

=

b - c. 12.

c

=

b . c. c

of=-

=

=

band c

of=-

0, then!!

b, then b can replace

=

a

in any expression.

a(b + c)

ab + ac

118

Chapter 2

Chapter Review

Properties

of Congruence

Reflexive Property

AB=AB LA =LA

Symmetric

Property

If AB If

== CD, then CD == ABLA == LB, then LB == LA. == CD and CD == EF, then AB == EF. LA == LB and LB == LC, then LA == LC.

x+3 Q a.~ R 2x

Transitive

Property

If AB If

I£l 24.

Algebra Fill in the reason that justifies each step. Given: QS QR x

=

42

s

+ RS = QS + 3 + 2x = 42 3x + 3 = 42

3x

b.~ c.~ d.~

= 39 x = 13

-:«.

each statement. 26. Division Property

Use the given property

to complete

25. Addition Property of Equality If x = 5, then x + 3 = ~. 27. Reflexive Property of Equality

If2(AX)

= 2(BY),

of Equality then AX

=~.

mLY=~

29. Transitive Property of Equality Ifx = 5 and 5 = y, then x =~. 31. Distributive 3p - 6q Property

28. Symmetric Property If XY = RS, then~. 30. Distributive 2( 4x + 5) Property = 8x + ~

of Equality

32. Reflexive Property

of Congruence

= 3(~)

NM==~

2·5 Objective " To prove and apply theorems about angles

A statement that you prove true is a theorem. A proof written as a paragraph is a paragraph proof. You can prove that vertical angles are congruent; that supplements of the same angle are congruent, and that complements of the same angle are congruent. 33. Algebra Find the value of y. 34. For the diagram at the left, find each of the following. a. mLAEC b. mLBED c. mLAEB

A

(3y

B

+

20t

(5y - 16)0

E C D

35. Complete Given: Prove:

the following paragraph

proof.

L1 L2

== L4 == L3 ==~.

L1 ~ L4 is .

Proof: By the Vertical Angles Theorem, z.I. ==~ and L4 given, so L2 == L3 by the ~ Property of Congruence.

Chapter 2

Chapter Review

119

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