Geometry

B Apply 18. Think About a Plan  If it is the night of your weekly basketball game, Monday
your family eats at your favorite restaurant. When your family eats salads $4.99
at your favorite restaurant, you always get chicken fingers. If it is
Tuesday, then it is the night of your weekly basketball game. How Tuesday
much do you pay for chicken fingers after your game? Use the chicken fingers $5.99
specials board at the right to decide. Explain your reasoning.
• How can you reorder and rewrite the sentences to help you? Wednesday
• How can you use the Law of Syllogism to answer the question? burgers $6.99

Beverages  For Exercises 19–24, assume that the following statements are true.

A. If Maria is drinking juice, then it is breakfast time.
B. If it is lunchtime, then Kira is

drinking milk and nothing else.
C. If it is mealtime, then Curtis is drinking water and nothing else.
D. If it is breakfast time, then Julio is drinking juice and nothing else.
E. Maria is drinking juice.

Use only the information given above. For each statement, write must be true,
may be true, or is not true. Explain your reasoning.

19. Julio is drinking juice. 20. Curtis is drinking water. 21. Kira is drinking milk.

22. Curtis is drinking juice. 23. Maria is drinking water. 24. Julio is drinking milk.

STEM 25. Physics  Quarks are subatomic particles identified by electric charge and rest

energy. The table shows how to categorize quarks by their flavors. Show how the

Law of Detachment and the table are used to identify the flavor of a quark with a
1
charge of - 3 e and rest energy 540 MeV.

Rest Energy and Charge of Quarks

Rest Energy (MeV ) 360 360 1500 540 173,000 5000

Electric Charge (e) ϩ 2 Ϫ 1 ϩ 2 Ϫ 1 ϩ 2 Ϫ 1
3 3 3 3 3 3

Flavor Up Down Charmed Strange Top Bottom

Write the first statement as a conditional. If possible, use the Law of Detachment
to make a conclusion. If it is not possible to make a conclusion, tell why.

26. Ahlslmna1ti1ognmal psear_k0s2a0re4i_ntt0er0e7st3in7g. 27. All squares are rectangles.

Mammoth Cave is a national park. ABCD is a square.

28. The temperature is always above 32°F 29. Every high school student likes art.

in Key West, Florida. Ling likes art.

The temperature is 62°F.

30. Writing  Give an example of a rule used in your school that could be written as a
conditional. Explain how the Law of Detachment is used in applying that rule.

Lesson 2-4  Deductive Reasoning 111

C Challenge 31. Reasoning  Use the following algorithm: Choose an integer. Multiply the integer
by 3. Add 6 to the product. Divide the sum by 3.

a. Complete the algorithm for four different integers. Look for a pattern in the
chosen integers and in the corresponding answers. Make a conjecture that
relates the chosen integers to the answers.

b. Let the variable x represent the chosen integer. Apply the algorithm to x. Simplify
the resulting expression.

c. How does your answer to part (b) confirm your conjecture in part (a)? Describe
how inductive and deductive reasoning are exhibited in parts (a) and (b).

STEM 32. Biology  Consider the following given statements and conclusion.

Given:  If an animal is a fish, then it has gills.
A turtle does not have gills.

You conclude:  A turtle is not a fish.

a. Make a Venn diagram to illustrate the given information.
b. Use the Venn diagram to help explain why the argument uses good reasoning.

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

Look back at the information about 2-by-2 calendar squares on page 81. MP 7

Choose from the following words and expressions to complete the sentences

below.

1 more than 7 more than a-1 a-7 d-7

1 less than 7 less than a+1 a+7 d-1
b+7
Syllogism Detachment c+1 c-1

a. Fact 1: A date in a monthly calendar is ? a date directly to the left of it.

b. Fact 2: A date in a monthly calendar is ? a date directly above it.
Let a, b, c and d represent the numbers in a 2-by-2 calendar square, with a, b, c,

d listed from least to greatest.
c. You can use Fact 1 above to conclude that b is equivalent to ? .

d. You can use Fact 2 above to conclude that c is equivalent to ? .

e. You can use Fact 1 above to conclude that d is equivalent to ? .

f. Each of these conclusions is an example of using the Law of ? .

112 Chapter 2  Reasoning and Proof

2-5 Reasoning in Algebra MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
and Geometry
Prepares for MG-ACFOS.C.9.192  .PGro-CveOt.h3e.9o rePmrosvaebout lines
tahnedoarenmglsesa.bAolustolinperseapnadreasngfolers.GA-lCsOo.pCr.1e0p,aGre-CsOfo.Cr.11
MMAPF1S,.M91P2.3G-CO.3.10, MAFS.912.G-CO.3.11
MP 1, MP 3

Objective To connect reasoning in algebra and geometry

Think about Follow the steps of the brainteaser Write down your age.
how each step is using your age. Then try it using a Multiply it by 10.
related to the family member’s age. What do you Add 8 to the product.
steps before it. notice? Explain how the brainteaser Double that answer and
works. then subtract 16.
MATHEMATICAL Finally, divide the result by 2.

PRACTICES

Lesson In the Solve It, you logically examined a series of steps. In this lesson, you will apply
logical reasoning to algebraic and geometric situations.
Vocabulary
• Reflexive Essential Understanding  Algebraic properties of equality are used in geometry.
They will help you solve problems and justify each step you take.
Property
• Symmetric In geometry you accept postulates and properties as true. Some of the properties that
you accept as true are the properties of equality from algebra.
Property
• Transitive Key Concept  Properties of Equality

Property
• proof
• two-column

proof

Let a, b, and c be any real numbers.

Addition Property If a = b, then a + c = b + c.

Subtraction Property # #If a = b, then a - c = b - c.
Multiplication Property
If a = b, then a c = b c.

Division Property If a = b and c ≠ 0, then ac = bc .
Reflexive Property a=a
Symmetric Property If a = b, then b = a.

Transitive Property If a = b and b = c, then a = c.

Substitution Property If a = b, then b can replace a in any expression.

Lesson 2-5  Reasoning in Algebra and Geometry 113

Key Concept  The Distributive Property

Use multiplication to distribute a to each term of the sum or difference
within the parentheses.

Sum: Di erence:
a(b ϩ c) ϭ a(b ϩ c) ϭ ab ϩ ac a(b Ϫ c) ϭ a(b Ϫ c) ϭ ab Ϫ ac

You use deductive reasoning when you solve an equation. You can justify each step
with a postulate, a property, or a definition. For example, you can use the Distributive
Property to justify combining like terms. If you think of the Distributive Property as
ab + ac = a(b + c) or ab + ac = (b + c)a, then 2x + x = (2 + 1)x = 3x.

Problem 1 Justifying Steps When Solving an Equation

How can you use the Algebra  What is the value of x? Justify each step. M
given information?
Use what you know ∠AOM and ∠MOC are supplementary. ⦞ that form a linear (2x ϩ 30)Њ xЊ
about linear pairs to pair are supplementary.
relate the two angles. AO
m∠AOM + m∠MOC = 180 Definition of supplementary ⦞ C

(2x + 30) + x = 180 Substitution Property

3x + 30 = 180 Distributive Property

3x = 150 Subtraction Property of Equality hsm11gmse_0205_t00588

x = 50 Division Property of Equality

Got It? 1. What is tAhBe> value of x? Justify each step. B
Given:  bisects ∠RAN . R xЊ (2x Ϫ 75)Њ

AN

Some properties of equality have corresponding properties of congruence.

hsm11gmse_0205_t00589

Key Concept  Properties of Congruence

Reflexive Property AB ≅ AB ∠A ≅ ∠A
Symmetric Property
If AB ≅ CD, then CD ≅ AB.
Transitive Property If ∠A ≅ ∠B, then ∠B ≅ ∠A.

If AB ≅ CD and CD ≅ EF , then AB ≅ EF .
If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.
If ∠B ≅ ∠A and ∠B ≅ ∠C, then ∠A ≅ ∠C.

114 Chapter 2  Reasoning and Proof

Problem 2 Using Properties of Equality and Congruence

Is the justification a What is the name of the property of equality or congruence that justifies going from
the first statement to the second statement?
property of equality

or congruence? A 2x + 9 = 19
N umbers are equal (=) 2x = 10
and you can perform
Subtraction Property of Equality
operations on them, so

o( Af )eaqnudal(itCy). are properties B  ∠O ≅ ∠W and ∠W ≅ ∠L
Figures and
 ∠O ≅ ∠L Transitive Property of Congruence
their corresponding parts
are congruent (≅),
s o (B) is a property of C m∠E = m∠T
m∠T = m∠E
congruence. Symmetric Property of Equality

Got It? 2. For parts (a)–(c), what is the name of the property of equality or congruence

that justifies going from the first statement to the second statement?
c. 41x
a. AR ≅ TY b. 3(x + 5) = 9 x = 7
TY ≅ AR 3x + 15 = 9 = 28

d. Reasoning  What property justifies the statement m∠R = m∠R?

A proof is a convincing argument that uses deductive reasoning. A proof logically
shows why a conjecture is true. A two-column proof lists each statement on the left.
The justification, or the reason for each statement, is on the right. Each statement must
follow logically from the steps before it. The diagram below shows the setup for a two-
column proof. You will find the complete proof in Problem 3.

Given: m/―1 = m/―3 AB C
Prove: m/―AEC = m/―DEB D
12
3

E

The first statement is usually Statements Reasons
the given statement.
1) m/―1 = m/―3 1) Given
Each statement should 2) 2)
follow logically from the
previous statements. 3) 3)

The last statement is what 4) 4)
you want to prove. 5) m/―AEC = m/―DEB 5)

hsm11gmse_0205_t07147 115

Lesson 2-5  Reasoning in Algebra and Geometry

Proof Problem 3 Writing a Two-Column Proof
Write a two-column proof.
Given:  m∠1 = m∠3 AB C
Prove:  m∠AEC = m∠DEB D
12
3

E

m∠1 = m∠3 To prove that Add m∠2 to both m∠1 and
m∠AEC = m∠DEB
m∠3. The resulthinsgman1g1legs wmillshea_ve0205_t00591

equal measure.

Statements Reasons
1) Given
1) m∠1 = m∠3 2) Reflexive Property of Equality
2) m∠2 = m∠2 3) Addition Property of Equality
3) m∠1 + m∠2 = m∠3 + m∠2 4) Angle Addition Postulate
4) m∠1 + m∠2 = m∠AEC
m∠3 + m∠2 = m∠DEB 5) Substitution Property
5) m∠AEC = m∠DEB

Got It? 3. a. Write a two-column proof.

Given:  AB ≅ CD ABCD

Prove:  AC ≅ BD

b. Reasoning  In Problem 3, why is Statement 2 necessary in the proof?

hsm11gmse_0205_t00592

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Name the property of equality or congruence
that justifies going from the first statement to 4. Developing Proof  Fill in the reasons for this
the second statement. algebraic proof.

1. m∠A = m∠S and m∠S = m∠K Given:  5x + 1 = 21
m∠A = m∠K Prove:  x = 4

2. 3x + x + 7 = 23 Statements Reasons
4x + 7 = 23
1) 5x + 1 = 21 1) a. ?
3. 4x + 5 = 17 2) 5x = 20 2) b. ?
4x = 12 3) x = 4 3) c. ?

116 Chapter 2  Reasoning and Proof

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Algebra  Fill in the reason that justifies each step. See Problem 1.

( ) 5. 21x - 5 = 10   Given 6.   5(x + 3) = -4  Given

 2 21x - 5 = 20   a. ?     5x + 15 = -4  a. ?
    x - 10 = 20   b. ?
    5x = -19  b. ?
      x = 30   c. ? 19
    x = - 5   c. ?

7. ∠CDE and ∠EDF are supplementary. ⦞ that form a linear pair E
are supplementary

m∠CDE + m∠EDF = 180 a. ? xЊ (3x ϩ 20)Њ
b. ? C DF
    x + (3x + 20) = 180

      4x + 20 = 180 c. ?

        4x = 160 d. ?

         x = 40 e. ? hsm11gmse_0205_t00593

8.  XY = 42 Given  3(n ϩ 4) 3n
ZY
    XZ + ZY = XY a. ? X

  3(n + 4) + 3n = 42 b. ?

  3n + 12 + 3n = 42 c. ?

     6n + 12 = 42 d. ? hsm11gmse_0205_t00594
       6n = 30 e. ?

       n = 5 f. ?

Name the property of equality or congruence that justifies going from the first See Problem 2.

statement to the second statement.

9. 2x + 1 = 7 10. 5x = 20 11. ST ≅ QR 12. AB - BC = 12
QR ≅ ST AB = 12 + BC
  2x = 6   x = 4

13. Developing Proof  Fill in the missing statements or reasons for the following See Problem 3.
two‑column proof.

Given:  C is the midpoint of AD. 4x 2x ϩ 12
Prove:  x = 6 A CD

Statements Reasons

1) C is the midpoint of AD. 1) a. ?
2) AC ≅ CD
2) b. ? hsm11gmse_0205_t00595

3) AC = CD 3) ≅ segments have equal length.

4) 4x = 2x + 12 4) c. ?

5) d. ? 5) Subtraction Property of Equality

6) x = 6 6) e. ?

Lesson 2-5  Reasoning in Algebra and Geometry 117

B Apply Use the given property to complete each statement.

14. Symmetric Property of Equality 15. Symmetric Property of Congruence
If AB = YU , then ? . If ∠H ≅ ∠K , then ? ≅ ∠H.

16. Reflexive Property of Congruence 17. Distributive Property
∠POR ≅ ? 3(x - 1) = 3x - ?

18. Substitution Property 19. Transitive Property of Congruence
If ∠XYZ ≅ ∠AOB and
If LM = 7 and EF + LM = NP, ∠AOB ≅ ∠WYT , then ? .
then ? = NP.

20. Think About a Plan  A very important part in writing proofs is analyzing E
the diagram for key information. What true statements can you make

based on the diagram at the right? 12

• What theorems or definitions relate to the geometric figures A BC

in the diagram?

• What types of markings show relationships between parts D
of geometric figures?

21. Writing  Explain why the statements LR ≅ RL and ∠CBA ≅ ∠ABC are both true
by the Reflexive Property of Congruence.

22. Reasoning  Complete the following statement. Describe the reasoning thaht sm11gmse_0205_t00596

supports your answer.

The Transitive Property of Falling Dominoes: If Domino A causes Domino B to fall,
and Domino B causes Domino C to fall, then Domino A causes Domino ? to fall.

Write a two-column proof.

23. Given:  KM = 35 2x Ϫ 5 2x 24. Given:  m∠GFI = 128
M Proo f Prove: m∠EFI = 40
Proo f Prove:  KL = 15 K L G (9x Ϫ 2)Њ E
4xЊ
C Challenge 25. Error Analysis  The steps below “show” that 1 = 2. Describe the error.
FI
a = b Given
ab = b2 hsm11gmse_0205_t00599
hsm11gMmuslteip_li0ca2ti0on5_Prto0p0er5ty9o8f Equality

ab - a2 = b2 - a2 Subtraction Property of Equality

a(b - a) = (b + a)(b - a) Distributive Property

a = b + a Division Property of Equality

a = a + a Substitution Property

a = 2a Simplify.

1 = 2 Division Property of Equality

118 Chapter 2  Reasoning and Proof

Relationships  Consider the following relationships among people. Tell whether
each relationship is reflexive, symmetric, transitive, or none of these. Explain.

Sample:  The relationship “is younger than” is not reflexive because Sue is not
younger than herself. It is not symmetric because if Sue is younger than Fred, then
Fred is not younger than Sue. It is transitive because if Sue is younger than Fred and
Fred is younger than Alana, then Sue is younger than Alana.

26. has the same birthday as 27. is taller than 28. lives in a different state than

Standardized Test Prep

SAT/ACT 29. You are typing a one-page essay for your English class. You set 1-in. margins 8.5 in.
on all sides of the page as shown in the figure at the right. How many square 11 in.
inches of the page will contain your essay? 1 in.

30. Given 2(m∠A) + 17 = 45 and m∠B = 2(m∠A), what is m∠B? hsm11gmse_0205_t12000.

31. A circular flowerbed has circumference 14p m. What is its area in square
meters? Use 3.14 for p.

32. The measure of the supplement of ∠1 is 98. What is m∠1?

33. What is the next term in the sequence 2, 4, 8, 14, 22, 32, 44, c ?

Mixed Review

34. Reasoning  Use logical reasoning to draw a conclusion. See Lesson 2-4.

If a student is having difficulty in class, then that student’s
teacher is concerned.

Walt is having difficulty in science class.

Use the diagram at the right. Find each measure. A O E See Lesson 1-4.

35. m∠AOC 36. m∠DOB 60Њ 45Њ 25Њ

37. m∠AOD 38. m∠BOE B CD

20Њ

Get Ready!  To prepare for Lesson 2-6, do Exercises 39–41.

Find the value of each variable. 40. See Lesson 1-5.
39.
hsm11gmse4_10. 205_t00yЊ600

130Њ xЊ zЊ 55Њ

hsm11gmse_0205_t00601 hsm11gmse_0205_t00602
Lesson 2-5  Reasoning in Algebrahasnmd 1G1egommestery_ 0205_t00601319

2-6 Proving Angles CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Congruent
MG-ACFOS.C.9.192  .PGro-CveOt.h3e.9o rePmrosvaebtohuetolrienmes anbdouatnlginleess. and
aThnegolerse;musseintchlueodree:m. .s.avbeortuictalilnaensgalensdaarnegcloens gtoruseonltve. . .
pMroPb1le,mMs.PTh3e,oMrePm4s,inMclPud6e: . . . vertical angles are
congruent . . .

MP 1, MP 3, MP 4, MP 6

Objective To prove and apply theorems about angles

Use what you’ve A quilter wants to duplicate this quilt but knows the measure of
learned about only two angles. What are the measures of angles 1, 2, 3, and 4?
congruent angle How do you know?
pairs.
1
MATHEMATICAL 60° 2

PRACTICES 3
4

Lesson In the Solve It, you may have noticed a relationship between vertical angles. You can
prove that this relationship is always true using deductive reasoning. A theorem is a
Vocabulary conjecture or statement that you prove true.
• theorem
• paragraph proof Essential Understanding  You can use given information, definitions,
properties, postulates, and previously proven theorems as reasons in a proof.

Theorem 2-1  Vertical Angles Theorem

Vertical angles are congruent. 1 2 3
∠1 ≅ ∠3 and ∠2 ≅ ∠4 4

When you are writing a geometric proof, it may help to separate the theorem you want

to prove into a hypothesis and conclusion. Another way to whristme t1h1egVmertsieca_l0A2n0g6le_st00748

Theorem is “If two angles are vertical, then they are congruent.” The hypothesis

becomes the given statement, and the conclusion becomes what you want to prove.

A two-column proof of the Vertical Angles Theorem follows.

120 Chapter 2  Reasoning and Proof

Proof Proof of Theorem 2-1: Vertical Angles Theorem

Given: ∠1 and ∠3 are vertical angles. 1 2 3
Prove: ∠1 ≅ ∠3

Statements Reasons

1) ∠1 and ∠3 are vertical angles. 1) Given
2) ∠1 and ∠2 are supplementary.
∠2 and ∠3 are supplementary. 2) ⦞ thhastmfo1r1mgamlisnee_ar0p2a0i6r _arte0s0u7p5p0lementary.
3) m∠1 + m∠2 = 180
m∠2 + m∠3 = 180 3) The sum of the measures of supplementary
4) m∠1 + m∠2 = m∠2 + m∠3 ⦞ is 180.
5) m∠1 = m∠3
6) ∠1 ≅ ∠3 4) Transitive Property of Equality
5) Subtraction Property of Equality
6) ⦞ with the same measure are ≅.

How do you get Problem 1 Using the Vertical Angles Theorem
started?
Look for a relationship What is the value of x?
in the diagram that
allows you to write (2x ϩ 21)Њ 4xЊ
an equation with the
variable. The two labeled angles 2x + 21 = 4x
are vertical angles, so set
them equal. 21 = 2x hsm11gmse_0206_t00751

Solve for x by subtracting 21 = x
2x from each side and 2
then dividing by 2.
21 /2
Grid the answer as 21/2 −
or 10.5.
00000 0
11111 1
22222 2
33333 3
44444 4
55555 5
66666 6
77777 7
88888 8
99999 9

Got It? 1. What is the value of x? hsm11gm3xsЊe_0206_t00752

(2x ϩ 40)Њ

Lessonh2s-6m  1P1rgovminsgeA_n0g2l0es6C_otn0g0r7u5e3nt 121

Proof Problem 2 Proof Using the Vertical Angles Theorem

Why does the Given: ∠1 ≅ ∠4 12
Transitive Property Prove: ∠2 ≅ ∠3 43
work for statements
3 and 5? Statements Reasons
In each case, an angle is
congruent to two other 1) ∠1 ≅ ∠4 1) Given hsm11gmse_0206_t00754
angles, so the two angles 2) ∠4 ≅ ∠2 2) Vertical angles are ≅.
are congruent to each 3) ∠1 ≅ ∠2 3) Transitive Property of Congruence
other. 4) ∠1 ≅ ∠3 4) Vertical angles are ≅.
5) ∠2 ≅ ∠3 5) Transitive Property of Congruence

Got It? 2. a. Use the Vertical Angles Theorem to prove the following. 12
Given:  ∠1 ≅ ∠2 43
Prove:  ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4
b. Reasoning  How can you prove ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4

without using the Vertical Angles Theorem? Explain.

TphaerapgrroaopfhinpProroobfliesmwr2ititsetnwaos-csoenlutmennc,ebsuint tahepraeraagrerampha.nByewloawysistothdeispprloayofafrphormsomof.1A1gmse_0206_t00755

Problem 2 in paragraph form. Each statement in the Problem 2 proof is red in the

paragraph proof.

Proof Given: ∠1 ≅ ∠4 12
Prove: ∠2 ≅ ∠3 43

Proof: ∠1 ≅ ∠4 is given. ∠4 ≅ ∠2 because vertical angles are congruent.

By the Transitive Property of Congruence, ∠1 ≅ ∠2. ∠1 ≅ ∠3 because

vertical angles are congruent. By the Transitive Property of Congruence,

∠2 ≅ ∠3. hsm11gmse_0206_t00754

The Vertical Angles Theorem is a special case of the following theorem.

Theorem 2-2  Congruent Supplements Theorem

Theorem If . . . Then . . .
If two angles are supplements ∠1 and ∠3 are supplements and ∠1 ≅ ∠2
of the same angle (or of ∠2 and ∠3 are supplements
congruent angles), then the
two angles are congruent.

31 2

You will prove Theorem 2-2 in Problem 3.

hsm11gmse_0206_t00756

122 Chapter 2  Reasoning and Proof

Proof Problem 3 Writing a Paragraph Proof

How can you use the Given: ∠1 and ∠3 are supplementary. 31 2
given information? ∠2 and ∠3 are supplementary.
Both ∠1 and ∠2 are
supplementary to ∠3. Prove: ∠1 ≅ ∠2
Use their relationship
with ∠3 to relate ∠1 Proof: ∠1 and ∠3 are supplementary because it is given. So m∠1 + m∠3 = 180
and ∠2 to each other. by the definition of supplementary angles. ∠2 and ∠3 are supplementary

because it is given, so m∠2 + m∠3 = 180 by the same hdesfmin1it1iognm. Bsyet_h0e206_t00756

Transitive Property of Equality, m∠1 + m∠3 = m∠2 + m∠3. Subtract m∠3
from each side. By the Subtraction Property of Equality, m∠1 = m∠2. Angles
with the same measure are congruent, so ∠1 ≅ ∠2.

Got It? 3. Write a paragraph proof for the Vertical Angles Theorem.

The following theorems are similar to the Congruent Supplements Theorem.

Theorem 2-3  Congruent Complements Theorem

Theorem If . . . Then . . .
If two angles are ∠1 ≅ ∠3
complements of the same ∠1 and ∠2 are complements
angle (or of congruent and ∠3 and ∠2 are
angles), then the two
angles are congruent. complements



21 3

You will prove Theorem 2-3 in Exercise 13.

Theorem 2-4

Theorem hIsfm. .1.1gmse_0206_t00758 Then . . .
All right angles are ∠1 ≅ ∠2
congruent. ∠1 and ∠2 are right angles


12

You will prove Theorem 2-4 in Exercise 18.

Theorem 2-5

Theorem If .h. s. m11gmse_0206_t00759 Then . . .
If two angles are congruent
and supplementary, then ∠1 ≅ ∠2, and ∠1 and ∠2 m∠1 = m∠2 = 90
each is a right angle.
are supplements



12

You will prove Theorem 2-5 in Exercise 23.

Leshsosnm21-61 gmProsvei_ng02A0n6g_lets0C0o7n6g0ruent 123

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
1. What are the measures of ∠1, ∠2, and ∠3?
3. Reasoning  If ∠A and ∠B are supplements, and ∠A
21 40Њ and ∠C are supplements, what can you conclude
3 50Њ about ∠B and ∠C? Explain.

2. What is the value of x? (6x ϩ 16)Њ 4. Error Analysis  Your friend knows that ∠1 and∠2
are complementary and that ∠1 and ∠3 are
complementary. He concludes that ∠2 and ∠3 must

be complementary. What is his error in reasoning?

12 120 136Њ 5. Compare and Contrast  How is a theorem different
from a postulate?
h 2s0m 11gmse_0 210366_t00761

hsm11gmse_0206_t00764

Practice and Problem-Solving Exercises MATHEMATICAL

A Practice Find the value of each variable. PRACTICES See Problem 1.
8.
6. 7.

3xЊ (80 Ϫ x)Њ yЊ (x ϩ 90)Њ 4xЊ
2xЊ 76Њ


Find the measures of the labeled angles in each exercise.

9. Ehxsemrc1is1eg6m se_0206_t00765 10. Ehxsemrc1is1eg7m se_0206_t00767 11. Exercise 8
12. Developing Proof  Complete the following proof by filling in the blanks. hsm11gmseS_e0e20P6ro_bt0le0m7628.

Given:  ∠1 ≅ ∠3 61 2
Prove:  ∠6 ≅ ∠4 3

Statements Reasons 54

1) ∠1 ≅ ∠3 1) Given hsm11gmse_0206_t00771
2) ∠3 ≅ ∠6 2) a. ?
3) b. ? 3) Transitive Property of Congruence
4) ∠1 ≅ ∠4 4) c. ?
5) ∠6 ≅ ∠4 5) d. ?

124 Chapter 2  Reasoning and Proof

13. Developing Proof  Fill in the blanks to complete this proof of See Problem 3.
the Congruent Complements Theorem (Theorem 2-3).

If two angles are complements of the same angle, then 1 2
the two angles are congruent. 3

Given:  ∠1 and ∠2 are complementary.
∠3 and ∠2 are complementary.

Prove:  ∠1 ≅ ∠3

Proof:  ∠1 and ∠2 are complementary and ∠3 and ∠2 are complementary

because it is given. By the definition of complementary angles,

m∠1 + m∠2 = a. ? and m∠3 + m∠2 = b. ? . Then

m∠1 + m∠2 = m∠3 + m∠2 by the Transitive Propehrtsymof1E1qgumalistye._0206_t00772

Subtract m∠2 from each side. By the Subtraction Property of Equality, you
get m∠1 = c. ? . Angles with the same measure are d. ? , so ∠1 ≅ ∠3.

B Apply 14. Think About a Plan  What is the measure of the angle 116th St.
Elm St.
formed by Park St. and 116th St.? Main St.
• Can you make a connection between the angle you

need to find and the labeled angle?
• How are angles that form a right angle related?

15. Open-Ended  Give an example of vertical angles in your Park St.
home or classroom.
35°

Algebra  Find the value of each variable and the measure of each labeled angle.

16. 17.
(x ϩ 10)Њ (4x Ϫ 35)Њ
(3x ϩ 8)Њ (5x Ϫ 20)Њ

(5x ϩ 4y)Њ

18. Developing Proof  Fill in the blanks to complete this proof of Theorem 2-4.

Ahlslmrig1h1tganmgslees_a0re20co6n_gtr0u0e7n7t.5 hsm11gmse_0206_t00776

Given:  ∠X and ∠Y are right angles.

Prove:  ∠X ≅ ∠Y XY

Proof:  ∠X and a. ? are right angles because it is given. hsm11gmse_0206_t00777
By the definition of b.  ? , m∠X = 90 and m∠Y = 90.
By the Transitive Property of Equality, m∠X = c. ? . 40°
Because angles of equal measure are congruent, d. ? . x° y°

19. Miniature Golf  In the game of miniature
golf, the ball bounces off the wall at the
same angle it hit the wall. (This is the angle
formed by the path of the ball and the line
perpendicular to the wall at the point of
contact.) In the diagram, the ball hits the
wall at a 40° angle. Using Theorem 2-3,
what are the values of x and y?

hsm11gmse_0206_a01525 125

Lesson 2-6  Proving Angles Congruent

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

20. A B 21. F 22. K
E

O IG JP L
DC H M

23. Developing Proof  Fill in the blanks to complete this proof of Theorem 2-5.

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

hGsimve1n1: g∠mWsea_n0d2∠0V6_atr0e0co7n7g8ruent ahnsdms1up1pglmemseen_t0a2ry0.6_t00779 hsm11gmse_0206_t00780

Prove:  ∠W and ∠V are right angles. WV

Proof:  ∠W and ∠V are congruent because a. ? . Because hsm11gmse_0206_t00781
congruent angles have the same measure, m∠W =
b. ? . ∠W and ∠V are supplementary because it 31 24

is given. By the definition of supplementary angles,
m∠W + m∠V = c. ? . Substituting m∠W for m∠V,
you get m∠W + m∠W = 180, or 2m∠W = 180. By
the d. ? Property of Equality, m∠W = 90. Since
m∠W = m∠V , m∠V = 90 by the Transitive Property
of Equality. Both angles are e. ? angles by the

definition of right angles.

24. Design  In the photograph, the legs of the table are constructed

so that ∠1 ≅ ∠2. What theorem can you use to justify the
statement that ∠3 ≅ ∠4?

25. Reasoning  Explain why this statement is true: If
m∠ABC + m∠XYZ = 180 and ∠ABC ≅ ∠XYZ, then ∠ABC
and ∠XYZ are right angles.

Algebra  Find the measure of each angle. 27. ∠A is half as large as its complement, ∠B.
26. ∠A is twice as large as its complement, ∠B. 29. ∠A is half as large as twice its supplement, ∠B.
28. ∠A is twice as large as its supplement, ∠B.

30. Write a proof for this form of Theorem 2-2. 12
Proof If two angles are supplements of congruent angles, 34

then the two angles are congruent.

Given:  & 1 and ∠2 are supplementary.
∠3 and ∠4 are supplementary.
∠2 ≅ ∠4

Prove:  ∠1 ≅ ∠3

C Challenge 31. tChoeocrodoinrdatineaGteesoomf eatproyi n∠t FDOsoEthcoatnOtaFin> isspaosiindtes D(2, 3), O(0, 0), and E(5, 1). Find
of an angle that is adjacent and

supplementary to ∠DOE. hsm11gmse_0206_t00782

126 Chapter 2  Reasoning and Proof

32. Coordinate Geometry  ∠AOX contains points A(1, 3), O(0, 0), and X(4, 0).

a. Find the coordinates of a point B so that ∠BOA and ∠AOX are adjacent

complementary angles. point C so that OC> is a side of a different angle that is
b. Find the coordinates of a

adjacent and complementary to ∠AOX .

Algebra  Find the value of each variable and the measure of each angle.

33. 34. (x ϩ y ϩ 5)Њ 35.
yЊ 2xЊ
(y ϩ x)Њ 2xЊ
2xЊ (y Ϫ x)Њ 4yЊ (x ϩ y ϩ 10)Њ

Standardized Test Prep
hsm11gmse_0206_t00783
SAT/ACT m∠h1sm= 1631ganmdsme_∠022=064_xt-0097, 84 hsm11gmse_0206_t00785
36. ∠1 and ∠2 are vertical angles. If

what is the value of x?

37. What is the area in square centimeters of a triangle with a base of 5 cm
and a height of 8 cm?

38. In the figure at the right, m∠1 = 1 (m∠2), m∠2 = 2 (m∠3). If
2 3
m∠3 = 72, what is m∠4?
1 23 4
39. What is the measure of an angle with a supplement that is four

times its complement?

Mixed Review hsm11gmse_0206_t12157.ai

Which property of equality or congruence justifies going from the first See Lesson 2-5.

statement to the second?

40. 3x + 7 = 19 41.  4x = 20 42. ∠1 ≅ ∠2 and ∠3 ≅ ∠2
  3x = 12 x = 5 ∠1 ≅ ∠3

Get Ready!  To prepare for Lesson 3-1, do Exercises 43–48.

Refer to the figure at the right. AB tr See Lesson 1-2.
43. Name four points on line t. EG H C
44. Are points G, A, and B collinear?

45. Are points F, I, and H collinear? FI
46. Name the line on which point E lies. D
47. Name line t in three other ways.

48. Name the point at which lines t and r intersect.

Leshsosnm21-61 gmProsvei_n0g2A0n6g_lets0C0o7n8g7ruent 127

2 Pull It All Together

RMANCPERFOE TASKCompleting the Performance Task

To solve these Look back at your results from the Apply What You’ve Learned sections in Lessons 2-1,
problems, you 2-2, and 2-4. Use the work you did to complete the following.
will pull together 1. Solve the problem in the Task Description on page 81 by using deductive reasoning to
ideas about
inductive and prove your conjecture from Lesson 2-1. Show all your work and explain each step of
deductive your solution.
reasoning.
2. Reflect  Choose one of the Mathematical Practices below and explain how you applied
it in your work on the Performance Task.

MP 3: Construct viable arguments and critique the reasoning of others.

MP 6: Attend to precision.

MP 7: Look for and make use of structure.

On Your Own

You can choose a 3-by-3 calendar square on a page of a monthly calendar by drawing a 3-by-3
square around nine dates, as shown below.

MARCH

SUN MON TUE WED THU FRI SAT
123

4 5 6 7 8 9 10

11 12 13 14 15 16 17

18 19 20 21 22 23 24

25 26 27 28 29 30 31

a. Use inductive reasoning to make a conjecture about the relationship between the
center number of a 3-by-3 calendar square and the three calendar-square numbers that
lie on any row, column, or diagonal containing the center number.

b. Use deductive reasoning to prove your conjecture.

128 Chapter 2  Pull It All Together

2 Chapter Review

Connecting and Answering the Essential Questions

Reasoning Inductive Reasoning (Lesson 2-1) Proofs (Lessons 2-5 and 2-6)
and Proof Inductive reasoning is the process of
You can observe patterns making conjectures based on patterns. A two-column proof lists the
to make a conjecture; statements to the left and the
you can prove it is true 3 9 27 81 corresponding reasons to the right.
using given information, In a paragraph proof, the statements
definitions, properties, ϫ3 ϫ3 ϫ3 and reasons are written as
postulates, and theorems. sentences.
Deductive Reasoning (Lessons 2-2, 2-3,
Statements Reasons
2D-e4d)uctivhe sremas1o1nigngmissteh_e0p2ro0c1es_sto0f0m6a0k6in.gai
• given information • definitions
logical conclusions from given statements or
facts. • information from • properties

a diagram • postulates

Law of Detachment Law of Syllogism • logical reasoning • previously

If  p S q is true If p S q is true • statement to proven

and p  is true, and  q S r is true, prove theorems

then q is true. then p S r is true.

Chapter Vocabulary • deductive reasoning (p. 106) • negation (p. 91)
• equivalent statements (p. 91) • paragraph proof (p. 122)
• biconditional (p. 98) • hypothesis (p. 89) • proof (p. 115)
• conclusion (p. 89) • inductive reasoning (p. 82) • theorem (p. 120)
• conditional (p. 89) • inverse (p. 91) • truth value (p. 90)
• conjecture (p. 83) • Law of Detachment (p. 106) • two-column proof (p. 115)
• contrapositive (p. 91) • Law of Syllogism (p. 108)
• converse (p. 91)
• counterexample (p.84)

Choose the correct vocabulary term to complete each sentence.
1. The part of a conditional that follows “then” is the ? .
2. Reasoning logically from given statements to a conclusion is ? .
3. A conditional has a(n) ? of true or false.
4. The ? of a conditional switches the hypothesis and conclusion.
5. When a conditional and its converse are true, you can write them as a single true

statement called a(n) ? .
6. A statement that you prove true is a(n) ? .
7. The part of a conditional that follows “if” is the ? .

Chapter 2  Chapter Review 129

2-1  Patterns and Inductive Reasoning

Quick Review Exercises

You use inductive reasoning when you make conclusions Find a pattern for each sequence. Describe the pattern
based on patterns you observe. A conjecture is a conclusion and use it to show the next two terms.
you reach using inductive reasoning. A counterexample is
an example that shows a conjecture is incorrect. 8. 1000, 100, 10, c
9. 5, -5, 5, -5, c
Example 10. 34, 27, 20, 13, c
11. 6, 24, 96, 384, c
Describe the pattern. What are the next two terms
in the sequence? Find a counterexample to show that each conjecture
1, − 3, 9, − 27, . . . is false.
Each term is -3 times the previous term. The next two
terms are -27 * ( -3) = 81 and 81 * ( -3) = -243. 12. The product of any integer and 2 is greater than 2.

13. The city of Portland is in Oregon.

2-2  Conditional Statements Exercises

Quick Review Rewrite each sentence as a conditional statement.
14. All motorcyclists wear helmets.
A conditional is an if-then statement. The symbolic form of 15. Two nonparallel lines intersect in one point.
a conditional is p S q, where p is the hypothesis and q is 16. Angles that form a linear pair are supplementary.
the conclusion. 17. School is closed on certain holidays.

• To find the converse, switch the hypothesis and Write the converse, inverse, and contrapositive of the
conclusion of the conditional (q S p). given conditional. Then determine the truth value
of each statement.
• To find the inverse, negate the hypothesis and the 18. If an angle is obtuse, then its measure is greater than
conclusion of the conditional (∼ p S ∼ q).
90 and less than 180.
• To find the contrapositive, negate the hypothesis 19. If a figure is a square, then it has four sides.
and the conclusion of the converse (∼ q S ∼ p). 20. If you play the tuba, then you play an instrument.
21. If you baby-sit, then you are busy on Saturday night.
Example

What is the converse of the conditional statement below?
What is its truth value?
  If you are a teenager, then you are younger than 20.

Converse: If you are younger than 20, then you are a
teenager.
A 7-year-old is not a teenager. The converse is false.

130 Chapter 2  Chapter Review

2-3  Biconditionals and Definitions Exercises

Quick Review Determine whether each statement is a good definition.
If not, explain.
When a conditional and its converse are true, you can 22. A newspaper has articles you read.
combine them as a true biconditional using the phrase if 23. A linear pair is a pair of adjacent angles whose
and only if. The symbolic form of a biconditional is p 4 q.
You can write a good definition as a true biconditional. noncommon sides are opposite rays.
24. An angle is a geometric figure.
Example
25. Write the following definition as a biconditional.
Is the following definition reversible? If yes, write it as a An oxymoron is a phrase that contains
true biconditional. contradictory terms.

A hexagon is a polygon with exactly six sides. 26. W rite the following biconditional as two statements,
a conditional and its converse.
Yes. The conditional is true: If a figure is a hexagon, then it
is a polygon with exactly six sides. Its converse is also true: Two angles are complementary if and only if the
If a figure is a polygon with exactly six sides, then it is a sum of their measures is 90.
hexagon.
Biconditional: A figure is a hexagon if and only if it is a Exercises
polygon with exactly six sides.
Use the Law of Detachment to make a conclusion.
2-4  Deductive Reasoning 27. I f you practice tennis every day, then you will

Quick Review become a better player. Colin practices tennis
every day.
Deductive reasoning is the process of reasoning logically 28. ∠1 and ∠2 are supplementary. If two angles are
from given statements to a conclusion. supplementary, then the sum of their measures
Law of Detachment: If p S q is true and p is true, is 180.
then q is true.
Law of Syllogism: If p S q and q S r are true, Use the Law of Syllogism to make a conclusion.
then p S r is true. 29. I f two angles are vertical, then they are congruent.

Example If two angles are congruent, then their measures
are equal.
What can you conclude from the given information? 30. I f your father buys new gardening gloves, then he
Given: If you play hockey, then you are on the team. will work in his garden. If he works in his garden,
If you are on the team, then you are a varsity athlete. then he will plant tomatoes.

The conclusion of the first statement matches the
hypothesis of the second statement. Use the Law of
Syllogism to conclude: If you play hockey, then you are
a varsity athlete.

Chapter 2  Chapter Review 131

2-5  Reasoning in Algebra and Geometry

Quick Review Exercises

You use deductive reasoning and properties to solve 3 1. Algebra  Fill in the reason that justifies each step.
equations and justify your reasoning.
A proof is a convincing argument that uses deductive Given:  QS = 42 xϩ3 2x
reasoning. A two-column proof lists each statement on the Prove:  x = 13 QR
left and the justification for each statement on the right. S

Example Statements Reasons

What is the name of the property that justifies going from 1) QS = 42 1) a. ?
the first line to the second line? 2) QR + RS = QS
3) (x + 3) + 2x = 42 hs m2)1 1bg. m? se_02cr_t00581
jA @ jB and jB @ jC 4) 3x + 3 = 42
jA @ jC 5) 3x = 39 3) c. ?
Transitive Property of Congruence 6) x = 13 4) d. ?
5) e. ?
6) f. ?

Use the given property to complete the statement.

32. Division Property of Equality
If 2(AX) = 2(BY), then AX = ? .

33. Distributive Property: 3p - 6q = 3( ? )

2-6  Proving Angles Congruent

Quick Review Exercises

A statement that you prove true is a theorem. A proof Use the diagram for Exercises 34–37.
written as a paragraph is a paragraph proof. In geometry,
each statement in a proof is justified by given information, a 34. Find the value of y. AB
property, postulate, definition, or theorem. 35. Find m∠AEC. (3y ϩ 20)Њ E (5y Ϫ 16)Њ
36. Find m∠BED.
Example
23 4 37. Find m∠AEB. C D
Write a paragraph proof.

Given:  ∠1 ≅ ∠4 1 38. Given:  ∠1 and ∠2 are
Prove:  ∠2 ≅ ∠3
complementary. 12
∠3 and ∠4 are
∠1 ≅ ∠4 because it is given. ∠1 ≅ ∠2 because vertical complementary.hsm11gmse_02cr_t00583

angles are congruent. ∠4 ≅ ∠2 by the Transitive Property

of Congruence. ∠4 ≅ ∠3 because vertical angles are ∠2 ≅ ∠4 3
Prove:  ∠1 ≅ ∠3 4
congruent. ∠2 ≅ ∠3 by the TransitivhesPmro1p1egrtmy se_02cr_t0058 2

of Congruence.

hsm11gmse_02cr_t01761

132 Chapter 2  Chapter Review

2 Chapter Test MathX

OLMathXL® for School
R SCHO Go to PowerGeometry.com
L®

FO

Do you know HOW? 16. The measure of an angle is 52 more than the measure
of its complement. What is the measure of the angle?
Use inductive reasoning to describe each pattern and
find the next two terms of each sequence. 17. Rewrite this biconditional as two conditionals.
A fish is a bluegill if and only if it is a bluish,
1. -16, 8, -4, 2, c 2. 1, 4, 9, 16, 25, c freshwater sunfish.

For Exercises 3 and 4, find a counterexample. For each diagram, state two pairs of angles that are
congruent. Justify your answers.
3. All snakes are poisonous.
18. P 19. E F
4. If two angles are complementary, then they are L
not congruent. 62Њ

5. Identify the hypothesis and conclusion: N BCD
If x + 9 = 11, then x = 2.
VM
6. Write “all puppies are cute” as a conditional.

Write the converse, inverse, and contrapositive for Use the Law of Detachment and thhesmLa1w1ogfmSyslelo_g0is2mcat_ot00586
each statement. Determine the truth value of each.
make any possible conclusion. Write not possible if you
7. If a figure is a square, then it has at least two
right angles. cannhostmm1a1kge mansyec_o0n2ccluas_iot0n0. 584

8. If a square has side length 3 m, then its perimeter 20. People who live in glass houses should not throw
is 12 m. stones. Emily should not throw stones.

Writing  Explain why each statement is not 21. James wants to be a chemical engineer. If a student
a good definition. wants to be a chemical engineer, then that student
must graduate from college.
9. A pen is a writing instrument.
Do you UNDERSTAND?
10. Supplementary angles are angles that form a
straight line. 22. Open-Ended  Write two different sequences whose
first three terms are 1, 2, 4. Describe each pattern.
11. Vertical angles are angles that are congruent.
23. Developing Proof  Complete this FD
Name the property that justifies each statement. proof by filling in the blanks. EW
12. If UV = KL and KL = 6, then UV = 6.
13. If m∠1 + m∠2 = m∠4 + m∠2, Given:  ∠FED and ∠DEW are
complementary.
then m∠1 = m∠4.
14. ∠ABC ≅ ∠ABC Prove:  ∠FEW is a right angle.
15. If ∠DEF ≅ ∠HJK , then ∠HJK ≅ ∠DEF .
∠FED and ∠DEW are complementary because it is

given. By the Definition of Complementary Angles,
hsm11gmse_02ca_t
m∠FED + m∠DEW = a. ? .
m∠FED + m∠DEW = m∠FEW by the b. ? .
90 = m∠FEW by the c. ? Property of Equality.
Then ∠FEW is a right angle by the d. ? .

Chapter 2  Chapter Test 133

2 Common Core Cumulative ASSESSMENT
Standards Review

Some questions ask you to The first four figures in a sequence are TIP 2
extend a pattern. Read the shown below. How many dots will be in the
sample question at the right. sixth figure of this sequence? Use the relationship between
Then follow the tips to answer it. the figures to extend the
6 15 pattern.
TIP 1 11 21
Think It Through
Look for a relationship hsm11gmse_02cu_t00559
between consecutive The second figure has 2 more
dots than the first figure, the third
TrfeigIlautrPieosn.s1Mhiapkheosludrseftohre figure has 3 more dots than the
second figure, and the fourth
each pair of consecutive figure has 4 more dots than the
figures, not just the third figure. So, the fifth figure
first two figures. will have 5 more dots than the
fourth figure, or 10 + 5 = 15
dots. The sixth figure will have 6
more dots than the fifth figure, or
15 + 6 = 21 dots.
The correct answer is D.

LVVeooscsacoabnubluarlayry Builder Selected Response

As you solve test items, you must understand Read each question. Then write the letter of the correct
the meanings of mathematical terms. Choose the answer on your paper.
correct term to complete each sentence.
1. Which pair of angles must be congruent?
I. R easoning that is based on patterns you observe is supplementary angles
called (inductive, deductive) reasoning. complementary angles
adjacent angles
II. T he (Law of Syllogism, Law of Detachment) allows vertical angles
you to state a conclusion from two true conditional
statements when the conclusion of one statement is 2. Which of the following best defines a postulate?
the hypothesis of the other statement. a statement accepted without proof
a conclusion reached using inductive reasoning
III. A conditional, or if-then, statement has two parts. an example that proves a conjecture false
The part following if is the (conclusion, hypothesis). a statement that you prove true

IV. T he (Reflexive Property, Symmetric Property) says
that if a = b, then b = a.

V. The (inverse, converse) of a conditional negates both
the hypothesis and the conclusion.

134 Chapter 2  Common Core Cumulative Standards Review

3. What is the second step in constructing ∠S, an angle 6. Which counterexample shows that the following
congruent to ∠A? conjecture is false?

A Every perfect square number has exactly three factors.

S hsmT 11gmse_02cu_t00560 The factors of 2 are 1, 2.
The factors of 4 are 1, 2, 4.
The factors of 8 are 1, 2, 4, 8.
The factors of 16 are 1, 2, 4, 8, 16.

7. How many rays are in the next two terms in
the sequence?

hS sm11gmse_R02cu_t00561
16 and 33 rays

17 and 33 rays
hsmS 11gmse_R02cu_t00562 17 and 34 rays
18 and 34 rays

B 8. hWonshmtihc1he1foogflltmohweseisnt_ag0tie2nmcfouern_mttsa0ct0ioo5un6l?d7be a conclusion based

A hsm11gmse_02cu_t00564 If a polygon is a pentagon, then it has one more side
than a quadrilateral. If a polygon has one more side
C than a quadrilateral, then it has two more sides than
a triangle.
4. What is the converse of the following statement?
If a polygon is a pentagon, then it has many sides.
If a whole number has 0 as its last digit, then the
If a polygon has two more sides than a
numbhesrmis 1ev1egnmlysdeiv_i0si2bcleub_yt0100.565 quadrilateral, then it is a hexagon.

If a number is evenly divisible by 10, then it is a If a polygon has more sides than a triangle, then it
whole number. is a pentagon.

If a whole number is divisible by 10, then it is an If a polygon is a pentagon, then it has two more
even number. sides than a triangle.

If a whole number is evenly divisible by 10, then it 9. The radius of each of the circular sections in the
has 0 as its last digit. dumbbell-shaped table below is 3 ft. The rectangular
portion has an area of 32 ft2 and the length is twice
If a whole number has 0 as its last digit, then it the width. What is the area of the entire table to the
must be evenly divisible by 10. nearest tenth?

5. The sum of the measures of the complement and the

supplement of an angle is 114. What is the measure of

the angle?

12 78

66 102 88.5 ft2 132.5 ft2
124.5 ft2 166.5 ft2

hsm11gmse_02cu_t00568

Chapter 2  Common Core Cumulative Standards Review 135

Constructed Response 17. What is the y-coordinate of the midpoint of a segment
with endpoints (0, -4) and ( -4, 7)?
1 0. An athletic field is a rectangle, 120 yd by 60 yd, with
a semicircle at each of the ends. A running track 15 yd 1 8. On a number line, P is at -4 and R is at 8. What is the
wide surrounds the field. How many yards of fencing 3
do you need to surround the outside edge of the track? coordinate of the point Q, which is 4 the way from R to P?
Round your answer to the nearest tenth of a yard. Use
3.14 for p. 1 9. Write the converse, inverse, and contrapositive

of the following statement. Determine the truth

value of each.

If you live in Oregon, then you live in the United States.

60 yd 2 0. AB has endpoints A(3, 6) and B(9, -2) and
midpoint M. Justify each response.
120 yd
a. What are the coordinates of M?
1 1. What is the next number in the pattern? b. What is AB?
1, -4, 9, -16, c
21. Draw a net for the figure below.
12. The basehosfma r1e1ctgamngslee_is072ccmu_lets0s0t5ha6n9three times
Extended Response
its height. If the base is 5 cm, what is the area of the
rectangle in square centimeters? 2 2. The sequence hbeslmow12li_stds hth2ecfuir_stt0ei5g0h0t p.aoiwers of 7.

1 3. m∠BZD = 107 71, 72, 73, 74, 75, 76, 77, 78, . . .
m∠FZE = 2x + 5
m∠CZD = x a. Make a table that lists the digit in the ones place
for each of the first eight powers of 7. For example,
What is the measure of ∠CZD? 74 = 2401. The digit 1 is in the ones place.

B C b. What digit is in the ones place of 734? Explain
AZ D your reasoning.

FE 23. Write a proof.

1 4. The measure of an angle is three more than twice its Given: ∠1 and ∠2 are 1
supplement. What is the measure of the angle? supplementary. 2

15. A square and rectangle have equal area. The rectangle Prove: ∠1 and ∠2 are
is 32 cm by 18 cm. What is the perimeter of the square right angles.
in centimegteerosm? 12_se_ccs_c02_csr_c02.ai

1 6. What is the value of x?

hsm11gmse_02cu_t00573

΀ x ϩ 5΁Њ (2x Ϫ 70)Њ
3

136 Chaptehr s2m  1C1omgmmosne_C0o2recCuu_mt0u0la5t7iv2e Standards Review

Get Ready! CHAPTER

Lesson 1-5 Identifying Angle Pairs 3

Identify all pairs of each type of angles in the diagram.

1. linear pair

2. complementary angles 1 52 3
3. vertical angles 4

4. supplementary angles

Lesson 2-5 Justifying Statements

Name the property that justifies each statement.

5. If 3x = 6, then x = 2. hsm11gmse_03co_t00552.ai

6. If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then ∠1 ≅ ∠3.

Skills Solving Equations

Handbook, Algebra  Solve each equation.
p. 894

7. 3x + 11 = 7x - 5 8. (x - 4) + 52 = 109 9. (2x + 5) + (3x - 10) = 70

Lesson 1-7 Finding Distances in the Coordinate Plane

Find the distance between the points.

10. (1, 3) and ( 5, 0) 11. ( -4, 2) and (4, 4) 12. (3, -1) and (7, -2)

Looking Ahead Vocabulary

13. The core of an apple is in the interior of the apple. The peel is on the exterior. How
can the terms interior and exterior apply to geometric figures?

14. A ship sailing from the United States to Europe makes a transatlantic voyage. What
does the prefix trans- mean in this situation? A transversal is a special type of line in
geometry. What might a transversal do? Explain.

15. People in many jobs use flowcharts to describe the logical steps of a particular
process. How do you think you can use a flow proof in geometry?

Chapter 3  Parallel and Perpendicular Lines 137

CHAPTER Parallel and

3 Perpendicular Lines

Download videos VIDEO Chapter Preview 1 Reasoning and Proof
connecting math Essential Question  How do you prove
to your world.. 3-1 Lines and Angles that two lines are parallel?
3-2 Properties of Parallel Lines
Interactive! YNAM IC 3-3 Proving Lines Parallel 2 Measurement
Vary numbers, T I V I TIAC 3-4 Parallel and Perpendicular Lines Essential Question  What is the sum of
graphs, and figures D 3-5 Parallel Lines and Triangles the measures of the angles of a triangle?
to explore math ES 3-6 Constructing Parallel and
concepts.. 3 Coordinate Geometry
Perpendicular Lines Essential Question  How do you write an
3-7 Equations of Lines in the equation of a line in the coordinate plane?

The online Coordinate Plane
Solve It will get 3-8 Slopes of Parallel and
you in gear for
each lesson. Perpendicular Lines

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Congruence
Spanish English/Spanish Vocabulary Audio Online:
• Expressing Geometric Properties with
English Spanish Equations

Online access alternate exterior ángulos alternos externos • Modeling with Geometry
to stepped-out angles, p. 142
problems aligned
to Common Core alternate interior ángulos alternos internos
Get and view angles, p. 142
your assignments
online. NLINE corresponding ángulos correspondientes
ME WO angles, p. 142

O exterior angle of a ángulo exterior de un polígono
RK polygon, p. 173
HO

parallel lines, p. 140 rectas paralelas

same-side interior ángulos internos del mismo
angles, p. 142 lado
Extra practice
and review skew lines, p. 140 rectas cruzadas
online
transversal, p. 141 transversal

Virtual NerdTM
tutorials with
built-in support

PERFORMANCE TASK

Common Core Performance Task

Planning the Paths for a Park

Kiana works for a city’s planning department. The city is developing a new park,
and Kiana is reviewing the plan for the builders. The rectangular park has two
pairs of parallel walkways, one of which is also parallel to two sides of the park. In
the blueprint of the park plan below, the walkways are shown in blue and parallel
lines are indicated by red arrowheads.

Kiana notices that the blueprint provides only a few angle measures, shown in
red. She would like to add additional angle measures to help the builders.

12 3
4

56 7 89

10 95° 11 12 13 14

15 16 17
18

19 20 17° 21 22 23 24

25 26 27 121° 32

28

29 30 31

Task Description

Find the measure of each numbered angle in the blueprint.

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll analyze a complex diagram and identify types of angle pairs.
(MP 1, MP 7)

• You’ll use theorems related to parallel lines to find angle measures. (MP 6)

• You’ll construct a viable argument to explain your method of finding angle
measures. (MP 3)

Chapter 3  Parallel and Perpendicular Lines 139

3-1 Lines and Angles CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

MG-ACFOS.A.9.112  .KGn-oCwO.p1r.e1c isKendoewfinpirteiocnisseodfe.fi.n.iptiaornaslloelf l.in. e. .
Pparerapllaerl elisnef.or G-CO.C.9  Prove theorems about lines
Panredpaanrgeless.for MAFS.912.G-CO.3.9  Prove theorems
aMbPou1t,liMnePs 3a,nMd aPn6gles.
MP 1, MP 3, MP 6

Objectives To identify relationships between figures in space
To identify angles formed by two lines and a transversal

You want to assemble a bookcase. You have all the pieces, but you
misplaced the instructions that came with the box. How would you write
the instructions?

Try visualizing CE
how the bookcase
looks in two AB
dimensions.
D F
MATHEMATICAL

PRACTICES

Lesson In the Solve It, you used relationships among planes in space to write the instructions.

Vocabulary In Chapter 1, you learned about intersecting lines and planes. In this lesson, you will
• parallel lines
• skew lines explore relationships of nonintersecting lineHsSaMnd11p_lGanMeSsE._0301_a_00833
• parallel planes Essential Understanding  Not all lineD3srudarknpedasnso1t2a-0ll7p-0la8nes intersect.
• transversal
• alternate interior Key Concept  Parallel and Skew

angles Definition S<<AAyEDm>> }}b<B<BoFCl>s> Diagram C
• same-side interior Parallel lines are coplanar D
lines that do not intersect.
angles The symbol } means “is
• corresponding parallel to.”

angles
• alternate exterior

angles

Skew lines are <AB> and <CG> are skew. AB
noncoplanar; they are HG
not parallel and do not
intersect. EF

Use arrows to show
AE ʈ BF and AD ʈ BC.

Parallel planes are planes plane ABCD } plane EFGH
that do not intersect.

hsm11gmse_0301_t00227.ai

140 Chapter 3  Parallel and Perpendicular Lines

A line and a plane that do not intersect are parallel. Segments and rays can also be
parallel or skew. They are parallel if they lie in parallel lines and skew if they lie in
skew lines.

Problem 1 Identifying Nonintersecting Lines and Planes

Parallel lines are In the figure, assume that lines and planes that appear to be parallel are parallel.

cp olapnlaensacro.nWtahiinchAB? A Which segments are parallel to AB? B C

P lanes ABCD, ABFE, and EF , DC, and HG

A BGH contain AB. You B Which segments are skew to CD? A D
need to visualize plane F G
A BGH. BF , AE, EH, and FG

B C What are two pairs of parallel planes?

A G plane ABCD } plane EFGH E H

H plane DCG } plane ABF

D What are two segments parallel to plane BCGF?

AD and DH hsm11gmse_0301_t00228.ai

hsm11gmse_0301_t1055G4.oati It? 1. Use the figure in Problem 1.

a. Which segments are parallel to AD?

b. Reasoning  Explain why FE and CD are not skew.

c. What is another pair of parallel planes?

d. What are two segments parallel to plane DCGH?

Essential Understanding  When a line intersects two or more lines, the
angles formed at the intersection points create special angle pairs.

A transversal is a line that intersects two or more coplanar lines at distinct points. The

diagram below shows the eight angles formed by a transversal t and two lines / and m.

t ᐍ

Exterior 12
43

Interior

56 m
87

Exterior

Notice that angles 3, 4, 5, and 6 lie between / and m. They are interior angles. Angles
1, 2, 7, and 8 lie outside of / and m. They are exterior angles.

hsm11gmse_0301_t00229.ai

Lesson 3-1  Lines and Angles 141

Pairs of the eight angles have special names as suggested by their positions.

Key Concept  Angle Pairs Formed by Transversals

Definition Example t ഞ
Alternate interior angles are 12 m
nonadjacent interior angles ∠4 and ∠6 43
that lie on opposite sides of the ∠3 and ∠5 56
transversal. 87

Same-side interior angles are ∠4 and ∠5 tᐍ
interior angles that lie on the ∠3 and ∠6 12
same side of the transversal.
hsm11gms4e_30301_t00230.ai
56
87 m

Corresponding angles ∠1 and ∠5 tᐍ
lie on the same side of ∠4 and ∠8 12
the transversal t and in ∠2 and ∠6 hsm11gm4se_30301_t00232.ai
corresponding positions. ∠3 and ∠7 56
87 m

Alternate exterior angles are ∠1 and ∠7 tഞ
nonadjacent exterior angles ∠2 and ∠8 12
that lie on opposite sides of the hsm11gms4e_30301_t00233.ai
transversal. 56
87 m

Problem 2 Identifying an Angle Pair hsm11gmse_0301_t00234.ai

Which choices can Multiple Choice  Which is a pair of alternate interior angles? m n
y ou eliminate? r
iY notuernioere adnaglpeasi.r∠of1, ∠4, ∠1 and ∠3 ∠2 and ∠6 12
and ∠8 are exterior ∠6 and ∠7 ∠4 and ∠8 8 7 34
angles. You can eliminate
∠2 and ∠6 are alternate interior angles because they lie on 65
choices A and D. opposite sides of the transversal r and in between m and n.
The correct answer is C.

Got It? 2. Use the figure in Problem 2. What are three pairs of corresponding angles?

hsm11gmse_0301_t00235.ai

142 Chapter 3  Parallel and Perpendicular Lines

Problem 3 Classifying an Angle Pair STEM

Architecture  The photo below shows the Royal Ontario Museum in Toronto, Canada.
Are angles 2 and 4 alternate interior angles, same-side interior angles, corresponding
angles, or alternate exterior angles?

How do the positions 1
of j2 and j4 2
compare?
∠2 and ∠4 are both 3
interior angles and they 4

lie on opposite sides of

a line.

Angles 2 and 4 are alternate interior angles.

Got It? 3. In Problem 3, are angles 1 and 3 alternate interior angles, same-side interior
angles, corresponding angles, or alternate exterior angles?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

Name one pair each of the segments, planes, or angles. 8. Vocabulary  Why is the word coplanar included in

Lines and planes that appear F the definition for parallel lines?
BH 9. Vocabulary  How does the phrase alternate interior
to be parallel are parallel. E G

1. parallel segments AD C angles describe the positions of the two angles?
2. skew segments Exercises 1−3
3. parallel planes 10. Error Analysis  In the figure at D C
4. alternate interior the right, lines and planes that H
5. same-side interior A
6. corresponding hsm5 1134gm68se12_07301_t00333.ai appear to be parallel are parallel. G B
Carly says AB } HG. Juan says E
AB and HG are skew. Who is F
correct? Explain.

7. alternate exterior Exercises 4−7

Lesson 3-1  Lines and Angles hsm11gmse_0310413_t003

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Use the diagram to name each of the following. Assume that lines and planes See Problem 1.
that appear to be parallel are parallel. See Problem 2.

11. a pair of parallel planes C D
12. all lines that are parallel to <AB> H
13. all lines that are parallel to <DH> JL
14. two lines that are skew to <EJ> FG
15. all lines that are parallel to plane JFAE
16. a plane parallel to <LH> E
AB

Identify all pairs of each type of angles in the diagram. Name the two lines and
hsm11gmse_0301_t00336.ai
the transversal that form each pair. bc
a
17. corresponding angles d 8

18. alternate interior angles 7 346

19. same-side interior angles e 12 5
20. alternate exterior angles

Are the angles labeled in the same color alternate interior angles, same-side See Problem 3.
interior angles, corresponding angles, or alternate exterior angles?
35
21. 22. hsm11gmse_0301_t00337.2a3i . 41
62
5 4 1 3 35 1
6 42
2 6

24. Aviation  The photo shows an overhead view of airport runways. Are ∠1 and 

∠2 alternate interior angles, same-side interior angles, corresponding angles, or

ahlstemrn1a1tegemxtseeri_or0a3n0g1le_st?00338.ai hsm11gmse_0301_t00339.ai hsm11gmse_0301_t00340.ai

1

2

144 Chapter 3  Parallel and Perpendicular Lines

B Apply How many pairs of each type of angles do two lines and a transversal form?

25. alternate interior angles 26. corresponding angles

27. alternate exterior angles 28. vertical angles

29. Recreation  You and a friend are driving go-karts on two different tracks. As you
drive on a straight section heading east, your friend passes above you on a straight
section heading south. Are these sections of the two tracks parallel, skew, or
neither? Explain.

In Exercises 30–35, describe the statement as true or false. If false, explain. H
B IG
Assume that lines and planes that appear to be parallel are parallel. A CJ
30. <CB> } <HG> 31. <ED> } <HG>
F
32. plane AED } plane FGH 33. plane ABH } plane CDF ED
34. <AB> and <HG> are skew lines. 35. <AE> and <BC> are skew lines.

36. Think About a Plan  A rectangular rug covers the floor in a living room. One of

the walls in the same living room is painted blue. Are the rug and the blue wall

parallel? Explain. hsm11gmse_0301_t00341.ai

• Can you visualize the rug and the wall as geometric figures?

• What must be true for these geometric figures to be parallel?

In Exercises 37–42, determine whether each statement is always, sometimes, or
never true.

37. Two parallel lines are coplanar.
38. Two skew lines are coplanar.
39. Two planes that do not intersect are parallel.
40. Two lines that lie in parallel planes are parallel.
41. Two lines in intersecting planes are skew.
42. A line and a plane that do not intersect are skew.

43. a. Writing  Describe the three ways in which two lines may be related.
b. Give examples from the real world to illustrate each of the relationships you

described in part (a).

44. Open-Ended  The letter Z illustrates alternate interior angles. Find at least two other
letters that illustrate pairs of angles presented in this lesson. Draw the letters. Then
mark and describe the angles.

45. a. Reasoning  Suppose two parallel planes A and B are each intersected by a third
plane C. Make a conjecture about the intersection of planes A and C and the
intersection of planes B and C.

b. Find examples in your classroom to illustrate your conjecture in part (a).

Lesson 3-1  Lines and Angles 145

C Challenge Use the figure at the right for Exercises 46 and 47. A
B
46. tDoo<CpDla>?nEexspAlaainnd. B have other lines in common that are parallel
47. lViniseusapliazraatliloenl t oA<CreDt>h?eDrerapwlaanseksetthcahttiontseurpsepcotrpt ylaonuersaAnsawnedr.B in CD

48. Draw a Diagram  A transversal r intersects lines / and m. If /
and r form ∠1 and ∠2 and m and r form ∠3 and ∠4, sketch

a diagram that meets the following conditions.
• ∠3 and ∠4 are supplementary. hsm11gmse_0301_t00342.ai
• ∠1 ≅ ∠2

• ∠3 is an interior angle. • ∠2 and ∠4 lie on opposite sides of r.

• ∠4 is an exterior angle.

PERFORMANCE TASK MATHEMATICAL

Apply What You’ve Learned PRACTICES
MP 1, MP 7

Look back at the information on page 139 about the plan for a city park.
The blueprint of the park plan is shown again below.

12 3
4

56 7 89

10 95° 11 12 13 14

15 16 17
18

19 20 17° 21 22 23 24

25 26 27 121° 32

28

29 30 31

The blueprint contains many examples of the types of angle pairs you learned
about in Lesson 3-1, some formed by parallel lines and a transversal, and some
formed by nonparallel lines and a transversal. Name all numbered angles in the
blueprint that fit the given description.

a. an angle that forms a pair of alternate interior angles with ∠2

b. an angle that forms a pair of corresponding angles with the angle that measures 121°

c. an angle that forms a pair of same-side interior angles with the angle that measures 121°

d. an angle that forms a pair of alternate interior angles with the angle that measures 95°

146 Chapter 3  Parallel and Perpendicular Lines

Concept Byte Parallel Lines and CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Related Angles
Use With Lesson 3-2 MG-ACFOS.D.9.12. GM-CaOke.4fo.1rm2 aMl gaekoemfoertmricalcognesotmruecttriiocns
cwointhstaruvcatiroientsy wofitthooalsvaarniedtymoefthtooodls..a. nAdlsmoethods...
TECHNOLOGY APrlseopaPrreespfaorresGf-oCrOM.CA.9FS.912.G-CO.3.9

MP 5

1
22 44
Use geometry software to construct two parallel lines. Check that the lines remain 3
parallel as you manipulate them. Construct a point on each line. Then construct
the transversal through these two points. 5
66 88
1. Measure each of the eight angles formed by the parallel lines and the 7
transversal. Record the measurements.

2. Manipulate the lines. Record the new measurements.

3. When a transversal intersects parallel lines, what are the relationships among
the angle pairs formed? Make as many conjectures as possible.

Exercises
4. Construct three or more parallel lines. Then construct a line that intersects all the hsm11gmse_03fa_t00500.ai

parallel lines.
a. What relationships can you find among the angles formed?
b. How many different angle measures are there?

5. Construct two parallel lines and a transversal perpendicular to one of the parallel
lines. What angle does the transversal form with the second line?

6. Construct two lines and a transversal, making sure that the two lines are not
parallel. Locate a pair of alternate interior angles. Manipulate the lines so that
these angles have the same measure.

a. Make a conjecture about the relationship between the two lines.
b. How is this conjecture different from the conjecture(s) you made in the

Activity?

7. Again, construct two lines and a transversal, making sure that the two lines are not
parallel. Locate a pair of same-side interior angles. Manipulate the lines so that
these angles are supplementary.

a. Make a conjecture about the relationship between the two lines.
b. How is this conjecture different from the conjecture(s) you made in

the Activity?

8. Construct perpendicular lines a and b. At a point that is not the intersection of
a and b, construct line c perpendicular to line a. Make a conjecture about lines
b and c.

Concept Byte  Parallel Lines and Related Angles 147

3-2 Properties of CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Parallel Lines
GM-ACFOS.C.9.192  .PGro-CveOt.h3e.9o rePmrosvaebtohuetolrienmes anbdouatnlginleess. and
Tahnegolerse;musseintchlueodree:m. .s.awbhoeunt lainterasnasnvderasnagl lcersostosessoplvaerallel
lpinroebs,leamltes.rTnhaeteorienmtesriionrcalundgele: s. .a.rwe hcoengaruterannts.v.e.rsal
McroPss1e,sMpaPra3l,leMl lPine4s,,MaltPer6nate interior angles are
congruent . . .

MP 1, MP 3, MP 4, MP 6

Objectives To prove theorems about parallel lines
To use properties of parallel lines to find angle measures

Look at the map of streets in Palm Bluff St. 12
Clearwater, Florida. Nicholson Street 43
and Cedar Street are parallel. N. Garden Ave.
You see the Which pairs of angles appear to be 19
vertical angles, congruent?
right? Keep
looking! There are Cedar St. 56
other angle pairs. 87

MATHEMATICAL 9 10

PRACTICES Nicholson St.

12 11

In the Solve It, you identified several pairs of angles that appear congruent. You
already know the relationship between vertical angles. In this lesson, you will explore
the relationships between the angles you learned about in Lesson 3-1 when they are
formed by parallel lines and a transversal.

Essential Understanding  The special angle pairs formed by parallel lines and a
transversal are congruent, supplementary, or both.

Postulate 3-1  Same-Side Interior Angles Postulate

Postulate If . . . Then . . .
If a transversal intersects /}m m∠4 + m∠5 = 180
two parallel lines, then ᐍ m∠3 + m∠6 = 180
same-side interior angles
are supplementary. m 43
56

148 Chapter 3  Parallel and Perpendicular Lines hsm11gmse_0302_t00238.ai

Problem 1 Identifying Supplementary Angles

How do you find The measure of j3 is 55. Which angles are 12 56
angles supplementary supplementary to j3? How do you know?
to a given angle? 43 87
Look for angles whose By definition, a straight angle measures 180.
measure is the difference
of 180 and the given angle If m∠a + m∠b = 180, then ∠a and ∠b are
measure. supplementary by definition of supplementary angles.

180 - 55 = 125, so any angle x, where m∠x = 125, is supplementary to ∠3.

m∠4 = 125 by the definition of a straight angle. hsm11gmse_0302_t00237.ai
m∠8 = 125 by the Same-Side Interior Angles Postulate.

m∠6 = m∠8 by the Vertical Angles Theorem, so m∠6 = 125.

m∠2 = m∠4 by the Vertical Angles Theorem, so m∠2 = 125.

Got It? 1. Reasoning  If you know the measure of one of the angles, can you always
find the measures of all 8 angles when two parallel lines are cut by a
transversal? Explain.

You can use the Same-Side Interior Angles Postulate to prove other angle relationships.

Theorem 3-1  Alternate Interior Angles Theorem

Theorem If . . . Then . . .
If a transversal intersects /}m
two parallel lines, then ᐍ ∠4 ≅ ∠6
alternate interior angles ∠3 ≅ ∠5
are congruent. m
43
56

Theorem 3-2  Corresponding Angles Theorem

Theorem Ifh.s.m. 11gmse_0302_t00238T.ahien . . .
/ } m ∠1 ≅ ∠5
If a transversal intersects
two parallel lines, then ∠2 ≅ ∠6
corresponding angles are ᐍ 12
congruent. ∠3 ≅ ∠7
4 3 ∠4 ≅ ∠8

m 56
87

You will prove Theorem 3-2 in Exercise 25.

hsm11gmse_0302_t00236.ai 149

Lesson 3-2  Properties of Parallel Lines

Proof Proof of Theorem 3-1: Alternate Interior Angles Theorem

Given: / } m ᐍ
Prove: ∠4 ≅ ∠6 43

m6

Statement Reasons

1) / } m hsm11gm 1s)e _G0iv3e0n2_t00239.ai
2) m∠3 + m∠4 = 180
2) Supplementary Angles

3) m∠3 + m∠6 = 180 3) Same-Side Interior Angles Postulate

4) m∠3 + m∠4 = m∠3 + m∠6 4) Transitive Property of Equality

5) m∠4 = m∠6 5) Subtraction Property of Equality

6) ∠4 ≅ ∠6 6) Definition of Congruence

Proof Problem 2 Proving an Angle Relationship

Given: a } b a 1
Prove: ∠1 and ∠8 are supplementary.

•a}b b5
From the diagram you know 87
• ∠1 and ∠5 are corresponding
• ∠5 and ∠8 form a linear pair

∠1 and ∠8 are supplementary, hsm11gmse_0302_t00240.ai
or m∠1 + m∠8 = 180.
Show that ∠1 ≅ ∠5 and that m∠5 + m∠8 = 180.
Then substitute m∠1 for m∠5 to prove that ∠1
and ∠8 are supplementary.

Statements Reasons

1) a } b 1) Given
2) ∠1 ≅ ∠5 2) If lines are }, then corresp. ⦞ are ≅.
3) Congruent ⦞ have equal measures.
3) m∠1 = m∠5 4) ⦞ that form a linear pair are suppl.
4) ∠5 and ∠8 are supplementary. 5) Def. of suppl.⦞

5) m∠5 + m∠8 = 180 6) Substitution Property
6) m∠1 + m∠8 = 180 7) Def. of suppl.⦞
7) ∠1 and ∠8 are supplementary.

Got It? 2. Using the same given information and diagram in Problem 2, prove that
∠1 ≅ ∠7.

150 Chapter 3  Parallel and Perpendicular Lines

In the diagram for Problem 2, ∠1 and ∠7 are alternate exterior angles. In Got It 2, you
proved the following theorem.

Theorem 3-3  Alternate Exterior Angles Theorem

Theorem If . . . 12 Then . . .
If a transversal intersects two /}m
parallel lines, then alternate ᐍ ∠1 ≅ ∠7
exterior angles are congruent. ∠2 ≅ ∠8

m
87

If you know the measure of one of thehasnmgl1es1fgomrmseed_b0y3t0w2o_pt0ar0a2ll4e1l l.ianies and a

transversal, you can use theorems and postulates to find the measures of the other
angles.

How do j3 and j4 Problem 3 Finding Measures of Angles
relate to the given
What are the measures of j3 and j4? Which theorem or postulate justifies
105° angle? each answer?
∠3 and the given angle
are alternate interior ᐍm
angles. ∠4 and the given p 25
angle are same-side
18 3
interior angles. q 4 105Њ

67

Since p } q, m∠3 = 105 by the Alternate Interior Angles Theorem.

Since / } m= ,1m80∠-h4s1+m05110=15g7=5m.1s8e0_b0y3th0e2S_atm00e2-S4id2e.aIinterior Angles Postulate.
So, m∠4

Got It? 3. Use the diagram in Problem 3. What is the measure of each angle?

Justify each answer.
a. ∠1 b. ∠2
c. ∠5 d. ∠6
e. ∠7 f. ∠8

Lesson 3-2  Properties of Parallel Lines 151

You can combine theorems and postulates with your knowledge of algebra to find
angle measures.

Problem 4 Finding an Angle Measure
Algebra  What is the value of y?

What do you know yЊ 80Њ 60
40Њ −
from the diagram?
By the Angle Addition Postulate, y + 40 is the measure of an 00000 0
You have one pair of interior angle. 11111 1
parallel lines. The 80° 22222 2
angle and the angle (y + 40) +hs8m0 =111g80m seSa_m0e3-s0id2e_int0te0rio2r4⦞3.oafi} lines are suppl. 33333 3
formed by the 40° and 44444 4
y° angles are same-side y + 120 = 180 Simplify. 55555 5
interior angles. y = 60 Subtract 120 from each side. 66666 6
77777 7
88888 8
99999 9

Got It? 4. a. In the figure at the right, what are the values 2xЊ 3yЊ
of x and y?
hsm11gmse_0302_t00331.ai
b. What are the measures of the four angles in
(x Ϫ 12)Њ (y ϩ 20)Њ
the figure?

Lesson Check hsm11gmse_0302_t00244.ai

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Use the diagram for Exercises 1–4.
PRACTICES
1. Identify four pairs of congruent
angles. (Exclude vertical angle 5. Compare and Contrast  How are the Alternate
pairs.)
fg Interior Angles Theorem and the Alternate Exterior
2. Identify two pairs of supplementary 12
angles. (Exclude linear pairs.) 34 56 Angles Theorem alike? How are they different?

3. If m∠1 = 70, what is m∠8? 78 6. In Problem 2, you proved that ∠1 and ∠8, in the
4. If m∠4 = 70 and m∠7 = 2x, diagram below, are supplementary. What is a good
name for this pair of angles? Explain.
what is the value of x?
a1

hsm11gmse_0302_t00350.ai b5
87

152 Chapter 3  Parallel and Perpendicular Lines hsm11gmse_0302_t00240.ai

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Identify all the numbered angles that are congruent to the given angle. See Problem 1.
Justify your answers.

7. 8. 9.
31
52 4 1 2
65Њ 2 34 3
51Њ 120Њ
67 1 5
45
67

10. Developing Proof  Supply the missing reasons in the two-column proof. See Problem 2.
hsm11gmse_0302_t00353.ai
Ghivsemn:1  1ag}mb,sce}_d0302_t00351.ai c d

Prove:  ∠1 ≅ ∠3 hasm111gmse_02302_t00352.ai

43
b

Statements Reasons

1) a } b 1) Given
2) ∠3 and ∠2 are supplementary.
hsm 21) 1ag.m?se_0302_t00354.ai
3) c } d
4) ∠1 and ∠2 are supplementary. 3) Given
4) b. ?
5) ∠1 ≅ ∠3 5) c. ?

11. Write a two-column proof for Exercise 10 that does not use ∠2. See Problem 3.
B
Proof
70Њ
Find mj1 and mj2. Justify each answer.
2
12. t 13. ab 14. A C
1ᐍ q 80Њ

2 21 1
75Њ m D
120Њ

Algebra  Find the value of x. Then find the measure of each labeled angle. See Problem 4.

15. hsm11gmse_03x0Њ 2_t003551.a6i. hsm11gm(s3xeϪ_0130)0Њ2_t003561.a7i. hsm11gmse_0302_t00357.ai

(x ϩ 40)Њ

(x Ϫ 50)Њ

5xЊ 4xЊ

hsm11gmse_0302_t00358.ai

Lesson 3-2  Properties of Parallel hLisnmes 11gmse_0302_t0036105.a3i

B Apply Algebra  Find the values of the variables.

18. 19. 20. 42Њ wЊ yЊ 87Њ
xЊ yЊ 20Њ vЊ
xЊ yЊ

(3p Ϫ 6)Њ

3y Њ

21. Think About a Plan  People in ancient Rome played a game called terni lapilhlis.m11gmse_0302_t00363.ai
Thhesmex1a1ctgrmulesseo_f0t3h0is2g_atm0e03ar6e2n.aoit known. Etchings on floors and walls in Rome
suggest that the game required a gridhosfmtw1o1ingtmersseec_t0in3g0p2a_irts0o0f3p6a4ra.alliel lines,

similar to the modern game tick-tack-toe. The measure of one of the angles formed
by the intersecting lines is 90°. Find the measure of each of the other 15 angles.
Justify your answers.
• How can you use a diagram to help?
• You know the measure of one angle. How does the position of that angle relate to

the position of each of the other angles?
• Which angles formed by two parallel lines and a transversal are congruent?

Which angles are supplementary?

22. Error Analysis  Which solution for the value of x in the figure at the right is (x ϩ 75)Њ
incorrect? Explain.

A. B. 2x Њ
2x ‫ ؍‬x ؉ 75 2x ؉ (x ؉ 75) ‫ ؍‬180
x ‫ ؍‬75 3x ؉ 75 ‫ ؍‬180
3x ‫ ؍‬105
x ‫ ؍‬35

23. Outhdsomor1R1egcmresaeti_o0n3  C0a2m_tp0e0rs3o6f7te.aniuse a “bear hsm11gmse_0302_t00366.ai

bag” at night to avoid attracting animals to their food 63˚

supply. In the bear bag system at thehrisgmht1, a1gcammspee_r0302_t00368.ai

pulls one end of the rope to raise and lower the food

bag.
a. Suppose a camper pulls the rope taut between the

two parallel trees, as shown. What is m∠1?
b. Are ∠1 and the given angle alternate interior

angles, same-side interior angles, or corresponding

angles?

24. Writing  Are same-side interior angles ever
congruent? Explain.

1

154 Chapter 3  Parallel and Perpendicular Lines

25. Write a two-column proof to prove the 26. Write a two-column proof.
Proof Corresponding Angles Theorem P roo f Given:  a } b, ∠1 ≅ ∠4
(Theorem 3-2).
Prove:  ∠2 ≅ ∠3
Given:  / } m
Prove:  ∠2 and ∠6 are congruent. a

ഞ 1 2 2 3
3
b1 4
m6

C Challenge Use the diagram at the right for Exercises 27 and 28. 1 2
(2x − 60)Њ
27. Algebra  Suppose the measures of ∠1 and ∠2 are in a
4h∶s1m1 r1a1tigo.mFisned_t0h3e0ir2m_et0as0u3r6es9..(aDi iagram is not to scalhe.s)m11gm(s6e0_−023x)0Њ 2_t00370.ai
28. Error Analysis  The diagram contains contradictory

information. What is it? Why is it contradictory?

PERFORMANCE TASK hsm11gmse_0302_t00371.ai
MATHEMATICAL
Apply What You’ve Learned PRACTICES

MP 6

Look back at the blueprint on page 139 of the plan for a city park. Choose

from the following words, numbers, and expressions to complete the following

sentences.

alternate interior corresponding same-side interior

alternate exterior congruent supplementary

vertical 59 90

95 121 ∠12

∠13 ∠14 ∠25

The measure of ∠22 is a. ? , because alternate interior angles formed by
parallel lines and a transversal are b. ? .

The measure of ∠14 is c. ? , because d. ? angles formed by parallel lines
and a transversal are congruent.

The measure of ∠32 is e. ? , because f. ? angles formed by parallel lines
and a transversal are g. ? .

The measure of h. ? is 17, because i. ? angles formed by parallel lines and
a transversal are congruent.

Lesson 3-2  Properties of Parallel Lines 155

3-3 Proving Lines Parallel CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Extends MG-ACFOS..C9.192  .PGro-CveO.t3h.e9o rePmrosveabthoeuotrleinmess aabnodut
alinegsleasn. Tdhaenogrelems.sTihnecoluredme:s. i.n.cwluhden: .a. .trwanhsevnearstarlacnrsovsesressal
cpraorsaslelesl plianreasl,lealltlienrensa,taeltienrtnearitoerianntegrlieosr arnegcleosnagrreuent and
conrrgersupeonntdaingd aconrgrleesspaornedcinogngarnugelnest .a.re. congruent . . .

MP 1, MP 3, MP 7

Objective To determine whether two lines are parallel

How can you use The maze below has two intersecting sets of parallel paths. A mouse makes
theorems you five turns in the maze to get to a piece of cheese. Follow the mouse’s path
already know to through the maze. What are the number of degrees at each turn? Explain
solve this maze how you know.
problem?
1
MATHEMATICAL 2 60Њ
3
PRACTICES

4
5

Lesson

Vocabulary
• flow proof

In the Solve It, you used parallel lines to find congruent and supplementary
relationships of special angle pairs. In this lesson, you will do the converse. You will
use the congruent and supplementary relationships of the special angle pairs to prove
lines parallel.

Essential Understanding  You can use certain angle pairs to decide whether two
lines are parallel.

Theorem 3-4  Converse of the Corresponding Angles Theorem

Theorem If . . . Then . . .
If two lines and a transversal ∠2 ≅ ∠6 /}m
form corresponding angles
that are congruent, then the ᐍ2
lines are parallel.
m6

156 Chapter 3  Parallel and Perpendicular Lines You will prove Theorem 3-4 in Lesson 5-5.

hsm11gmse_0303_t00245.ai

Problem 1 Identifying Parallel Lines

Which line is the Which lines are parallel if j1 @ j2? Justify your answer. ᐍ a b
transversal for j1 ∠1 and ∠2 are corresponding angles. If ∠1 ≅ ∠2, then a } b 3
and j2?
by the Converse of the Corresponding Angles Theorem. 45 6
Line m is the transversal
m1 2
because it forms one side
87
o f both angles. Got It? 1. Which lines are parallel if ∠6 ≅ ∠7? Justify

your answer.

You can use the Converse of the Corresponding Angles Theorem to phrsomve1th1egfmolsloew_i0n3g03_t00246.ai

converses of the theorems and postulate you learned in Lesson 3-2.

Theorem 3-5  Converse of the Alternate Interior Angles Theorem

Theorem If . . . Then . . .
If two lines and a ∠4 ≅ ∠6 /}m
transversal form alternate
interior angles that are
congruent, then the two ᐍ
lines are parallel.
4

m6

Theorem 3-6  Converse of the Same-Side Interior Angles Postulate

Theorem Imf ∠. h.3s. m+ m11∠g6m=s1e8_00303_t00247.T/ah}iemn . . .
If two lines and a
transversal form same-
side interior angles that ᐍ
are supplementary, then 3
the two lines are parallel. m6

Theorem 3-7  Converse of the Alternate Exterior Angles Theorem

Theorem I∠f 1. h.≅s. m∠171gmse_0303_t00248T./ah}iemn . . .
If two lines and a
transversal form alternate ᐍ 1
exterior angles that are
congruent, then the two m
lines are parallel. 7

hsm11gmse_0303_t00249.ai 157

Lesson 3-3  Proving Lines Parallel

The proof of the Converse of the Alternate Interior Angles Theorem below looks
different than any proof you have seen so far in this course. You know two forms of
proof—paragraph and two-column. In a third form, called flow proof, arrows show the
logical connections between the statements. Reasons are written below the statements.

Proof Proof of Theorem 3-5: Converse of the Alternate Interior Angles Theorem

Given:  ∠4 ≅ ∠6 ᐍ2
Prove:  / } m 4

m6

∠4 ≅ ∠6 ∠2 ≅ ∠6 ᐉʈm
Given
Transitive hsm11Itfhgcemonrrtsehesep_l.in0⦞e3sa0are3re≅_ʈt.,00250.ai
∠2 ≅ ∠4 Property of ≅
Vertical ⦞ are ≅.

Proof Problem 2 Writing a Flow Proof of Theorem 3-7
Ghivsemn:1  1∠g1m≅se∠_70303_t00251
ᐍ 1
Prove:  / } m 3

m 56

7

• ∠1 ≅ ∠7 One pair of Use a pair of congruent
From the diagram you know
• ∠1 and ∠3 are vertical corresponding angles vertical angles to relate
• ∠5 and ∠7 are vertical conhgrsument1t1ogprmovsee_0303_eitt0he0r2∠512or ∠7 to its
• ∠1 and ∠5 are corresponding /}m corresponding angle.
• ∠3 and ∠7 are corresponding

∠1 ≅ ∠7 ∠3 ≅ ∠7 ᐉʈm
Given Transitive
Property of ≅ If corresp. ⦞ are ≅,
∠3 ≅ ∠1 then the lines are ʈ.
Vertical ⦞ are ≅.

Got It? 2. Use the same diagram from Problem 2 to Prove Theorem 3-6.
Given:  m∠3 + m∠6 = 180

hsm11gmPsroev_e0:3  /03} _mt00253

158 Chapter 3  Parallel and Perpendicular Lines

The four theorems you have just learned provide you with four ways to determine if two
lines are parallel.

How do j1 and j2 Problem 3 Determining Whether Lines are Parallel rs
relate to each other
in the diagram? The fence gate at the right is made up of pieces of 3
∠1 and ∠2 are both wood arranged in various directions. Suppose 2
exterior angles and they j1 @ j2. Are lines r and s parallel?
Explain. 1
lie on opposite sides of
Yes, r } s. ∠1 and ∠2 are alternate exterior
the transversal. angles. If two lines and a transversal form
congruent alternate exterior angles, then the
lines are parallel (Converse of the Alternate
Exterior Angles Theorem).

Got It? 3. In Problem 3, what is another way to
explain why r } s? Justify your answer.

You can use algebra along with the postulates and theorems from Lesson 3-2 and
Lesson 3-3 to help you solve problems involving parallel lines.

Work backward. Problem 4 Using Algebra a
Think about what must (2x ϩ 9)Њ
be true of the given Algebra  What is the value of x for which a } b?
angles for a and b to be b 111Њ
parallel. The two angles are same-side interior angles. By the Converse
of the Same-Side Interior Angles Postulate, a } b if the angles hsm11gmse_0303_t00254.ai
are supplementary.

(2x + 9) + 111 = 180 Def. of supplementary angles
2x + 120 = 180 Simplify.
2x = 60 Subtract 120 from each side.
x = 30 Divide each side by 2.

Got It? 4. What is the value of w for which c } d?

55Њ c

(3w Ϫ 2)Њ d

hsm11gmse_0303_t00255 159

Lesson 3-3  Proving Lines Parallel

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

State the theorem or postulate that proves a } b. 4. Explain how you know when to use the Alternate

1. a Interior Angles Theorem and when to use the

Converse of the Alternate Interior Angles Theorem.

b 5. Compare and Contrast  How are flow proofs and
two‑column proofs alike? How are they different?

2. a 6. EsaryrosrthAanta<AlyBs> i}s <DACc> lbaasssemdaoten the A B
diagram at the right. Explain your
yЊ classmate’s error. 83Њ 97Њ

hsm11gm65sЊe_0303_t00b372.ai D C

3. What is the value of y for which a } b in Exercise 2?

hsm11gmse_0303_t00373.ai MATHEMATICAL hsm11gmse_0303_t00374.a
Practice and Problem-Solving Exercises
PRACTICES See Problem 1.
Q
A Practice Which lines or segments are parallel? Justify your answer.

7. 8. P
BC

EG S T
9. C H 10. J
KL M

hsm11gmse_0303_t00375.ai hsm11gmse_0303_t00376.ai

45Њ 45Њ Q R ST

MA R

11. Developing Proof  Complete the flow proof below. See Problem 2.

Given:  ∠1 and ∠3 are supplementary. 12 a

Phrsomve1: 1ag}mbse_0303_t00377.ai hsm11gm3se_0303b_t10398.ai

s∠u1ppalnedm∠en3taarrey. d. aʈb
a. e.
Supplements of the
same ∠ are ≅. hsm11gmse_0303_t00378.ai

b. ∠1 and ∠2 are
Def. of linear pair supplementary.

c.

160 Chapter 3  Parallel and Perpendicular Lines

hsm11gmse_0303_t00379.ai