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Transcript
```Reasoning and Proof
Chapter Overview and Pacing
Year-long pacing: pages T20–T21.
PACING (days)
Regular
Block
LESSON OBJECTIVES
Basic/
Average
Basic/
Average
Inductive Reasoning and Conjecture (pp. 62–66)
• Make conjectures based on inductive reasoning.
• Find counterexamples.
1
1
0.5
0.5
Logic (pp. 67–74)
• Determine truth values of conjunctions and disjunctions.
• Construct truth tables.
2
1
1
0.5
Conditional Statements (pp. 75–80)
• Analyze statements in if-then form.
• Write the converse, inverse, and contrapositive of if-then statements.
2
1
1
0.5
(with 2-3
(with 2-3
(with 2-3 (with 2-3
Follow-Up) Follow-Up) Follow-Up) Follow-Up)
Deductive Reasoning (pp. 82–88)
• Use the Law of Detachment.
• Use the Law of Syllogism.
Follow-Up: Use a table and deductive reasoning to solve a logic problem.
1
1
0.5
0.5
(with 2-4
(with 2-4
(with 2-3 (with 2-3
Follow-Up) Follow-Up) Follow-Up) Follow-Up)
Postulates and Paragraph Proofs (pp. 89–93)
• Identify and use basic postulates about points, lines, and planes.
• Write paragraph proofs.
1.5
1
1
0.5
Algebraic Proof (pp. 94–100)
• Use algebra to write two-column proofs.
• Use properties of equality in geometry proofs.
1.5
1
1
0.5
Proving Segment Relationships (pp. 101–106)
• Write proofs involving segment addition.
• Write proofs involving segment congruence.
2
2
1
1
Proving Angle Relationships (pp. 107–114)
• Write proofs involving supplementary and complementary angles.
• Write proofs involving congruent and right angles.
2
2
1
1
Study Guide and Practice Test (pp. 115–121)
Standardized Test Practice (pp. 122–123)
1
1
0.5
0.5
Chapter Assessment
1
1
0.5
0.5
15
12
8
6
TOTAL
An electronic version of this chapter is available on StudentWorksTM. This backpack solution CD-ROM
allows students instant access to the Student Edition, lesson worksheet pages, and web resources.
60A
Chapter 2 Reasoning and Proof
Timesaving Tools
™
All-In-One Planner
and Resource Center
Chapter Resource Manager
See pages T5 and T21.
57–58
59–60
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62
63–64
65–66
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101–102
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104
119
119, 121
120
GCC 19
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GCC 20
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2-5
2-6
2-6
2-7
2-7
2-8
2-8
SC 3
41–44,
83–86,
93–94
89–90,
101–104
120
Ap
plic
atio
ns*
5-M
Tra inute
nsp
C
are heck
nci
es
Int
e
Cha racti
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)
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Pre
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Wo isite
rkb Ski
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Enr
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and tudy
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(Sk Pra
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and tice
Ave
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Rea
di
Ma ng to
the
ma Learn
tics
CHAPTER 2 RESOURCE MASTERS
SC 4
81–82,
85–86
Materials
protractor
6
patty paper, protractor, paper
105–118,
122–124
*Key to Abbreviations: GCC Graphing Calculator and Computer Masters
SC School-to-Career Masters
Chapter 2 Reasoning and Proof
60B
Mathematical Connections
and Background
Continuity of Instruction
Prior
Prior Knowledge
Knowledge
In algebra, students learned to solve for a
variable. In Chapter 1, students learned
about points, lines and planes. They were
introduced to adjacent and vertical angles,
complementary and supplementary angles,
and right angles. They can identify
congruent segments and angles as
well as perpendicular
lines.
This Chapter
In this chapter, students explore methods of
reasoning and learn to apply those methods to
geometry. They make conjectures, determine
the truth values of compound statements,
and construct truth tables. They also analyze
conditional statements and write related
conditionals. The terms postulate and
theorem are introduced. Algebraic properties
of equality are applied to geometry, enabling
students to write formal and informal proofs
proving segment and angle relationships.
Inductive Reasoning and
Conjecture
A conjecture is an educated guess based on known
information. Examining several specific situations to
arrive at a conjecture is called inductive reasoning.
Meteorologists use inductive reasoning to predict
weather conditions.
Just because a conjecture is true in most circumstances does not make it a true conjecture. If just one
example contradicts the conjecture, the conjecture is not
true. The false example is called a counterexample.
Logic
A statement is any sentence that is either true or
false, but not both. The truth or falsity of a statement is
called its truth value. The negation of a statement has
the opposite meaning as well as an opposite truth value.
This means that if a statement is represented by p, then
not p is the negation of the statement. You could also
write not p as p.
Two or more statements can be joined to form a
compound statement. A conjunction is a compound
statement formed by joining two or more statements
with the word and. The symbol can be used instead of
and. Two or more statements can also be joined to form
a disjunction. A disjunction is a compound statement
formed by joining two or more statements with the
word or. You might see the symbol instead of the
word or in a disjunction.
Conjunctions and disjunctions can be illustrated
with Venn diagrams. Truth tables can also be helpful in
evaluating the truth values of statements. A truth table
will show that a conjunction is true only when both
statements are true. A disjunction, on the other hand, is
true unless both statements are false.
Conditional Statements
Future Connections
In Chapter 4, students will build on their
knowledge of proofs when they investigate
triangles. Students will also write proofs in
Chapter 7. Thinking logically is a crucial skill for
daily living. Logic is used to make informed
choices and to examine a statement for truth.
60C
Chapter 2 Reasoning and Proof
A conditional statement is a statement that can be
written in if-then form: if p, then q. The phrase immediately following the word if is called the hypothesis. The
phrase immediately following the word then is called
the conclusion. An arrow pointing to the right is written
between p and q to symbolize an if-then situation. A
conditional statement is true in all cases except where
the hypothesis is true and the conclusion is false.
Related conditionals are statements constructed
from an if-then statement. A converse statement is
formed by exchanging the hypothesis and the conclusion:
if q, then p. An inverse statement is formed by negating
both the hypothesis and the conclusion of the original
statement: if p, then q. A contrapositive is formed
by negating both the hypothesis and the conclusion
of the converse statement: if q, then p.
Deductive Reasoning
Deductive reasoning uses facts, rules, definitions, or properties to reach logical conclusions. A
form of deductive reasoning that is used to draw conclusions from true conditional statements is called the
Law of Detachment. This law states that if p → q is
true and p is true, then q is also true.
The Law of Syllogism is another law of logic.
It states that if p → q and q → r are true, then p → r is
also true. You may see the similarity between this law
and the Transitive Property of Equality from algebra.
Postulates and
Paragraph Proofs
In geometry, a postulate is a statement that
describes a fundamental relationship between the
basic terms of geometry. Postulates are accepted as
true without proof. Several postulates based on the
relationship among points, lines, and planes were
introduced in Chapter 1, but were not labeled as
postulates.
In this course, you will learn to use various
methods to justify the truth of a statement or conjecture. Once a statement or conjecture has been shown
to be true, it is called a theorem. A theorem can be
used like a definition or postulate to justify that other
statements are true.
A proof is a logical argument in which each
statement you make is supported by a statement that
is accepted as true. One type of proof is called a paragraph or informal proof. It is a written explanation of
why a conjecture for a given situation is true. A good
proof states the theorem or conjecture to be proven. It
lists the given information and, if possible, supplies a
diagram to illustrate the given information. The proof
states what is to be proved and develops a system of
deductive reasoning.
Algebraic Proof
In algebra, you learned to use properties of
equality to solve algebraic equations and to verify
relationships. These properties can be used to justify
each step when solving an equation. A group of algebraic steps used to solve problems form a deductive
argument. This argument can be demonstrated by
writing the solution to the equation in the first column
and listing the property justifying each step in the
second column.
In geometry, a similar format is used to prove
conjectures and theorems. A two-column, or formal,
proof contains statements and reasons organized in
two columns. Each step is called a statement, and the
properties that justify each step are called reasons.
Proving Segment
Relationships
As you learned in Chapter 1, a segment can be
measured, and measures can be used in calculations
because they are real numbers. One postulate about
segments is called the Ruler Postulate. It states that the
points on any line or line segment can be paired with
real numbers so that, given any two points A and B
on a line, A corresponds to 0, and B corresponds to a
positive real number. That number is the length of
the segment. Another postulate states that if point B
lies between points A and C on the same line,
AB BC AC. The converse statement holds true
as well.
The Reflexive, Symmetric, and Transitive
Properties of Equality can be used to write proofs
about segment congruence. The theorem resulting
from the proofs states that congruence of segments is
reflexive, symmetric, and transitive.
Proving Angle Relationships
This lesson introduces postulates and theorems
about angle relationships. The Protractor Postulate
and a number r between 0 and 180,
states, “Given AB
there is exactly one ray with endpoint A, extending
, such that the measure of the
on either side of AB
angle formed is r.” The Angle Addition Postulate
states that if R is in the interior of PQS, then
mPQR mRQS mPQS. If mPQR mRQS mPQS, then R is in the interior of PQS. This
postulate can be used with other angle relationships
to prove other theorems relating to angles.
Some of these theorems relate to supplementary
and complementary angles. Another theorem extends
the Reflexive, Transitive, and Symmetric properties to
angle congruence. There is also a series of theorems
about perpendicular lines and right angles.
Chapter 2 Reasoning and Proof
60D
and Assessment
Key to Abbreviations:
TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters
ASSESSMENT
INTERVENTION
Type
Student Edition
Teacher Resources
Ongoing
Prerequisite Skills, pp. 61, 80,
87, 93, 100, 106
Practice Quiz 1, p. 80
Practice Quiz 2, p. 100
5-Minute Check Transparencies
Prerequisite Skills Workbook, pp. 41–44, 81–86,
89–90, 93–94, 101–104
Quizzes, CRM pp. 119–120
Mid-Chapter Test, CRM p. 121
Study Guide and Intervention, CRM pp. 57–58, 63–64,
69–70, 75–76, 81–82, 87–88, 93–94, 99–100
Mixed
Review
pp. 66, 74, 80, 93, 100, 106, 114
Cumulative Review, CRM p. 122
Error
Analysis
Find the Error, pp. 84, 111
Common Misconceptions, p. 76
Find the Error, TWE pp. 84, 111
Unlocking Misconceptions, TWE p. 91
Tips for New Teachers, TWE p. 70
Standardized
Test Practice
pp. 66, 74, 80, 86, 87, 93, 96,
97, 99, 106, 114, 121, 122
TWE pp. 122–123
Standardized Test Practice, CRM pp. 123–124
Open-Ended
Assessment
Writing in Math, pp. 66, 74, 79,
86, 93, 99, 106, 114, 123
Open Ended, pp. 63, 71, 78, 84,
91, 97, 103, 111
Standardized Test, p. 123
Modeling: TWE pp. 74, 87, 106
Speaking: TWE pp. 80, 93
Writing: TWE pp. 66, 100, 114
Open-Ended Assessment, CRM p. 117
Chapter
Assessment
Study Guide, pp. 115–120
Practice Test, p. 121
Multiple-Choice Tests (Forms 1, 2A, 2B),
CRM pp. 105–110
Free-Response Tests (Forms 2C, 2D, 3),
CRM pp. 111–116
Vocabulary Test/Review, CRM p. 118
Yearly ProgressPro, see p. 2.
Geometry Lesson
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
Yearly ProgressPro Skill Lesson
Inductive Reasoning and Conjecture
Logic
Conditional Statements
Deductive Reasoning
Postulates and Paragraph Proofs
Algebraic Proof
Proving Segment Relationships
Proving Angle Relationships
Chapter 2 Reasoning and Proof
GeomPASS: Tutorial Plus,
Lesson 6
www.geometryonline.com/
self_check_quiz
www.geometryonline.com/
extra_examples
Standardized Test Practice
CD-ROM
www.geometryonline.com/
standardized_test
ExamView® Pro (see below)
MindJogger Videoquizzes
www.geometryonline.com/
vocabulary_review
www.geometryonline.com/
chapter_test
ExamView® Pro
Use the networkable ExamView® Pro to:
• Create multiple versions of tests.
• Create modified tests for Inclusion students.
• Use built-in state curriculum correlations to create
tests aligned with state standards.
• Apply art to your test from a program bank of artwork.
60E
Technology/Internet
Glencoe Geometry provides numerous opportunities to incorporate reading and writing
into the mathematics classroom.
Student Edition
• Foldables Study Organizer, p. 61
• Concept Check questions require students to verbalize
and write about what they have learned in the lesson.
(pp. 63, 71, 78, 84, 91, 97, 103, 111)
• Writing in Math questions in every lesson, pp. 66, 74,
79, 86, 93, 99, 106, 114
• Reading Study Tip, p. 75
• WebQuest, p. 65
Teacher Wraparound Edition
• Foldables Study Organizer, pp. 61, 115
• Study Notebook suggestions, pp. 64, 72, 78, 81, 84, 88,
91, 97, 104, 111
• Modeling activities, pp. 74, 87, 106
• Speaking activities, pp. 80, 93
• Writing activities, pp. 66, 100, 114
• Differentiated Instruction (Verbal/Linguistic), p. 83
• ELL Resources, pp. 60, 65, 73, 79, 81, 83, 86, 92, 99,
105, 113, 115
• Vocabulary Builder worksheets require students to
define and give examples for key vocabulary terms as
they progress through the chapter. (Chapter 2 Resource
Masters, pp. vii-viii)
• Proof Builder helps students learn and understand
theorems and postulates from the chapter. (Chapter 2
Resource Masters, pp. ix–x)
• Reading to Learn Mathematics master for each lesson
(Chapter 2 Resource Masters, pp. 61, 67, 73, 79, 85, 91,
97, 103)
• Vocabulary PuzzleMaker software creates crossword,
jumble, and word search puzzles using vocabulary lists
that you can customize.
• Teaching Mathematics with Foldables provides
suggestions for promoting cognition and language.
• Reading Strategies for the Mathematics Classroom
• WebQuest and Project Resources
Many of the vocabulary terms introduced in
Chapter 2 can be represented by symbols. Threecolumn notes can be a helpful way for students
to organize new vocabulary terms. To reinforce
understanding, students can write an explanation
of each term in their own words and provide the
appropriate symbol. The table at the right shows
notes for Lesson 2-2. Students can add on to this
sample with other terms from Chapter 2.
Term
Explanation
Symbol
negation
the opposite of the given
statement
conjunction
a compound statement formed
with the word “and”
pq
disjunction
a compound statement formed
with the word “or”
pq
Chapter 2 Reasoning and Proof
60F
Reasoning
and Proof
Notes
Have students read over the list
of objectives and make a list of
any words with which they are
not familiar.
• Lessons 2-1 through 2-3 Make conjectures,
determine whether a statement is true or false,
and find counterexamples for statements.
• Lesson 2-4 Use deductive reasoning to reach
valid conclusions.
• Lessons 2-5 and 2-6 Verify algebraic and
geometric conjectures using informal and
formal proof.
• Lessons 2-7 and 2-8 Write proofs involving
segment and angle theorems.
Point out to students that this is
only one of many reasons why
each objective is important.
Others are provided in the
introduction to each lesson.
Key Vocabulary
•
•
•
•
•
inductive reasoning (p. 62)
deductive reasoning (p. 82)
postulate (p. 89)
theorem (p. 90)
proof (p. 90)
Logic and reasoning are used throughout geometry to solve
problems and reach conclusions. There are many professions
that rely on reasoning in a variety of situations. Doctors,
for example, use reasoning to diagnose and treat patients.
You will investigate how doctors use reasoning in Lesson 2-4.
Lesson
2-1
2-2
2-3
2-4
2-4
Follow-Up
2-5
2-6
2-7
2-8
NCTM
Standards
Local
Objectives
1, 6, 7, 8, 9, 10
6, 7, 8, 9, 10
3, 6, 7, 8, 9, 10
3, 6, 7, 8, 9, 10
6, 7
3, 6, 7, 8, 9, 10
2, 3, 6, 7, 8, 9,
10
3, 6, 7, 8, 9, 10
3, 6, 7, 8, 9, 10
Key to NCTM Standards:
1=Number & Operations, 2=Algebra,
3=Geometry, 4=Measurement,
5=Data Analysis & Probability, 6=Problem
Solving, 7=Reasoning & Proof,
8=Communication, 9=Connections,
10=Representation
60
Chapter 2 Reasoning and Proof
60
Chapter 2
Reasoning and Proof
B. Busco/Getty Images
Vocabulary Builder
ELL
The Key Vocabulary list introduces students to some of the main vocabulary terms
included in this chapter. For a more thorough vocabulary list with pronunciations of
new words, give students the Vocabulary Builder worksheets found on pages vii and
viii of the Chapter 2 Resource Masters. Encourage them to complete the definition
of each term as they progress through the chapter. You may suggest that they add
these sheets to their study notebooks for future reference when studying for the
Chapter 2 test.
Prerequisite Skills To be successful in this chapter, you’ll need to master
these skills and be able to apply them in problem-solving situations. Review
these skills before beginning Chapter 2.
For Lesson 2-1
This section provides a review of
the basic concepts needed before
beginning Chapter 2. Page
references are included for
the Prerequisite Skills Workbook,
pages 41–44, 81–86, 89–90, 93–94,
101–104.
Evaluate Expressions
Evaluate each expression for the given value of n. (For review, see page 736.)
1. 3n 2; n 4 10
2. (n 1) n; n 6 13
3. n2 3n; n 3 0
4. 180(n 2); n 5 540
5. n; n 10 50
n(n 3)
6. ; n 8 20
n
2
2
For Lessons 2-6 through 2-8
Solve Equations
Prerequisite Skills in the Getting
Ready for the Next Lesson section
at the end of each exercise set
review a skill needed in the next
lesson.
Solve each equation. (For review, see pages 737 and 738.)
7. 6x 42 4x 21
10. 12 7x x 18 ⫺5
8. 8 3n 2 2n 2
18
1
11. 3x 4 x 5 ⫺
2
5
For Lesson 2-8
9. 3(y 2) 12 y ⫺9
3
2
12. 2 2x x 2 3
2
For Exercises 13–14, refer to the figure at the right. (For review, see Lesson 1-5.)
13. If mAGB 4x 7 and mEGD 71, find x. 16
A
14. If mBGC 45, mCGD 8x 4, and mDGE 15x 7,
find x. 6
F
B
C
G
E
Prerequisite
Skill
2-3
Evaluating Algebraic
Expressions, p. 74
Solving Equations, p. 80
Information from Figures, p. 87
Solving Equations, p. 93
Segment Measures, p. 100
Complementary and
Supplementary Angles, p. 106
2-4
2-5
2-6
2-7
2-8
D
with eight sheets of 8 12 ” by 11” grid paper.
Staple
For
Lesson
Cut Tabs
Cut the bottom of each
sheet to form a tabbed
book.
Stack and staple the eight
sheets together to form a
booklet.
Label
Reasoning & Proof
2-1
Label each of the tabs with
chapter title to the first tab.
2-2
2-3
2-4
2-5
2-6
2-7
2-8
Reading and Writing As you read and study each lesson, use the corresponding page to write proofs and
record examples of when you used logical reasoning in your daily life.
Chapter 2 Reasoning and Proof
61
TM
Teaching Mathematics
with Foldables.
Organization of Data Use this Foldable for student writing about
reasoning and proofs. After students make their Foldable, have them
label the tabs to correspond to the eight lessons in this chapter.
Students use their Foldable to take notes, define terms, record
concepts, write statements in if-then form, and write paragraph
proofs. On the back of the Foldable, have students record examples
of ways in which they use reasoning and proofs in their daily lives.
Note how columnists and authors present their reasoning and
ways in which they try to prove or disprove their points of view.
Chapter 2 Reasoning and Proof 61
Inductive Reasoning
and Conjecture
Lesson
Notes
• Make conjectures based on inductive reasoning.
1 Focus
5-Minute Check
Transparency 2-1 Use as a
quiz or review of Chapter 1.
• Find counterexamples.
• conjecture
• inductive reasoning
• counterexample
Mathematical Background notes
are available for this lesson on
p. 60C.
can inductive reasoning
help predict weather
conditions?
• What are normal temperatures
for the month of January?
in January are usually in the 30s or
40s. (Answers will vary in different
parts of the country.)
• How do people benefit from
the inductive reasoning
techniques of meteorologists?
for outdoor events a few days in
appropriately for daily weather
conditions and carry weatherrelated items, such as umbrellas,
sunglasses, and so on.
can inductive reasoning help
predict weather conditions?
Vocabulary
Meteorologists use science and weather
patterns to make predictions about future
weather conditions. They are able to make
accurate educated guesses based on past
weather patterns.
MAKE CONJECTURES A conjecture is an educated guess based on known
information. Examining several specific situations to arrive at a conjecture is called
inductive reasoning. Inductive reasoning is reasoning that uses a number of specific
examples to arrive at a plausible generalization or prediction.
Example 1 Patterns and Conjecture
The numbers represented below are called triangular numbers. Make a
conjecture about the next triangular number based on the pattern.
1
3
10
15
Observe:
Each triangle is formed by adding another row of dots.
Find a Pattern:
1
3
2
6
3
10
4
15
5
The numbers increase by 2, 3, 4, and 5.
Study Tip
Conjectures
6
Conjecture:
identify patterns before
you make a conjecture.
The next number will increase by 6.
So, it will be 15 6 or 21.
In Chapter 1, you learned some basic geometric concepts. These concepts can be
used to make conjectures in geometry.
62
Chapter 2 Reasoning and Proof
Bob Daemmrich/Stock Boston
Resource Manager
Workbook and Reproducible Masters
Chapter 2 Resource Masters
• Study Guide and Intervention, pp. 57–58
• Skills Practice, p. 59
• Practice, p. 60
• Reading to Learn Mathematics, p. 61
• Enrichment, p. 62
Graphing Calculator and
Computer Masters, p. 19
Transparencies
5-Minute Check Transparency 2-1
Technology
Interactive Chalkboard
Example 2 Geometric Conjecture
2 Teach
For points P, Q, and R, PQ ⴝ 9, QR ⴝ 15, and PR ⴝ 12. Make a conjecture and
draw a figure to illustrate your conjecture.
Given:
points P, Q, and R; PQ 9, QR 15, and PR 12
MAKE CONJECTURES
Examine the measures of the segments.
Since PQ PR QR, the points cannot
be collinear.
Conjecture: P, Q, and R are noncollinear.
In-Class Examples
Teaching Tip
Tell students to
test all fundamental operations,
including powers and roots,
when they are looking for
patterns in a series of numbers.
two operations can be used.
Q
15
9
P
Power
Point®
12
R
1 Make a conjecture about the
FIND COUNTEREXAMPLES
A conjecture based on several observations may
be true in most circumstances, but false in others. It takes only one false example to
show that a conjecture is not true. The false example is called a counterexample .
next number based on the
pattern. 2, 4, 12, 48, 240 1440
2 For points L, M, and N,
LM 20, MN 6, and
LN 14. Make a conjecture
and draw a figure to
Example 3 Find a Counterexample
FINANCE Find a counterexample
for the following statement based
on the graph.
The rates for CDs are at least 1.5%
less than the rates a year ago.
Examine the graph. The statement
is true for 6-month, 1-year, and
1
2-year CDs. However, the
2
difference in the rate for a 5-year
CD is 0.74% less, which is less than
1.5%. The statement is false for a
5-year certificate of deposit. Thus,
the change in the 5-year rate is a
counterexample to the original
statement.
Log on for:
• Updated data
• More on finding
counterexamples
www.geometryonline.com/
usa_today
USA TODAY Snapshots®
Latest CD rates
L
Average certificate of deposit rates as of Wednesday:
This week
6-month Last week
Year ago
1-year
This week
Last week
Year ago
2 1⁄2-year
This week
Last week
Year ago
5-year
This week
Last week
Year ago
N
14
1.80%
1.80%
20
Conjecture: L, M, and N are
collinear.
4.55%
2.12%
2.11%
M
6
4.64%
2.96%
2.96%
FIND COUNTEREXAMPLES
4.74%
4.22%
4.23%
4.96%
In-Class Example
Power
Point®
3 UNEMPLOYMENT Based on
Source: Bank Rate Monitor, 800-327-7717, www.bankrate.com
Concept Check
1– 3. See p. 123A.
USA TODAY
1. Write an example of a conjecture you have made outside of school.
2. Determine whether the following conjecture is always, sometimes, or never true
based on the given information.
Given: collinear points D, E, and F
Conjecture: DE EF DF
3. OPEN ENDED
Interactive
Chalkboard
PowerPoint®
Presentations
County
Write a statement. Then find a counterexample for the statement.
www.geometryonline.com/extra_examples
Lesson 2-1 Inductive Reasoning and Conjecture 63
This CD-ROM is a customizable Microsoft® PowerPoint®
presentation that includes:
• Step-by-step, dynamic solutions of each In-Class Example
from the Teacher Wraparound Edition
• Additional, Try These exercises for each example
• The 5-Minute Check Transparencies
• Hot links to Glencoe Online Study Tools
the table showing
unemployment rates for
various cities in Kansas, find
a counterexample for the
following statement.
The unemployment rate is highest
in the cities with the most people.
Shawnee
Jefferson
Jackson
Douglas
Osage
Wabaunsee
Pottawatomie
Civilian
Rate
Labor Force
90,254
9,937
8,915
55,730
10,182
3,575
11,025
3.1%
3.0%
2.8%
3.2%
4.0%
3.0%
2.1%
Source: Labor Market Information Services—Kansas
Department of Human Resources
Osage has only 10,182 people on
its civilian labor force, and it has
a higher rate of unemployment
than Shawnee, which has 90,254
people on its civilian labor force.
Lesson 2-1 Inductive Reasoning and Conjecture 63
3 Practice/Apply
Study Notebook
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4–5
6–7
8–9
1
2
3
Have students—
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 2.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
Organization by Objective
• Make Conjectures: 11–28
• Find Counterexamples:
29–36
Odd/Even Assignments
Exercises 11–36 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
5. 8, 5, 2, 1, 4 7
Make a conjecture based on the given information. Draw a figure to illustrate
CD intersect at P.
6. PQ RS and RS TU
7. AB and Determine whether each conjecture is true or false. Give a counterexample for any
false conjecture.
8. Given: x is an integer.
Conjecture: x is negative. False; if x ⴝ 2, then x ⴝ (2) or 2.
9. Given: WXYZ is a rectangle.
Conjecture: WX YZ and WZ XY true
Application
Make a conjecture about the next item in each sequence.
4.
10. HOUSES Most homes in the northern United States have roofs made with
steep angles. In the warmer areas of the southern states, homes often have flat
roofs. Make a conjecture about why the roofs are different. Sample answer:
Snow will not stick on a roof with a steep angle.
★ indicates increased difficulty
Practice and Apply
For
Exercises
See
Examples
11–20
21–28
29–36
1
2
3
Make a conjecture about the next item in each sequence.
11.
12.
Extra Practice
See page 756.
13. 1, 2, 4, 8, 16 32
1 1 1 1 1
16. 1, , , , 2 4 8 16 32
14. 4, 6, 9, 13, 18 24
17. 2, 6, 18, 54 162
Assignment Guide
Basic: 11–41 odd, 43–67
Average: 11–41 odd, 43–67
(optional: 65–67)
6. PQ TU
P
Q
R
S
T
U
11
1
5 7
15. , 1, , , 3 3
3 3
3
18. 5, 25, 125, 625
3125
Make a conjecture about the number of blocks in the next item of each sequence.
19.
20.
21. Lines ᐉ and m
form four right angles.
22. A, B, and C are
collinear.
23. ⬔3 and ⬔4 are
supplementary.
24. ⬔ABD ⬵ ⬔DBC.
25. ∆PQR is a
scalene triangle.
26. HI IJ JK KH
19– 20. See p. 123A.
Make a conjecture based on the given information. Draw a figure to illustrate
your conjecture. 21 –28. See p. 123A for figures.
21. Lines ᐉ and m are perpendicular.
22. A(2, 11), B(2, 1), C(5, 10)
23. 3 and 4 are a linear pair.
is an angle bisector of ABC.
24. BD
25. P(1, 7), Q(6, 2), R(6, 5)
26. HIJK is a square.
27. PQRS is a rectangle.
28. B is a right angle in ABC.
PQ SR, QR PS
64 Chapter 2 Reasoning and Proof
(AB ) 2 (BC ) 2 (AC)2
7. A, B, C, and D are noncollinear.
D
A
P
C
64
B
Chapter 2 Reasoning and Proof
Differentiated Instruction
Naturalist Students can practice brainstorming conjectures and finding
counterexamples in nature. For example have students consider the
statement, “If plants don’t receive water daily, they will not survive.” A
counterexample would be a cactus, which can go weeks without water.
Nature topics could include plants, animals, predator/food supply
relationships, insects, weather, and so on.
You can use scatter
plots to make
relationships between
latitude, longitude,
degree distance, and
the monthly high
temperature.
Visit www.geometry
online.com/WebQuest
to continue work on
counterexample.
p. 57
(shown)
p. 58
Inductive
Reasoningand
and Conjecture
Exercises
1. 5, 10, 20, 40 Pattern: Each number is 2 times the previous number.
Conjecture: The next number is 80.
2. 1, 10, 100, 1000 Pattern: Each number is 10 times the previous number.
Conjecture: The next number is 10,000.
33. Given: n is a real number.
Conjecture: n2 is a nonnegative number. true
6 7 8
5 5 5
1
5
3. 1, , , Pattern: Each number is more than the previous number.
9
5
Conjecture: The next number is .
Make a conjecture based on the given information. Draw a figure to illustrate
4. A(1, 1), B(2, 2), C(4, 4)
5. 1 and 2 form a right angle.
1 and 2 are complementary.
Points A, B, and C are collinear.
y
collinear.
35. Given: JK KL LM MJ
Conjecture: JKLM forms a square. False; JKLM may not have a right angle.
P
C (4, 4)
B(2, 2)
O
A(–1, –1)
1
and error, a process of inductive reasoning
Structural Formula
The average mediumsized piano has about
230 strings. Each string
of tension. That’s a
combined tension of
H
C
H
C
H
Make a conjecture about the next item in each sequence.
5. ABC is a right angle.
H
C
C
H
H
H
H
H
H
H
C
C
C
H
H
H
4. 12, 6, 3, 1.5, 0.75 0.375
6. Point S is between R and T.
⊥ BC
BA
RS ST RT
R
A
S
T
C
7. P, Q, R, and S are noncollinear
and P
Q
QR
RS
SP
.
8. ABCD is a parallelogram.
The segments form a square.
C3H8
H
1 1
8 16
1 1
2 4
3. 2, 1, , , P
Q
S
R
A
D
B
C
Determine whether each conjecture is true or false. Give a counterexample for
any false conjecture.
9. Given: S, T, and U are collinear and ST TU.
Conjecture: T is the midpoint of S
U
.
H
10. Given: 1 and 2 are adjacent angles.
Conjecture: 1 and 2 form a linear pair.
False; 1 and 2 could each measure 60°.
38. Make a conjecture about butane, which is the next compound in the group.
Write its structural formula.
39. Write the chemical formula for the 7th compound in the group. C7H16
★ 40. Develop a rule you could use to find the chemical formula of the nth substance
in the alkane group. CnH2n 2
H
C
H
H
true
12. ALLERGIES Each spring, Rachel starts sneezing when the pear trees on her street blossom.
She reasons that she is allergic to pear trees. Find a counterexample to Rachel’s conjecture.
Sample answer: Rachel could be allergic to other types of plants that
blossom when the pear trees blossom.
Gl
NAME
______________________________________________
DATE
/M
G
Hill
60
____________
Gl PERIOD
G _____
2-1
to Learn
Learn Mathematics
Mathematics,
p. 61
Inductive Reasoning and Conjecture
Pre-Activity
41. CRITICAL THINKING The expression n2 n 41 has a prime value for
n 1, n 2, and n 3. Based on this pattern, you might conjecture that this
expression always generates a prime number for any positive integral value
of n. Try different values of n to test the conjecture. Answer true if you think
the conjecture is always true. Answer false and give a counterexample if you
think the conjecture is false. false; n 41
www.geometryonline.com/self_check_quiz
11. Given: G
H
and JK
form a right angle and intersect at P.
Conjecture: GH
⊥
JK
ELL
How can inductive reasoning help predict weather conditions?
Read the introduction to Lesson 2-1 at the top of page 62 in your textbook.
• What kind of weather patterns do you think meteorologists look at to
help predict the weather? Sample answer: patterns of high and
low temperatures, including heat spells and cold spells;
patterns of precipitation, including wet spells and dry spells
• What is a factor that might contribute to long-term changes in the
weather? Sample answer: global warming due to high usage
of fossil fuels
1. Explain in your own words the relationship between a conjecture, a counterexample, and
inductive reasoning.
Lesson 2-1 Inductive Reasoning and Conjecture 65
Mary Kate Denny/PhotoEdit
Sample answer: A conjecture is an educated guess based on specific
examples or information. A counterexample is an example that shows
that a conjecture is false. Inductive reasoning is the process of making a
conjecture based on specific examples or information.
2. Make a conjecture about the next item in each sequence.
NAME ______________________________________________ DATE
2-1
Enrichment
Enrichment,
29.
p. 62
Counterexamples
b. 1, , , c. 0, 1, 3, 6, 10 15
e. 1, 8, 27, 64 125
g.
d. 8, 3, 2, 7 12
f. 1, 2, 4, 8 16
h.
a. The sum of two odd integers is even.
true
b. The product of an odd integer and an even integer is odd.
Example
False; sample answer: 5 8 40, which is even.
1
x
Is the statement 1 true when you replace x with
c. The opposite of an integer is a negative integer. False; sample answer: The
1, 2, and 3? Is the statement true for all reals? If possible, find a
counterexample.
W
1 1 1 1
3 9 27 81
a. 5, 9, 13, 17 21
3. State whether each conjecture is true or false. If the conjecture is false, give a
counterexample.
When you make a conclusion after examining several specific
cases, you have used inductive reasoning. However, you must be
cautious when using this form of reasoning. By finding only one
counterexample, you disprove the conclusion.
2
1
31.
____________ PERIOD _____
X
Y
Z
opposite of the integer 5 is 5, which is a positive integer.
d. The perfect squares (squares of whole numbers) alternate between odd and even.
1
1
1
1
1
1, 1, and 1. But when x , then 2. This counterexample
1
2
3
2
x
true
shows that the statement is not always true.
Helping You Remember
1. The coldest day of the year in Chicago
occurred in January for five straight
years. Is it safe to conclude that the
coldest day in Chicago is always in
2. Suppose John misses the school bus
four Tuesdays in a row. Can you
safely conclude that John misses the
school bus every Tuesday? no
4. Write a short sentence that can help you remember why it only takes one counterexample
to prove that a conjecture is false.
Sample answer: True means always true.
Lesson 2-1 Inductive Reasoning and Conjecture 65
Lesson 2-1
C
H
p. 59 and
Practice,
p. 60 (shown)
Inductive Reasoning
and Conjecture
Make a conjecture based on the given information. Draw a figure to illustrate
Propane
C2H6
H
R
____________
Gl PERIOD
G _____
true
38. Butane will have
4 carbon atoms and
10 hydrogen atoms.
H
C
H
Source: www.pianoworld.com
H
H
F
Skills
Practice,
2-1
Practice
(Average)
B
Ethane
H
Music
Q
NAME
______________________________________________
DATE
/M
G
Hill
57
2. 5, 10, 15, 20 25
Alkanes
CH4
P
E
1.
CHEMISTRY For Exercises 38–40, use the following information.
Hydrocarbons are molecules composed of only carbon (C) and hydrogen (H)
atoms. The simplest hydrocarbons are called alkanes. The first three alkanes are
shown below.
Chemical Formula
D
T
37. MUSIC Many people learn to play the piano by ear. This means that they first
learned how to play without reading music. What process did they use? trial
Methane
E and F are congruent.
E
C
Gl
Compound Name
B
W
7. E and F are right angles.
ABC and DBE are congruent.
A
R
2
T
x
6. ABC and DBE are vertical angles.
36. Given: noncollinear points R, S, and T
S
, S
T
, and R
T
form a triangle. true
Conjecture: R
H
Example 2 Make
the number of small
squares in the next figure.
Observe a pattern: The sides of the squares
have measures 1, 2, and 3 units.
Conjecture: For the next figure, the side of
the square will be 4 units, so the figure
will have 16 small squares.
Describe the pattern. Then make a conjecture about the next number in the
sequence.
counterexample.
34. Given: DE EF
. False; D, E, and F do not have to be
Conjecture: E is the midpoint of DF
H
____________ PERIOD _____
Make Conjectures A conjecture is a guess based on analyzing information or
observing a pattern. Making a conjecture after looking at several situations is called
inductive reasoning.
Example 1 Make a conjecture about
the next number in the sequence 1, 3, 9,
27, 81.
Analyze the numbers:
Notice that each number is a power of 3.
1
3
9 27 81
30 31 32 33 34
Conjecture: The next number will be 35 or 243.
30. Given: m y 10, y 4
Conjecture: m 6 False; if y 7 and m 5, then 7 5 10
and 5 4, but 7 6.
31. Given: points W, X, Y, and Z
Conjecture: W, X, Y, and Z are noncollinear. False; see margin for
32. Given: A(4, 8), B(3, 8), C(3, 5)
Conjecture: ABC is a right triangle. true
NAME ______________________________________________ DATE
Study
Guide
andIntervention
Intervention,
2-1
Study
Guide and
Lesson 2-1
Determine whether each conjecture is true or false. Give a counterexample for any
false conjecture.
29. Given: 1 and 2 are complementary angles.
Conjecture: 1 and 2 form a right angle. False; see margin for
42. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson. See margin.
How can inductive reasoning help predict weather conditions?
4 Assess
• an explanation as to how a conjecture about a weather pattern in the summer
might be different from a similar weather pattern in the winter, and
• a conjecture about tomorrow’s weather based on your local weather over the
past several days.
Open-Ended Assessment
rules or activities. Then have
students swap papers with a
partner and try to come up with
a counterexample for each
conjecture. An example statement
could be: Students must attend
school Monday through Friday. A
counterexample for this would
be a holiday or a snow day.
Standardized
Test Practice
43. What is the next term in the sequence 1, 1, 2, 3, 5, 8? C
A 11
B 12
C 13
D
14
44. ALGEBRA If the average of six numbers is 18 and the average of three of the
numbers is 15, then what is the sum of the remaining three numbers? D
A 21
B 45
C 53
D 63
Lesson 2-2
Basic Skill Students will learn
2-2. They will determine the
truth value of various situations.
Use Exercises 65–67 to determine
determining which elements
make a statement true.
experience, when dark clouds
appear, there is a chance of rain.
following.
• When there is precipitation in
the summer, it is usually rain
because the temperature is
above freezing. When the
temperature is below freezing, as
in the winter, ice or snow forms.
• See students’ work.
Mixed Review
pentagon,
convex,
regular
hexagon, convex, irregular
48. Yes; the symbol
denotes that KJN is
a right angle.
49. No; we do not
the angle measures.
50. No; we do not
know whether MNP
is a right angle.
51. Yes; they form a
linear pair.
52. Yes; since the
other three angles in
rectangle KLPJ are
right angles, KLP
must also be a right
angle.
Determine whether each statement can be
assumed from the figure. Explain. (Lesson 1-5)
48. KJN is a right angle.
66
heptagon, concave,
irregular
K
L
M
J
P
N
49. PLN NLM
50. PNL and MNL are complementary.
51. KLN and MLN are supplementary.
52. KLP is a right angle.
Find the coordinates of the midpoint of a segment having the given endpoints.
(Lesson 1-3)
53. A
B
for A(1, 3), B(5, 5) (2, 1)
54. C
D
for C(4, 1), D(3, 7) (0.5, 4)
55. F
G
for F(4, 9), G(2, 15) (1, 12) 56. H
J for H(5, 2), J(7, 4) (1, 1)
57. K
L
for K(8, 1.8), L(3, 6.2) (5.5, 2.2) 58. M
N
for M(1.5, 6), N(4, 3)
(2.75, 1.5)
Find the value of the variable and MP, if P is between M and N. (Lesson 1-2)
59. MP 7x, PN 3x, PN 24 8; 56
60. MP 2c, PN 9c, PN 63 7; 14
61. MP 4x, PN 5x, MN 36 4; 16
62. MP 6q, PN 6q, MN 60 5; 30
63. MP 4y 3, PN 2y, MN 63
64. MP 2b 7, PN 8b, MN 43
10; 43
the Next Lesson
66 Chapter 2 Reasoning and Proof
Name each polygon by its number of sides and then classify it as convex or
concave and regular or irregular. (Lesson 1-6)
45.
46.
47.
5; 3
BASIC SKILL Determine which values in the given replacement set make the
inequality true.
66. 12 x 0 13, 14
67. 5x 1 25 5, 6, 7
65. x 2 5 4, 5
{2, 3, 4, 5}
{11, 12, 13, 14}
{4, 5, 6, 7}
Chapter 2 Reasoning and Proof
Lesson
Notes
Logic
• Determine truth values of conjunctions and disjunctions.
1 Focus
• Construct truth tables.
does logic apply to school?
Vocabulary
•
•
•
•
•
•
•
statement
truth value
negation
compound statement
conjunction
disjunction
truth table
questions on a test, you are using
a basic principle of logic. For
example, refer to the map, and
VIRGINIA
NORTH CAROLINA
Raleigh
Ashville
Gastonia
Charlotte
SOUTH
CAROLINA
Raleigh is a city in North Carolina.
Willmington
You know that there is only one
correct answer, either true or false.
Study Tip
Statements
A mathematical statement
with one or more
variables is called an open
sentence. The truth value
of an open sentence
cannot be determined
until values are assigned
to the variables. A
statement with only
numeric values is a closed
sentence.
DETERMINE TRUTH VALUES A statement , like the true-false example
above, is any sentence that is either true or false, but not both. Unlike a conjecture,
we know that a statement is either true or false. The truth or falsity of a statement
is called its truth value .
Statements are often represented using a letter such as p or q. The statement above
can be represented by p.
p:
Raleigh is a city in North Carolina. This statement is true.
The negation of a statement has the opposite meaning as well as an opposite
truth value. For example, the negation of the statement above is not p.
not p: Raleigh is not a city in North Carolina. In this case, the statement is false.
Negation
• Words
If a statement is represented by p, then not p is the negation of the
statement.
5-Minute Check
Transparency 2-2 Use as a
quiz or review of Lesson 2-1.
Mathematical Background notes
are available for this lesson on
p. 60C.
does logic apply to
school?
• Determine whether the
following statement is true or
false: “South Carolina borders
North Carolina, Georgia, and
Tennessee.” false
• Locate Wilmington on the map
of North Carolina. Is this a
coastal or inland city? Make a
might find ocean or lakes in
Wilmington given its location
on the map. Coastal; sample
answer: You would find ocean in
Wilmington, North Carolina.
• Symbols ~p, read not p
Two or more statements can be joined to form a compound statement . Consider
the following two statements.
p: Raleigh is a city in North Carolina.
q: Raleigh is the capital of North Carolina.
The two statements can be joined by the word and.
p and q: Raleigh is a city in North Carolina, and Raleigh is the capital of
North Carolina.
Lesson 2-2 Logic 67
Resource Manager
Workbook and Reproducible Masters
Chapter 2 Resource Masters
• Study Guide and Intervention, pp. 63–64
• Skills Practice, p. 65
• Practice, p. 66
• Reading to Learn Mathematics, p. 67
• Enrichment, p. 68
• Assessment, p. 119
Graphing Calculator and
Computer Masters, p. 20
Teaching Geometry With Manipulatives
Masters, p. 16
Transparencies
5-Minute Check Transparency 2-2
Technology
Interactive Chalkboard
Lesson x-x Lesson Title 67
The statement formed by joining p and q is an example of a conjunction.
2 Teach
Conjunction
• Words
DETERMINE TRUTH
VALUES
In-Class Example
A conjunction is a compound statement formed by joining two or
more statements with the word and.
• Symbols p q, read p and q
Power
Point®
A conjunction is true only when both statements in it are true. Since it is true that
Raleigh is in North Carolina and it is the capital, the conjunction is also true.
1 Use the following statements
to write a compound
statement for each
conjunction. Then find its
truth value.
p: One foot is 14 inches.
q: September has 30 days.
r: A plane is defined by three
noncollinear points.
Example 1 Truth Values of Conjunctions
Use the following statements to write a compound statement for each
conjunction. Then find its truth value.
p: January 1 is the first day of the year.
q: ⴚ5 ⴙ 11 ⴝ ⴚ6
r: A triangle has three sides.
a. p and q
One foot is 14 inches, and
September has 30 days; false.
a. p and q
January is the first day of the year, and 5 11 6.
p and q is false, because p is true and q is false.
b. r p
A plane is defined by three
noncollinear points, and one foot
is 14 inches; false.
b. r ⵩ p
A triangle has three sides, and January 1 is the first day of the year.
r p is true, because r is true and p is true.
c. q r
September does not have
30 days, and a plane is defined by
three noncollinear points; false.
d. p r
A foot is not 14 inches, and a
plane is defined by three
noncollinear points; true.
Study Tip
Negations
The negation of a
statement is not
necessarily false. It
has the opposite truth
value of the original
statement.
c. p and not r
January 1 is the first day of the year, and a triangle does not have three sides.
p and not r is false, because p is true and not r is false.
d. ~q ⵩ r
5 11 6, and a triangle has three sides
q r is true because q is true and r is true.
Statements can also be joined by the word or. This type of statement is a
disjunction. Consider the following statements.
p:
Ahmed studies chemistry.
q:
Ahmed studies literature.
p or q: Ahmed studies chemistry, or Ahmed studies literature.
Disjunction
• Words
A disjunction is a compound statement formed by joining two or
more statements with the word or.
• Symbols p q, read p or q
68 Chapter 2 Reasoning and Proof
68
Chapter 2 Reasoning and Proof
A disjunction is true if at least one of the statements is true. In the case of p or q
above, the disjunction is true if Ahmed either studies chemistry or literature or both.
The disjunction is false only if Ahmed studies neither chemistry nor literature.
In-Class Example
Power
Point®
2 Use the following statements
Example 2 Truth Values of Disjunctions
Use the following statements to write a compound statement for each
disjunction. Then find its truth value.
p:
100 ⴜ 5 ⴝ 20
q: The length of a radius of a circle is twice the length of its diameter.
r: The sum of the measures of the legs of a right triangle equals the measure of
the hypotenuse.
a. p or q
100 5 20, or the length of a radius of a circle is twice the length of its
diameter.
p or q is true because p is true. It does not matter that q is false.
b. q r
The length of a radius of a circle is twice the length of its diameter, or the sum
of the measures of the legs of a right triangle equals the measure of the
hypotenuse.
q r is false since neither statement is true.
Study Tip
Venn Diagrams
The size of the
overlapping region in a
Venn Diagram does not
indicate how many items
fall into that category.
Conjunctions can be illustrated with
Venn diagrams. Refer to the statement at
the beginning of the lesson. The Venn
diagram at the right shows that Raleigh (R)
is represented by the intersection of the set
of cities in North Carolina and the set of
state capitals. In other words, Raleigh must
be in the set containing cities in North
Carolina and in the set of state capitals.
to write a compound
statement for each
disjunction. Then find its
truth value.
is proper notation for
p: AB
“line AB.”
q: Centimeters are metric units.
r: 9 is a prime number.
a. p or q
B
A
is proper notation for “line
AB,” or centimeters are metric
units; true.
b. q r
Centimeters are metric units, or
9 is a prime number; true.
All U.S. Cities
U.S.
State
Capitals
R
Cities
in
North
Carolina
A disjunction can also be illustrated with
a Venn diagram. Consider the following statements.
p:
Jerrica lives in a U.S. state capital.
q:
Jerrica lives in a North Carolina city.
p q: Jerrica lives in a U.S. state capital, or Jerrica lives in a North Carolina city.
In the Venn diagrams, the disjunction is represented by the union of the two sets.
The union includes all U.S. capitals and all cities in North Carolina. The city in
which Jerrica lives could be located in any of the three regions of the union.
The three regions represent
A U.S. state capitals excluding the capital
of North Carolina,
B cities in North Carolina excluding the
state capital, and
C the capital of North Carolina,
which is Raleigh.
www.geometryonline.com/extra_examples
All U.S. Cities
U.S.
State
Capitals
A
C
Cities
in
North
Carolina
B
Lesson 2-2 Logic 69
Lesson 2-2 Logic
69
In-Class Example
Venn diagrams can be used to solve real-world problems involving conjunctions
and disjunctions.
Power
Point®
3 DANCING The Venn diagram
Example 3 Use Venn Diagrams
shows the number of
students enrolled in
Monique’s Dance School for
tap, jazz, and ballet classes.
RECYCLING The Venn diagram shows the number of neighborhoods that have
a curbside recycling program for paper or aluminum.
Curbside Recycling
Tap
28
17
Paper Aluminum
12 46 20
Jazz
43
13
9
25
29
Ballet
a. How many neighborhoods recycle both paper and aluminum?
The neighborhoods that have paper and aluminum recycling are represented
by the intersection of the sets. There are 46 neighborhoods that have paper and
aluminum recycling.
a. How many students are
enrolled in all three classes? 9
b. How many students are
enrolled in tap or ballet? 121
b. How many neighborhoods recycle paper or aluminum?
The neighborhoods that have paper or aluminum recycling are represented
by the union of the sets. There are 12 46 20 or 78 neighborhoods that have
paper or aluminum recycling.
c. How many students are
enrolled in jazz and ballet
and not tap? 25
c. How many neighborhoods recycle paper and not aluminum?
The neighborhoods that have paper and not aluminum recycling are
represented by the nonintersecting portion of the paper region. There
are 12 neighborhoods that have paper and not aluminum recycling.
TRUTH TABLES
Intervention
Tell students
New
that truth tables
must display
all combinations
of Ts and Fs to exhaust all
possible outcomes, so for each
statement, p, q, and r, they will
need to mix the occurrences of
true and false. First, they should
determine the number of rows
they need. Then fill the top half
of the p column with Ts and the
other half with Fs. For the q
column, they can alternate T
and F the whole way down. If
they need an r column, they can
alternate sets of 2 Ts and 2 Fs
all the way down, and so on.
Assure students that as long
as they initially set up the
basic structure correctly, they
should be able to produce the
rest of the table quite easily.
70
Chapter 2 Reasoning and Proof
TRUTH TABLES A convenient method for organizing the truth values of
statements is to use a truth table .
Negation
p
p
T
F
F
T
If p is a true statement, then p is a false statement.
If p is a false statement, then p is a true statement.
Truth tables can also be used to determine truth values of compound statements.
Study Tip
Conjunction
A conjunction is
true only when
both statements
are true.
Disjunction
p
q
pq
T
T
T
T
F
F
T
F
T
T
F
F
T
T
F
F
F
F
F
Tautology
p
q
pq
A compound sentence is a
tautology if its truth value
is always true. For
example, “It is snowing or
it is not snowing” is a
tautology.
T
T
T
F
F
A disjunction is
false only when
both statements
are false.
You can use the truth values for negation, conjunction, and disjunction to
construct truth tables for more complex compound statements.
70 Chapter 2 Reasoning and Proof
1. The conjunction (p and q) is represented by the intersection of the two circles.
2a. Sample answer: October has 31 days or 5 3 8.
2b. Sample answer: A square has five right angles and the Postal Service does not deliver
mail on Sundays.
2c. Sample answer: July 5th is not a national holiday.
3. A conjunction is a compound statement using the word and, while a disjunction is a
compound statement using the word or.
Example 4 Construct Truth Tables
In-Class Example
Construct a truth table for each compound statement.
a. p q
Step 1 Make columns with the headings p, q, q, and p q.
Step 2 List the possible combinations of truth values for p and q.
Step 3 Use the truth values of q to determine the truth values of q.
Step 4 Use the truth values for p and q to write the truth values for p q.
p
q
q
p q
T
T
F
F
T
F
T
T
F
T
F
F
F
F
T
F
4 Teaching Tip
Tell students
that they could potentially
interchange the columns for p,
q, and r in 4c, and as long as
they correctly fill in the columns
for p q and (p q) r, the
overall outcome would be the
same. They would still end up
with 5 Ts and 3 Fs, but the Ts
and Fs would be in a different
order.
Step 1
Construct a truth table for
each compound statement.
Step 2
Step 3
a. p q
Step 4
b. p q
Study Tip
Truth Tables
Use the Fundamental
Counting Principle to
determine the number
of rows necessary.
Power
Point®
p
q
p
q
p q
T
T
F
F
F
T
F
F
T
T
F
T
T
F
T
F
F
T
T
T
b. p (q r)
c. (p q) r
Make columns for p, q, p q, r, and (p q) r.
p
q
pq
r
(p q) r
T
T
T
T
T
T
F
F
T
T
T
T
T
F
T
T
F
F
F
F
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
F
F
F
F
F
q p p q
T F
T
F F
F
T T
T
F T
T
p
T
T
F
F
p
T
T
T
T
F
F
F
F
q
T
F
T
F
T
F
T
F
r
T
T
F
F
T
T
F
F
q q r p (q r)
F
F
T
T
T
T
F
F
T
T
F
T
F
F
F
T
T
T
F
F
F
T
F
F
c. (p q) r
Concept Check
1– 3. See margin.
1. Describe how to interpret the Venn diagram
for p q.
2. OPEN ENDED Write a compound statement
for each condition.
a. a true disjunction
b. a false conjunction
c. a true statement that includes a negation
pq
p
q
3. Explain the difference between a conjunction and a disjunction.
Lesson 2-2 Logic 71
p
T
T
T
T
F
F
F
F
q
T
F
T
F
T
F
T
F
r r p q (p q) r
T F
T
F
T F
T
F
F T
T
T
F T
T
T
T F
T
F
T F
F
F
F T
T
T
F T
F
F
Differentiated Instruction
Logical/Mathematical Have students examine the relationship between
the number of simple statements (p, q, and r) and the number of rows
necessary to exhaust all possible combinations in a truth table. Point out
that for Example 4a, there are 2 statements and 4 rows; for Example 4c,
there are 3 statements and 8 rows. Ask students to form a conjecture
about how many rows would be needed for 4, 5, and n statements.
Similarly, students can examine the relationship between the number of
circles and the number of intersecting areas of a Venn diagram.
Lesson 2-2 Logic
71
3 Practice/Apply
Study Notebook
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4–6
7–9
10–14
15–17
1
2
3
4
Have students—
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 2.
• include an example of a Venn
diagram and an example of a
truth table.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
Odd/Even Assignments
Exercises 18–40 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
the Internet or other research
materials.
Assignment Guide
Basic: 19–37 odd, 41–51 odd,
52–73
Average: 19–51 odd, 52–73
54–69 (optional: 70–73)
4. p and q false
7. p or ~q true
6. q r false
9. ~p ~r false
5. p and r true
8. q r true
10. Copy and complete the truth table.
p
q
q
p q
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
F
Construct a truth table for each compound statement. 11–14. See p. 123A.
11. p q
12. q r
13. ~p r
14. (p q) r
Application
Organization by Objective
• Determine Truth Values:
18–29, 42–48
• Truth Tables: 30–41
Use the following statements to write a compound statement for each conjunction
and disjunction. Then find its truth value. 4–9. See margin for statements.
p: 9 ⴙ 5 ⴝ 14
q: February has 30 days.
r: A square has four sides.
AGRICULTURE For Exercises 15–17, refer to the
Venn diagram that represents the states producing
more than 100 million bushels of corn or wheat
per year.
15. How many states produce more than 100 million
bushels of corn? 14
16. How many states produce more than 100 million
bushels of wheat? 7
17. How many states produce more than 100 million
bushels of corn and wheat? 3
Grain Production
Corn
Chapter 2 Reasoning and Proof
WA
MT
ID
★ indicates increased difficulty
Practice and Apply
Use the following statements to write a compound statement for each conjunction
and disjunction. Then find its truth value. 18– 29. See p. 123A for statements.
For
Exercises
See
Examples
18–29
30–41
42–48
1, 2
4
3
Extra Practice
See page 756.
p:
q:
r:
s:
ⴚ64 ⴝ 8
An equilateral triangle has three congruent sides.
00
An obtuse angle measures greater than 90° and less than 180°.
18.
21.
24.
27.
p and q false
r and s false
p s false
s q true
19.
22.
25.
28.
p or q true
q or r true
q r false
(p q) s true
Copy and complete each truth table.
30.
p
q
p
p q
T
T
T
F
F
T
F
F
F
F
T
T
31.
T
F
T
T
20.
23.
26.
29.
p and r false
q and s true
r p false
s (q and r) true
p
q
p
q
p q
T
T
F
T
F
F
F
F
T
T
F
T
T
F
F
F
T
T
F
F
72
ND
Source: U.S. Department of Agriculture
72 Chapter 2 Reasoning and Proof
4. 9 5 14 and February has
30 days.
5. 9 5 14 and a square has four
sides.
6. February has 30 days and a
square has four sides.
7. 9 5 14 or February does not
have 30 days.
Wheat
IA PA IL
NE IN
MN
MO CO SD
WI OH KS
MI KY
8. February has 30 days or
a square has four sides.
9. 9 5 14 or a square
does not have four
sides.
45.
Level of Participation
Among 310 Students
Sports
95
20
Clubs
60
★ 32. Copy and complete the truth table.
NAME ______________________________________________ DATE
p
q
r
pq
(p q) r
T
T
T
T
T
F
T
F
T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
F
F
F
F
T
T
F
T
F
F
F
T
F
F
F
p. 63
Logic(shown) and p. 64
Determine Truth Values
A statement is any sentence that is either true or false. The
truth or falsity of a statement is its truth value. A statement can be represented by using a
letter. For example,
Statement p: Chicago is a city in Illinois. The truth value of statement p is true.
Several statements can be joined in a compound statement.
Construct a truth table for each compound statement. 33– 40. See pp. 123A–123B.
33. q and r
34. p or q
35. p or r
36. p and q
38. p q
★ 39. p (q r)
★ 40. p (q r)
37. q r
MUSIC For Exercises 41–44, use the following information.
A group of 400 teens were asked what type of
Music Preference
music they listened to. They could choose among
pop, rap, and country. The results are shown in
the Venn diagram.
Pop 34 Country
175
45
41. How many teens said that they listened to none of
7
these types of music? 42
25
10
42. How many said that they listened to all three types
Rap
of music? 7
62
42
43. How many said that they listened to only pop and
rap music? 25
44. How many teens said that they listened to pop, rap, or country music? 358
Statement p and statement q joined
by the word and is a conjunction.
Statement p and statement q joined
by the word or is a disjunction.
Symbols: p q (Read: p and q )
Symbols: p q (Read: p or q)
Negation: not p is the negation of
the statement p.
The conjunction p q is true only
when both p and q are true.
The disjunction p q is true if p is
true, if q is true, or if both are true.
The statements p and p have
opposite truth values.
Example 1 Write a compound
statement for each conjunction. Then
find its truth value.
p: An elephant is a mammal.
q: A square has four right angles.
Example 2 Write a compound
statement for each disjunction. Then
find its truth value.
p: A diameter of a circle is twice the radius.
q: A rectangle has four equal sides.
a. p q
Join the statements with and: An elephant
is a mammal and a square has four right
angles. Both parts of the statement are
true so the compound statement is true.
a. p q
Join the statements p and q with the
word or: A diameter of a circle is twice
the radius or a rectangle has four equal
sides. The first part of the compound
statement, p, is true, so the compound
statement is true.
b. p q
p is the statement “An elephant is not a
mammal.” Join p and q with the word
and: An elephant is not a mammal and a
square has four right angles. The first
part of the compound statement, p, is
false. Therefore the compound statement
is false.
Lesson 2-2
T
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
2-2
Study
Guide and
b. p q
Join p and q with the word or: A
diameter of a circle is not twice the
radius or a rectangle has four equal
sides. Neither part of the disjunction is
true, so the compound statement is false.
Exercises
Write a compound statement for each conjunction and disjunction.
Then find its truth value.
p: 10 8 18
q: September has 30 days. r: A rectangle has four sides.
1. p and q 10 8 18 and September has 30 days; true.
2. p or r 10 8 18 or a rectangle has four sides; true.
3. q or r September has 30 days or a rectangle has four sides; true.
4. q and r September has 30 days and a rectangle does not have four
sides; false.
NAME
______________________________________________
DATE
/M
G
Hill
63
Gl
Skills
Practice,
p. 65 and
2-2
Practice
(Average)
Practice,
p. 66 (shown)
Logic
____________
Gl PERIOD
G _____
Use the following statements to write a compound statement for each conjunction
and disjunction. Then find its truth value.
p: 60 seconds 1 minute
q: Congruent supplementary angles each have a measure of 90.
r: 12 11 1
1. p q 60 seconds 1 minute and congruent supplementary angles each
have a measure of 90; true.
2. q r Congruent supplementary angles each have a measure of 90 or
12 11 1; true.
3. p q 60 seconds 1 minute or congruent supplementary angles each
SCHOOL For Exercises 45–47, use the following information.
In a school of 310 students, 80 participate in academic clubs, 115 participate in
sports, and 20 students participate in both.
45. Make a Venn diagram of the data. See margin.
46. How many students participate in either clubs or sports? 175
47. How many students do not participate in either clubs or sports? 135
have a measure of 90; true.
4. p r 60 seconds 1 minute and 12 11 1; false.
Copy and complete each truth table.
5.
6.
p
q
p
q
p q
p
q
p
p q
p (p q)
T
T
F
T
F
T
T
T
F
F
T
F
T
F
F
F
T
T
T
F
T
F
F
T
T
T
F
F
F
F
T
T
T
F
T
T
T
F
F
F
Construct a truth table for each compound statement.
RESEARCH For Exercises 48–50, use the Internet or another resource to determine
whether each statement about cities in New York is true or false.
48. Albany is not located on the Hudson river. false
49. Either Rochester or Syracuse is located on Lake Ontario. true
50. It is false that Buffalo is located on Lake Erie. false
Nationwide, approximately
80% of high school
seniors participate in
extracurricular activities.
Athletics, performing arts,
and clubs are the most
popular.
CRITICAL THINKING For Exercises 51 and 52, use the following information.
All members of Team A also belong to Team B, but only some members of
Team B also belong to Team C. Teams A and C have no members in common.
51.
2-2
Enrichment
Enrichment,
B
C
Solve the alphametic puzzle at the right.
F 8, O 3, U 1, R 0
Can you find other solutions to this puzzle?
HALF
9703
2.
TWO
T
F
F
T
F
T
F
F F T
F
F
F F
T
T
T
T
information.
The Venn diagram shows the number of students in the band
who work after school or on the weekends.
Work
After
School
5
3
Work
Weekends
17
NAME
______________________________________________
DATE
/M
G
Hill
66
____________
Gl PERIOD
G _____
ELL
How does logic apply to school?
73
____________ PERIOD _____
FOUR
ONE
F I VE
1. Supply one or two words to complete each sentence.
compound
a. Two or more statements can be joined to form a
statement.
b. A statement that is formed by joining two statements with the word or is called a
disjunction .
truth value .
c. The truth or falsity of a statement is called its
d. A statement that is formed by joining two statements with the word and is called a
conjunction .
e. A statement that has the opposite truth value and the opposite meaning from a given
negation
statement is called the
of the statement.
2. Use true or false to complete each sentence.
false
a. If a statement is true, then its negation is
.
true
b. If a statement is false, then its negation is
.
true
c. If two statements are both true, then their conjunction is
true
their disjunction is
.
false
d. If two statements are both false, then their conjunction is
false
their disjunction is
.
e. If one statement is true and another is false, then their conjunction is
false
true
and their disjunction is
.
and
and
3. Consider the following statements:
p: Chicago is the capital of Illinois. q: Sacramento is the capital of California.
Write each statement symbolically and then find its truth value.
a. Sacramento is not the capital of California. q; false
b. Sacramento is the capital of California and Chicago is not the capital of Illinois.
q p; true
Helping You Remember
Find a value for each letter in each alphametic. Sample answers are shown
1.
F
F
F T
standardized test if you are not sure of the correct answer? Sample answer:
Eliminate the choices that you know are wrong. Then choose the
one you think is most likely correct from the ones that are left.
8310
347
8657
N 4, E 7, I 6, and V 5.
T
T F
T
Read the introduction to Lesson 2-2 at the top of page 67 in your textbook.
An alphametic is a computation puzzle using letters instead of
digits. Each letter represents one of the digits 0–9, and two
different letters cannot represent the same digit. Some alphametic
puzzles have more than one answer.
Example
F
T
F
Mathematics,
p. 67
Logic
p. 68
Since R E E, the value of R must be 0. Notice that the
thousands digit must be the same in the first addend and the
sum. Since the value of I is 9 or less, O must be 4 or less. Use
trial and error to find values that work.
F
T
F T F
Pre-Activity
Letter Puzzles
A
T T
T F T
2-2
to Learn
Learn Mathematics
Bill Bachmann/PhotoEdit
NAME ______________________________________________ DATE
T
SCHOOL For Exercises 9 and 10, use the following
Gl
c. No person that is a member of Team A can be a member of Team C.
Lesson 2-2 Logic
F
9. How many students work after school and on weekends? 3
b. If a person is not a member of Team B, then the person is not a member
of Team A.
www.geometryonline.com/self_check_quiz
p q p q p q q (p q)
T T F
10. How many students work after school or on weekends? 25
51. Draw a Venn diagram to illustrate the situation. See margin.
52. Which of the following statements is true? b
a. If a person is a member of Team C, then the person is not a member of
Team A.
Source: National Center for
Education Statistics
8. q (p q)
p q q p q q (p q)
4. Prefixes can often help you to remember the meaning of words or to distinguish between
similar words. Use your dictionary to find the meanings of the prefixes con and dis and
explain how these meanings can help you remember the difference between a
conjunction and a disjunction. Sample answer: Con means together and dis
means apart, so a conjunction is an and (or both together) statement and
a disjunction is an or statement.
734
Lesson 2-2 Logic
73
Lesson 2-2
School
7. q (p q)
53. WRITING IN MATH
4 Assess
How does logic apply to school?
Open-Ended Assessment
Modeling Have students model
a Venn diagram and a truth table
with buttons or chips. For the
Venn diagram, students can
draw two large overlapping
circles on a piece of paper and
label them Science and English.
Then they can place buttons on
the diagram to represent the
number of students in the class
who like one, the other, or both
subjects. Students can also draw
a grid and use white buttons for
true and black buttons for false to
model one of the truth tables in
the lesson.
• an example of a conjunction using statements about your favorite subject
and your favorite extracurricular activity, and
• a Venn diagram showing various characteristics of the members of your
geometry class (for example, male/female, grade in school, and so on).
Standardized
Test Practice
A
Mixed Review
63. 34.4
64. 30.4
Make a conjecture about the next item in each sequence. (Lesson 2-1)
3 3 3
56. 3, 5, 7, 9 11
57. 1, 3, 9, 27 81
58. 6, 3, , 2 4 8
1
59. 17, 13, 9, 5 1
60. 64, 16, 4, 1 61. 5, 15, 45, 135 405
4
COORDINATE GEOMETRY Find the perimeter of each polygon. Round answers
to the nearest tenth. (Lesson 1-6)
62. triangle ABC with vertices A(6, 7), B(1, 3), and C(2, 7) 33.1
63. square DEFG with vertices D(10, 9), E(5, 2), F(2, 7), and G(3, 14)
64. quadrilateral HIJK with vertices H(5, 10), I(8, 9), J(5, 5), and K(2, 4)
65. hexagon LMNPQR with vertices L(2, 1), M(4, 5), N(6, 4), P(7, 4), Q(5, 8),
and R(3, 7) 29.5
Measure each angle and classify it as right,
acute, or obtuse. (Lesson 1-4)
66. ABC 145°, obtuse
67. DBC 55°, acute
68. ABD 90°, right
Quiz (Lessons 2-1 and 2-2) is
available on p. 119 of the Chapter 2
Resource Masters.
D
C
A
B
69. FENCING Michelle wanted to put a fence around her rectangular garden. The
front and back measured 35 feet each, and the sides measured 75 feet each. If
she wanted to make sure that she had enough feet of fencing, how much should
she buy? (Lesson 1-2) 222 ft
74 Chapter 2 Reasoning and Proof
C
Assessment Options
53. Sample answer: Logic can be
used to eliminate false choices on
should include the following.
• Math is my favorite subject and
drama club is my favorite
activity.
• See students’ work.
B
54. Which statement about ABC has the same truth value
as AB BC? A
A mA mC
B mA mB
C AC BC
D AB AC
55. ALGEBRA If the sum of two consecutive even integers
is 78, which number is the greater of the two integers? C
A 36
B 38
C 40
D 42
Lesson 2-3
Prerequisite Skill Students will
in Lesson 2-3. They will substitute
the hypothesis and conclusion for
the if and then parts of statements.
Use Exercises 70–73 to determine
substituting numbers for variables
in algebraic expressions.
Answer the question that was posed at the beginning
of the lesson. See margin.
the Next Lesson
74
PREREQUISITE SKILL Evaluate each expression for the given values.
(To review evaluating algebraic expressions, see page 736.)
70. 5a 2b if a 4 and b 3 14
72. 4e 3f if e 1 and f 2 ⴚ10
Chapter 2 Reasoning and Proof
71. 4cd 2d if c 5 and d 2 44
73. 3g2 h if g 8 and h 8 184
Lesson
Notes
Conditional Statements
• Analyze statements in if-then form.
• Write the converse, inverse, and contrapositive of if-then statements.
are conditional statements
Vocabulary
•
•
•
•
•
•
•
•
•
conditional statement
if-then statement
hypothesis
conclusion
related conditionals
converse
inverse
contrapositive
logically equivalent
5-Minute Check
Transparency 2-3 Use as a
quiz or review of Lesson 2-2.
them that they are getting something for
free in addition to their purchase.
Mathematical Background notes
are available for this lesson on
p. 60C.
-
IF-THEN STATEMENTS The statements above are examples of conditional
statements. A conditional statement is a statement that can be written in if-then
form. The first example above can be rewritten to illustrate this.
If you buy a car, then you get \$1500 cash back.
If-Then Statement
• Words
An if-then statement is written in the form if p, then q. The phrase
immediately following the word if is called the hypothesis , and the
phrase immediately following the word then is called the conclusion.
• Symbols
p → q, read if p then q, or p implies q.
Example 1 Identify Hypothesis and Conclusion
The word if is not part of
the hypothesis. The word
then is not part of the
conclusion.
hypothesis
are conditional
statements used in
What happens if you buy a new
car? How can you get a free
phone? You get \$1500 cash back;
enroll in phone service for one year.
• How effective are these types
effective; people like to get
have to pay for something else.
Identify the hypothesis and conclusion of each statement.
a. If points A, B, and C lie on line ᐉ, then they are collinear.
If points A, B, and C lie on line ᐉ, then they are collinear.
Study Tip
1 Focus
conclusion
Hypothesis: points A, B, and C lie on line ᐉ
Conclusion: they are collinear
b. The Tigers will play in the tournament if they win their next game.
Hypothesis: the Tigers win their next game
Conclusion: they will play in the tournament
Identifying the hypothesis and conclusion of a statement is helpful when
writing statements in if-then form.
Lesson 2-3 Conditional Statements 75
Resource Manager
Workbook and Reproducible Masters
Chapter 2 Resource Masters
• Study Guide and Intervention, pp. 69–70
• Skills Practice, p. 71
• Practice, p. 72
• Reading to Learn Mathematics, p. 73
• Enrichment, p. 74
Teaching Geometry With Manipulatives
Masters, p. 43
Transparencies
5-Minute Check Transparency 2-3
Real-World Transparency 2
Technology
GeomPASS: Tutorial Plus, Lesson 6
Interactive Chalkboard
Lesson x-x Lesson Title 75
Example 2 Write a Conditional in If-Then Form
IF-THEN STATEMENTS
In-Class Examples
Power
Point®
1 Identify the hypothesis and
conclusion of each statement.
a. If a polygon has 6 sides, then
it is a hexagon. Hypothesis: a
polygon has 6 sides; Conclusion:
it is a hexagon
Study Tip
If-Then Statements
When you write a statement
in if-then form, identify
the condition that causes the
result as the hypothesis.
The result is the conclusion.
b. Perpendicular lines intersect.
Sometimes you must add information to a statement. In this case, it is necessary
to know that perpendicular lines come in pairs.
Hypothesis: two lines are perpendicular
Conclusion: they intersect
If two lines are perpendicular, then they intersect.
b. Tamika will advance to the
next level of play if she
completes the maze in her
computer game. Hypothesis:
Tamika completes the maze in
her computer game; Conclusion:
she will advance to the next level
of play.
Recall that the truth value of a statement is either true or false. The hypothesis
and conclusion of a conditional statement, as well as the conditional statement itself,
can also be true or false.
2 Identify the hypothesis and
conclusion of each statement.
Then write each statement in
the if-then form.
a. Distance is positive.
Hypothesis: a distance is
determined; Conclusion: it is
positive; If a distance is
determined, then it is positive.
b. A five-sided polygon is a
pentagon. Hypothesis: a polygon
has five sides; Conclusion: it is a
pentagon; If a polygon has five
sides, then it is a pentagon.
Teaching Tip
Tell students to
use parentheses to identify the
hypothesis and conclusion in
each situation. Explain that if
the hypothesis in the situation
matches the hypothesis in the
original statement, students can
mark a T over the parentheses; if
not, they can mark an F. They can
do the same for the conclusions.
3 Determine the truth value of
a.
b.
c.
d.
76
the following statement for
each set of conditions. If Yukon
rests for 10 days, his ankle will
heal.
Yukon rests for 10 days, and
he still has a hurt ankle. false
Yukon rests for 3 days, and
he still has a hurt ankle. true
Yukon rests for 10 days, and
he does not have a hurt ankle
anymore. true
Yukon rests for 7 days, and
he does not have a hurt ankle
anymore. true
Chapter 2 Reasoning and Proof
Identify the hypothesis and conclusion of each statement. Then write each
statement in if-then form.
a. An angle with a measure greater than 90 is an obtuse angle.
Hypothesis: an angle has a measure greater than 90
Conclusion: it is an obtuse angle
If an angle has a measure greater than 90, then it is an obtuse angle.
Example 3 Truth Values of Conditionals
SCHOOL Determine the truth value of the following statement for each set of
conditions.
If you get 100% on your test, then your teacher will give you an A.
a. You get 100%; your teacher gives you an A.
The hypothesis is true since you got 100%, and the conclusion is true
because the teacher gave you an A. Since what the teacher promised is true,
the conditional statement is true.
Study Tip
Common
Misconception
A true hypothesis does not
necessarily mean that a
conditional is true. Likewise,
a false conclusion does not
guarantee that a conditional
is false.
b. You get 100%; your teacher gives you a B.
The hypothesis is true, but the conclusion is false. Because the result is not what
was promised, the conditional statement is false.
c. You get 98%; your teacher gives you an A.
The hypothesis is false, and the conclusion is true. The statement does not say
what happens if you do not get 100% on the test. You could still get an A. It is
also possible that you get a B. In this case, we cannot say that the statement is
false. Thus, the statement is true.
d. You get 85%; your teacher gives you a B.
As in part c, we cannot say that the statement is false. Therefore, the conditional
statement is true.
The resulting truth values in Example 3 can be used
to create a truth table for conditional statements. Notice
that a conditional statement is true in all cases except
where the hypothesis is true and the conclusion is false.
p
q
p→q
T
T
T
T
F
F
F
T
T
F
F
T
76 Chapter 2 Reasoning and Proof
Teacher to Teacher
Nancy Lee Keen
Martinsville High School, Martinsville, IN
To develop the concept of conditional statements, I made posters of each of the
four related conditionals. I wrote the hypotheses on yellow poster board, the
conclusions on blue poster board, and NOT on red poster board. As we
introduced each type of conditional, we placed the posters in the correct order.
CONVERSE, INVERSE, AND CONTRAPOSITIVE Other statements based
on a given conditional statement are known as related conditionals .
Related Conditionals
Statement
Formed by
Symbols
Examples
Conditional
given hypothesis and conclusion
p→q
If two angles have the same measure,
then they are congruent.
Converse
exchanging the hypothesis and
conclusion of the conditional
q→p
If two angles are congruent,
then they have the same measure.
Inverse
negating both the hypothesis and
conclusion of the conditional
p → q
If two angles do not have the same
measure, then they are not congruent.
Contrapositive
negating both the hypothesis and
conclusion of the converse statement
q → p
If two angles are not congruent, then
they do not have the same measure.
Study Tip
Contrapositive
The relationship of the
truth values of a
conditional and its
contrapositive is known as
the Law of Contrapositive.
q
Conditional
p→q
Converse
q→p
Inverse
p → q
Contrapositive
q → p
T
T
T
T
T
T
T
F
F
T
T
F
F
T
T
F
F
T
F
F
T
T
T
T
Concept Check
Write the converse, inverse, and contrapositive of the statement Linear pairs of
angles are supplementary. Determine whether each statement is true or false. If a
statement is false, give a counterexample.
First, write the conditional in if-then form.
Conditional: If two angles form a linear pair, then they are supplementary.
The conditional statement is true.
Write the converse by switching the hypothesis and conclusion of the conditional.
Converse: If two angles are supplementary, then they form a linear
pair. The converse is false. ABC and PQR are
C
supplementary, but are not a linear pair.
If two angles do not form a linear pair,
then they are not supplementary. The
A
inverse is false. ABC and PQR do not
form a linear pair, but they are supplementary.
P
110°
B
70°
Q
R
The contrapositive is the negation of the hypothesis and conclusion of the converse.
Contrapositive: If two angles are not supplementary, then they do not form a
linear pair. The contrapositive is true.
www.geometryonline.com/extra_examples
Power
Point®
and contrapositive of the
statement All squares are
rectangles. Determine whether
each statement is true or false.
If a statement is false, give a
counterexample.
Conditional: If a shape is a
square, then it is a rectangle.
Converse: If a shape is a
rectangle, then it is a square.
False; a rectangle with 2 and
w 4 is not a square. Inverse: If
a shape is not a square, then it is
not a rectangle. False; a 4-sided
polygon with side lengths 2, 2, 4,
and 4 is not a square.
Contrapositive: If a shape is not
a rectangle, then it is not a
square. true
Example 4 Related Conditionals
Inverse:
In-Class Examples
4 Write the converse, inverse,
If a given conditional is true, the converse and inverse are not necessarily
true. However, the contrapositive of a true conditional is always true, and the
contrapositive of a false conditional is always false. Likewise, the converse and
inverse of a conditional are either both true or both false.
Statements with the same truth values are said to be logically equivalent . So, a
conditional and its contrapositive are logically equivalent as are the converse and
inverse of a conditional. These relationships are summarized below.
p
CONVERSE, INVERSE, AND
CONTRAPOSITIVE
In Lesson 2-2, p and q
represented simple statements,
not necessarily related to one
another. In this lesson, they
become the hypothesis and
conclusion of a conditional
statement. Make sure students
know that separately, p and q are
still simple statements, but they
now have an interdependent
relationship. Before moving on,
students should feel very
comfortable identifying the
hypothesis and conclusion,
determining the truth value of
each one separately, and
determining their combined
truth value in various forms of
conditional statements.
Lesson 2-3 Conditional Statements 77
Differentiated Instruction
Kinesthetic Provide index cards for each student labeled “Hypothesis,”
“Conclusion,” and “Implies” (or an arrow pointing to the right). Give each
student two cards labeled “Not” in red ink. Ask students to use the cards
to form a conditional, a converse, an inverse, and a contrapositive.
Students should respond by placing the cards in the correct position and
order to reflect the requests. Students can also use the cards to work
some examples or exercises in this lesson by writing the parts of
conditional statements on corresponding cards.
Lesson 2-3 Conditional Statements 77
3 Practice/Apply
Concept Check
1 – 3. See margin.
Study Notebook
Have students—
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 2.
• include a simplified version of the
Related Conditionals chart and
the truth table on page 77.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4–6
7–9, 15
10–12
13, 14
1
2
3
4
7. If a pitcher is a
32-ounce pitcher,
then it holds a quart
of liquid.
8. If two angles are
supplementary, then
the sum of the
measures of the
angles is 180.
9. If an angle is formed
by perpendicular lines,
then it is a right angle.
Organization by Objective
• If-Then Statements: 16–39
• Converse, Inverse, and
Contrapositive: 40–45
Odd/Even Assignments
Exercises 16–45 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Assignment Guide
Basic: 17–47 odd, 48–68
Average: 17–47 odd, 48–68
(optional: 66–68)
All: Quiz 1 (1–5)
1. Writing a conditional in if-then
form is helpful so that the hypothesis and conclusion are easily
recognizable.
2. Sample answer: If you eat your
peas, then you will have dessert.
3. In the inverse, you negate both
the hypothesis and the conclusion
of the conditional. In the contrapositive, you negate the hypothesis
and the conclusion of the converse.
4. H: it rains on Monday; C: I will
stay home
5. H: x 3 7; C: x 10
78
Chapter 2 Reasoning and Proof
Application
1. Explain why writing a conditional statement in if-then form is helpful.
2. OPEN ENDED Write an example of a conditional statement.
3. Compare and contrast the inverse and contrapositive of a conditional.
Identify the hypothesis and conclusion of each statement. 4– 6. See margin.
4. If it rains on Monday, then I will stay home.
5. If x 3 7, then x 10.
6. If a polygon has six sides, then it is a hexagon.
Write each statement in if-then form.
7. A 32-ounce pitcher holds a quart of liquid.
8. The sum of the measures of supplementary angles is 180.
9. An angle formed by perpendicular lines is a right angle.
Determine the truth value of the following statement for each set of conditions.
If you drive faster than 65 miles per hour on the interstate, then you will receive
a speeding ticket.
10. You drive 70 miles per hour, and you receive a speeding ticket. true
11. You drive 62 miles per hour, and you do not receive a speeding ticket. true
12. You drive 68 miles per hour, and you do not receive a speeding ticket. false
Write the converse, inverse, and contrapositive of each conditional statement.
Determine whether each related conditional is true or false. If a statement is false,
find a counterexample. 13 –14. See margin.
13. If plants have water, then they will grow.
14. Flying in an airplane is safer than riding in a car.
15. FORESTRY In different regions of the country, different variations of trees
dominate the landscape. In Colorado, aspen trees cover high areas of the
mountains. In Florida, cypress trees rise from swamps. In Vermont, maple trees
are prevalent. Write these conditionals in if-then form. See p. 123B.
Practice and Apply
For
Exercises
See
Examples
16–21
22–27
28–39
40–45
1
2
3
4
Extra Practice
See page 756.
Identify the hypothesis and conclusion of each statement. 16 – 21. See p. 123B.
16. If 2x 6 10, then x 2.
17. If you are a teenager, then you are at least 13 years old.
18. If you have a driver’s license, then you are at least 16 years old.
19. If three points lie on a line, then they are collinear.
20. “If a man hasn’t discovered something that he will die for, he isn’t fit to live.”
(Martin Luther King, Jr., 1963)
21. If the measure of an angle is between 0 and 90, then the angle is acute.
Write each statement in if-then form. 22–27. See p. 123B.
22. Get a free visit with a one-year fitness plan.
23. Math teachers love to solve problems.
24. “I think, therefore I am.” (Descartes)
25. Adjacent angles have a common side.
26. Vertical angles are congruent.
27. Equiangular triangles are equilateral.
78 Chapter 2 Reasoning and Proof
6. H: a polygon has six sides; C: it is
a hexagon
13. Converse: If plants grow, then they
have water; true. Inverse: If plants
do not have water, then they will
not grow; true. Contrapositive: If
plants do not grow, then they do
not have water. False; they may
have been killed by overwatering.
14. Converse: If you are safer than riding in a car,
then you are flying in an airplane. False; there
are other places that are safer than riding in a
car. Inverse: If you are not flying in an airplane,
then you are not safer than riding in a car. False;
there are other places that are safer than riding
in a car. Contrapositive: If you are not safer than
riding in a car, then you are not flying in an
airplane; true.
Determine the truth value of the following statement for each set of conditions.
If you are over 18 years old, then you vote in all elections.
28. You are 19 years old and you vote. true
29. You are 16 years old and you vote. true
NAME ______________________________________________ DATE
p. 69
(shown)
and p. 70
Conditional
Statements
If-then Statements
An if-then statement is a statement such as “If you are reading
this page, then you are studying math.” A statement that can be written in if-then form is
called a conditional statement. The phrase immediately following the word if is the
hypothesis. The phrase immediately following the word then is the conclusion.
A conditional statement can be represented in symbols as p → q, which is read “p implies q”
or “if p, then q.”
30. You are 21 years old and do not vote. false
Example 1
31. You are 17 years old and do not vote. true
Identify the hypothesis and conclusion of the statement.
If X R and R S, then X S.
hypothesis
conclusion
Example 2
Identify the hypothesis and conclusion.
Write the statement in if-then form.
You receive a free pizza with 12 coupons.
33. Your dad is 45 years old and does not vote. false
If you have 12 coupons, then you receive a free pizza.
hypothesis
Identify the hypothesis and conclusion of each statement.
2. If x 8 32, then x 40. H: x 8 32; C: x 40
M
A
3. If a polygon has four right angles, then the polygon is a rectangle.
H: a polygon has four right angles; C: the polygon is a rectangle
Write each statement in if-then form.
Lesson 2-3
1. If it is Saturday, then there is no school. H: it is Saturday; C: there is no school
4. All apes love bananas.
If an animal is an ape, then it loves bananas.
5. The sum of the measures of complementary angles is 90. If two angles
36. Q lies in plane M . true
are complementary, then the sum of their measures is 90.
P
37. P, Q, A, and B are coplanar. false
38. AP contains Q. false
Q R
6. Collinear points lie on the same line.
If points are collinear, then they lie on the same line.
B
N
39. Planes M and N intersect at RQ . true
Determine the truth value of the following statement for each set of conditions.
If it does not rain this Saturday, we will have a picnic.
7. It rains this Saturday, and we have a picnic. true
8. It rains this Saturday, and we don’t have a picnic. true
9. It doesn’t rain this Saturday, and we have a picnic. true
10. It doesn’t rain this Saturday, and we don’t have a picnic. false
Write the converse, inverse, and contrapositive of each conditional statement.
Determine whether each related conditional is true or false. If a statement is false,
find a counterexample. 40–45. See p. 123B.
40. If you live in Dallas, then you live in Texas.
41. If you exercise regularly, then you are in good shape.
Gl
NAME
______________________________________________
DATE
/M
G
Hill
69
Skills
Practice,
2-3
Practice
(Average)
p. 71 and
Practice,
72 (shown)
Conditionalp.
Statements
____________
Gl PERIOD
G _____
Identify the hypothesis and conclusion of each statement.
1. If 3x 4 5, then x 3.
H: 3x 4 5; C: x 3
2. If you take a class in television broadcasting, then you will film a sporting event.
H: you take a class in television broadcasting;
C: you will film a sporting event
42. The sum of two complementary angles is 90.
Write each statement in if-then form.
3. “Those who do not remember the past are condemned to repeat it.” (George Santayana)
If you do not remember the past, then you are condemned to repeat it.
44. All right angles measure 90.
4. Adjacent angles share a common vertex and a common side.
45. Acute angles have measures less than 90.
If two angles are adjacent, then they share a common vertex and a
common side.
SEASONS For Exercises 46 and 47, use the following information.
Due to the movement of Earth around the sun, summer days in Alaska have more
hours of daylight than darkness, and winter days have more hours of darkness than
daylight.
46. Write two true conditional statements in if-then form for summer days and
Seasons
conclusion
Exercises
In the figure, P, Q, and R are collinear, P and A lie in
plane M , and Q and B lie in plane N . Determine the
truth value of each statement.
34. P, Q, and R lie in plane M . true
35. QB lies in plane N . true
summer, then there
are more hours of
daylight than darkness. In Alaska, if it is
winter, then there are
more hours of darkness than daylight.
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
2-3
Study
Guide and
At the poles, sunlight may
shine continuously for six
months during spring and
summer, but never rises
more than 23.5° above the
horizon. During the other
six months of the year, the
poles are in darkness.
47. Write the converse of the two true conditional statements. State whether each is
true or false. If a statement is false, find a counterexample. See p. 123B.
Source: U.S. Geological Survey
49. WRITING IN MATH
Determine the truth value of the following statement for each set of conditions.
If DVD players are on sale for less than \$100, then you buy one.
5. DVD players are on sale for \$95 and you buy one. true
6. DVD players are on sale for \$100 and you do not buy one. true
7. DVD players are not on sale for under \$100 and you do not buy one. true
8. Write the converse, inverse, and contrapositive of the conditional statement. Determine
whether each statement is true or false. If a statement is false, find a counterexample.
If (8) 2 0, then 8 0.
Converse: If 8 0, then (8)2 0; true.
Inverse: If (8)2 0, then 8 0; true.
Contrapositive: If 8 0, then (8)2 0; false.
SUMMER CAMP For Exercises 9 and 10, use the following information.
Older campers who attend Woodland Falls Camp are expected to work. Campers who are
juniors wait on tables.
9. Write a conditional statement in if-then form.
48. CRITICAL THINKING Write a false conditional statement. Is it possible to insert
the word not into your conditional to make it true? If so, write the true
conditional.
Answer the question that was posed at the beginning
of the lesson. See margin.
Sample answer: If you are a junior, then you wait on tables.
10. Write the converse of your conditional statement.
If you wait on tables, then you are a junior.
Gl
NAME
______________________________________________
DATE
/M
G
Hill
72
____________
Gl PERIOD
G _____
2-3
to Learn
Learn Mathematics
Mathematics,
p. 73
Conditional Statements
Pre-Activity
ELL
Read the introduction to Lesson 2-3 at the top of page 75 in your textbook.
Does the second advertising statement in the introduction mean that you
will not get a free phone if you sign a contract for only six months of
service? Explain your answer. No; it only tells you what happens if
• an example of a conditional statement in if-then form, and
• an example of a conditional statement that is not in if-then form.
1. Identify the hypothesis and conclusion of each statement.
a. If you are a registered voter, then you are at least 18 years old. Hypothesis: you
are a registered voter; Conclusion: you are at least 18 years old
b. If two integers are even, their product is even. Hypothesis: two integers are
even; Conclusion: their product is even
Lesson 2-3 Conditional Statements 79
Galen Rowell/CORBIS
49. Conditional statements can be used to
describe how to get a discount, rebate, or
refund. Sample answers should include the
following.
• If you are not 100% satisfied, then return
the product for a full refund.
• Wearing a seatbelt reduces the risk of
injuries.
NAME ______________________________________________ DATE
2-3
Enrichment
Enrichment,
____________ PERIOD _____
p. 74
2. Complete each sentence.
a. The statement that is formed by replacing both the hypothesis and the conclusion of a
inverse
conditional with their negations is the
.
b. The statement that is formed by exchanging the hypothesis and conclusion of a
converse
conditional is the
.
3. Consider the following statement:
You live in North America if you live in the United States.
a. Write this conditional statement in if-then form and give its truth value. If the
statement is false, give a counterexample. If you live in the United States, then
you live in North America; false: You live in Hawaii.
b. Write the inverse of the given conditional statement in if-then form and give its truth
value. If the statement is false, give a counterexample. If you do not live in the
United States, then you do not live in North America; false; sample
Venn Diagrams
A type of drawing called a Venn diagram can be useful in explaining conditional
statements. A Venn diagram uses circles to represent sets of objects.
c. Write the contrapositive of the given conditional statement in if-then form and give
its truth value. If the statement is false, give a counterexample. If you do not live
Consider the statement “All rabbits have long ears.” To make a Venn diagram for this
statement, a large circle is drawn to represent all animals with long ears. Then a
smaller circle is drawn inside the first to represent all rabbits. The Venn diagram
shows that every rabbit is included in the group of long-eared animals.
d. Write the converse of the given conditional statement in if-then form and give its
truth value. If the statement is false, give a counterexample. If you live in North
animals with
long ears
rabbits
The set of rabbits is called a subset of the set
of long-eared animals.
The Venn diagram can also explain how to write the
statement, “All rabbits have long ears,” in if-then form. Every
rabbit is in the group of long-eared animals, so if an animal is
a rabbit, then it has long ears.
in North America, then you do not live in the United States; false: You
live in Hawaii.
America, then you live in the United States; false; sample answer: You
Helping You Remember
4. When working with a conditional statement and its three related conditionals, what is
an easy way to remember which statements are logically equivalent to each other?
Sample answer: The two statements whose names contain verse (the
converse and the inverse) are a logically equivalent pair. The other two
(the original conditional and the contrapositive) are the other logically
equivalent pair.
For each statement, draw a Venn diagram. Then write the sentence in if-then form.
1 Every dog has long hair
2 All rational numbers are real
Lesson 2-3 Conditional Statements 79
Lesson 2-3
www.geometryonline.com/self_check_quiz
Standardized
Test Practice
4 Assess
50. Which statement has the same truth value as the following statement? C
If Ava and Willow are classmates, then they go to the same school.
A If Ava and Willow go to the same school, then they are classmates.
B If Ava and Willow are not classmates, then they do not go to the
same school.
C If Ava and Willow do not go to the same school, then they are
not classmates.
D If Ava and Willow go to the same school, then they are not classmates.
Open-Ended Assessment
Speaking Students can practice
their speaking skills by identifying
parts of statements and translating
statements into the converse,
inverse, and contrapositive aloud.
51. ALGEBRA In a history class with 32 students, the ratio of girls to boys is 5 to 3.
How many more girls are there than boys? B
A 2
B 8
C 12
D 20
Lesson 2-4
Mixed Review
Prerequisite Skill Students will
in Lesson 2-4. They will apply
concepts of solving equations to
deductive-reasoning techniques.
Use Exercises 66–68 to determine
solving equations.
Use the following statements to write a compound statement for each conjunction
and disjunction. Then find its truth value. (Lesson 2-2) 52–57. See margin.
p: George Washington was the first president of the United States.
q: A hexagon has five sides.
r: 60 ⴛ 3 ⴝ 18
53. q r false
54. p q true
52. p q false
55. ~q r true
56. p ~q true
57. ~p ~r false
Make a conjecture based on the given information. Draw a figure to illustrate
your conjecture. (Lesson 2-1) 58–61. See margin for sample figures.
58. ABCD is a rectangle.
59. In FGH, mF 45, mG 67, mH 68.
60. J(3, 2), K(1, 8), L(5, 2) 61. In PQR, mPQR 90 PQR is a right angle.
59. The sum of the
measures of the
angles in a triangle
is 180.
60. JKL has two
sides congruent.
Assessment Options
Practice Quiz 1 The quiz
provides students with a brief
review of the concepts and skills
in Lessons 2-1 through 2-3.
Lesson numbers are given to the
right of the exercises or
instruction lines so students can
review concepts not yet
mastered.
Use the Distance Formula to find the distance between each pair of points.
(Lesson 1-3)
62. C(2, 1), D(0, 3) 20
4.5
64. P(3, 1), Q(2, 3) 29
5.4
the Next Lesson
63. J(3, 5), K(1, 0) 41
6.4
65. R(1, 7), S(4, 3) 125
11.2
PREREQUISITE SKILL Identify the operation used to change Equation (1) to
Equation (2). (To review solving equations, see pages 737 and 738.) 66–68. See margin.
1
66. (1) 3x 4 5x 8
67. (1) (a 5) 12
68. (1) 8p 24
2
(2) 3x 5x 12
(2) a 5 24
(2) p 3
P ractice Quiz 1
52. George Washington was the first
president of the United States and
a hexagon has 5 sides.
53. A hexagon has five sides or
60 3 18.
54. George Washington was the first
president of the United States or a
hexagon has five sides.
55. A hexagon doesn’t have five sides
or 60 3 18.
56. George Washington was the first
president of the United States and
a hexagon doesn’t have five sides.
57. George Washington was not the
first president of the United States
and 60 3 18.
58. D
C
Determine whether each conjecture is true or false. Give a counterexample for any false conjecture.
Lessons 2-1 through 2-3
(Lesson 2-1)
1. Given: WX XY
Conjecture: W, X, and Y are collinear.
1 and 2 are complementary.
2 and 3 are complementary.
Conjecture: m1 m3 true
2. Given:
False; see p. 123B for counterexample.
Construct a truth table for each compound statement. (Lesson 2-2) 3–4. See p. 123B.
3. ~p q
4. p (q r)
5. Write the converse, inverse, and contrapositive of the following conditional
statement. Determine whether each related conditional is true or false. If a
statement is false, find a counterexample. (Lesson 2-3)
If two angles are adjacent, then the angles have a common vertex. See p. 123C.
80
Chapter 2 Reasoning and Proof
59.
G
60.
F
61.
y K
67
P
45
68
Q
H
A
B
L
J
O
80 Chapter 2 Reasoning and Proof
x
R
66. Subtract 4 from each side.
67. Multiply each side by 2.
68. Divide each side by 8.
Mathematics
Biconditional Statements
Getting Started
Ashley began a new summer job, earning \$10 an hour. If she works over 40 hours
a week, she earns time and a half, or \$15 an hour. If she earns \$15 an hour, she has
worked over 40 hours a week.
Explain that true biconditional
for writing proofs because they
can be used forwards or
backwards. Tell students,
however, that they are not as
common as regular conditionals,
and students will need to be
wary and thorough when they
are determining whether a
biconditional is true or false.
p: Ashley earns \$15 an hour
q: Ashley works over 40 hours a week
p → q:
q → p:
If Ashley earns \$15 an hour, she has worked over 40 hours a week.
If Ashley works over 40 hours a week, she earns \$15 an hour.
In this case, both the conditional and its converse are true. The conjunction of the
two statements is called a biconditional .
Biconditional Statement
• Words
A biconditional statement is the conjunction of a conditional
and its converse.
• Symbols
(p → q) (q → p) is written (p ↔ q) and read p if and only if q.
Teach
If and only if can be abbreviated iff.
Biconditional Statements
Students can also get some extra
practice writing the inverse and
contrapositive of each
biconditional statement.
So, the biconditional statement is as follows.
p ↔ q: Ashley earns \$15 an hour if and only if she works over 40 hours a week.
Examples
Write each biconditional as a conditional and its converse. Then determine
whether the biconditional is true or false. If false, give a counterexample.
a. Two angle measures are complements if and only if their sum is 90.
Conditional: If two angle measures are complements, then their sum is 90.
Converse:
If the sum of two angle measures is 90, then they are complements.
Both the conditional and the converse are true, so the biconditional is true.
Assess
b. x 9 iff x 0
Study Notebook
Conditional: If x 9, then x 0.
Converse:
If x 0, then x 9.
The conditional is true, but the converse is not. Let x 2. Then 2 0 but 2 ⬎ 9.
So, the biconditional is false.
biconditional statements.
Reading to Learn 1 –5. See margin.
Write each biconditional as a conditional and its converse. Then determine
whether the biconditional is true or false. If false, give a counterexample.
1. A calculator will run if and only if it has batteries.
2. Two lines intersect if and only if they are not vertical.
3. Two angles are congruent if and only if they have the same measure.
4. 3x 4 20 iff x 7.
5. A line is a segment bisector if and only if it intersects the segment at its midpoint.
ELL English Language
Learners may benefit from
writing key concepts from this
activity in their Study Notebooks
in their native language and then
in English.
1. Conditional: If a calculator runs, then it has
batteries. Converse: If a calculator has
batteries, then it will run. False; a calculator
may be solar powered.
2. Conditional: If two lines intersect, then they
are not vertical. Converse: If two lines are
not vertical, then they intersect. False; two
parallel horizontal lines will not intersect.
3. Conditional: If two angles are congruent,
then they have the same measure.
Converse: If two angles have the same
measure, then they are congruent. true
4. Conditional: If 3x 4 20, then x 7.
Converse: If x 7, then 3x 4 20.
False; 3x 4 17 when x 7.
5. Conditional: If a line is a segment
bisector, then it intersects the
segment at its midpoint. Converse:
If a line intersects a segment at
its midpoint, then it is a segment
bisector. true
Lesson
Notes
1 Focus
5-Minute Check
Transparency 2-4 Use as a
quiz or review of Lesson 2-3.
Deductive Reasoning
• Use the Law of Detachment.
• Use the Law of Syllogism.
Mathematical Background notes
are available for this lesson on
p. 60D.
does deductive
reasoning apply
to health?
• If you have a mass of 57.8 kg,
what dose will a doctor give
you based on the chart? 350 mg
• What might happen if a patient
used inductive reasoning to form
a conjecture about the dose of
an antidepressant based on the
chart above? Is this a safe
method for health situations?
Sample answer: The dose of the
antidepressant might be much less
than that of the antibiotic for the
same weight, so the patient could
potentially overdose on the
antidepressant; no.
does deductive reasoning
apply to health?
Vocabulary
• deductive reasoning
• Law of Detachment
• Law of Syllogism
Weigh
t
(kg)
10–20
20–30
30–40
40–50
50–60
60–70
When you are ill, your doctor may prescribe an
a dose chart like the one shown to determine the
correct amount of medicine you should take.
Dose
(mg)
150
200
250
300
350
400
LAW OF DETACHMENT The process that doctors use to determine the
amount of medicine a patient should take is called deductive reasoning . Unlike
inductive reasoning, which uses examples to make a conjecture, deductive reasoning
uses facts, rules, definitions, or properties to reach logical conclusions.
A form of deductive reasoning that is used to draw conclusions from true
conditional statements is called the Law of Detachment .
Law of Detachment
• Words
If p → q is true and p is true, then q is also true.
• Symbols [(p → q) p] → q
Study Tip
Validity
When you apply the Law
of Detachment, make sure
that the conditional is true
before you test the validity
of the conclusion.
Example 1 Determine Valid Conclusions
The following is a true conditional. Determine whether each conclusion is valid
based on the given information. Explain your reasoning.
If a ray is an angle bisector, then it divides the angle
into two congruent angles.
bisects ABC.
a. Given: BD
Conclusion: ABD CBD
A
D
B
is the bisector
The hypothesis states that BD
of ABC. Since the conditional is true and
the hypothesis is true, the conclusion is valid.
C
b. Given: PQT RQS
P
and QT
are angle bisectors.
Conclusion: QS
Knowing that a conditional statement and its conclusion
are true does not make the hypothesis true. An angle
bisector divides an angle into two separate congruent
angles. In this case, the given angles are not separated by
one ray. Instead, they overlap. The conclusion is not valid.
82
Chapter 2 Reasoning and Proof
Resource Manager
Workbook and Reproducible Masters
Chapter 2 Resource Masters
• Study Guide and Intervention, pp. 75–76
• Skills Practice, p. 77
• Practice, p. 78
• Reading to Learn Mathematics, p. 79
• Enrichment, p. 80
• Assessment, pp. 119, 121
School-to-Career Masters, p. 3
Teaching Geometry With Manipulatives
Masters, p. 47
Transparencies
5-Minute Check Transparency 2-4
Technology
Interactive Chalkboard
Q
S
T
R
LAW OF SYLLOGISM Another law of logic is the Law of Syllogism. It is
similar to the Transitive Property of Equality.
Law of Syllogism
• Words
If p → q and q → r are true, then p → r is also true.
2 Teach
LAW OF DETACHMENT
In-Class Example
• Symbols [(p → q) (q → r)] → (p → r)
Power
Point®
1 The following is a true
Study Tip
Conditional
Statements
Label the hypotheses
and conclusions of a
series of statements
before applying the
Law of Syllogism.
Example 2 Determine Valid Conclusions From Two Conditionals
CHEMISTRY Use the Law of Syllogism to determine whether a valid conclusion
can be reached from each set of statements.
a. (1) If the symbol of a substance is Pb, then it is lead.
(2) The atomic number of lead is 82.
Let p, q, and r represent the parts of the statement.
p: the symbol of a substance is Pb
r: the atomic number is 82
Statement (1): p → q
Statement (2): q → r
Since the given statements are true, use the Law of Syllogism to conclude p → r.
That is, If the symbol of a substance is Pb, then its atomic number is 82.
b. (1) Water can be represented by H2O.
(2) Hydrogen (H) and oxygen (O) are in the atmosphere.
There is no valid conclusion. While both statements are true, the conclusion of
each statement is not used as the hypothesis of the other.
conditional. Determine
whether each conclusion is
valid based on the given
information. Explain your
reasoning.
If two segments are congruent
and the second segment is
congruent to a third segment,
then the first segment is also
congruent to the third segment.
X
UV
; U
V
RT
a. Given: W
Conclusion: WX
RT
true
b. Given: UV
; W
X
RT
Conclusion: WX
UV
and
V
RT
false
U
LAW OF SYLLOGISM
In-Class Example
Example 3 Analyze Conclusions
Determine whether statement (3) follows from statements (1) and (2) by the Law
of Detachment or the Law of Syllogism. If it does, state which law was used. If
it does not, write invalid.
a. (1) Vertical angles are congruent.
(2) If two angles are congruent, then their measures are equal.
(3) If two angles are vertical, then their measures are equal.
p: two angles are vertical
q: they are congruent
r: their measures are equal
Statement (3) is a valid conclusion by the Law of Syllogism.
b. (1) If a figure is a square, then it is a polygon.
(2) Figure A is a polygon.
(3) Figure A is a square.
Statement (1) is true, but statement (3) does not follow from statement (2).
Not all polygons are squares.
Statement (3) is invalid.
www.geometryonline.com/extra_examples
Lesson 2-4 Deductive Reasoning 83
Differentiated Instruction
Power
Point®
2 PROM Use the Law of
Syllogism to determine
whether a valid conclusion
can be reached from each set
of statements.
a. (1) If Salline attends the prom,
she will go with Mark.
(2) Mark is a 17-year-old
student. not valid
b. (1) If Mel and his date eat at
the Peddler Steakhouse before
going to the prom, they will
miss the senior march.
(2) The Peddler Steakhouse
stays open until 10 P.M.
not valid
In-Class Example 3 is on p. 84.
ELL
Verbal/Linguistic Have students write a paragraph to explain and
provide an example for the Law of Detachment. Repeat for the Law of
Syllogism. Then students can write another paragraph to point out
similarities and differences between the two laws. They can place their
written explanations in their study notebooks.
Lesson 2-4 Deductive Reasoning 83
In-Class Example
Power
Point®
3 Determine whether statement
Concept Check
(3) follows from statements
(1) and (2) by the Law of
Detachment or the Law of
Syllogism. If it does, state
which law was used. If it
does not, write invalid.
a. (1) If the sum of the squares of
two sides of a triangle is equal
to the square of the third side,
then the triangle is a right
triangle. (2) For XYZ,
(XY)2 (YZ)2 (ZX)2.
(3) XYZ is a right triangle.
Law of Detachment
b. (1) If Ling wants to participate
in the wrestling competition,
he will have to meet an extra
three times a week to practice.
extra to his weekly schedule,
he cannot take karate lessons.
(3) If Ling wants to participate
in the wrestling competition,
he cannot take karate lessons.
Law of Syllogism
1. OPEN ENDED Write an example to illustrate the correct use of
the Law of Detachment. 1–3. See margin.
2. Explain how the Transitive Property of Equality is similar to the
Law of Syllogism.
3. FIND THE ERROR An article in a magazine states that if you get seasick, then
you will get dizzy. It also says that if you get seasick, you will get an upset
stomach. Suzanne says that this means that if you get dizzy, then you will get
an upset stomach. Lakeisha says that she is wrong. Who is correct? Explain.
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4, 5
6, 7
8, 9
1
2
3
Determine whether the stated conclusion is valid based on the given information.
If not, write invalid. Explain your reasoning.
If two angles are vertical angles, then they are congruent.
4. Given: A and B are vertical angles.
Conclusion: A B valid
5. Given: C D
Conclusion: C and D are vertical angles. Invalid; congruent angles
do not have to be vertical.
Use the Law of Syllogism to determine whether a valid conclusion can be reached
from each set of statements. If a valid conclusion is possible, write it. If not, write
no conclusion.
6. If you are 18 years old, you are in college.
You are in college. no conclusion
7. The midpoint of a
segment divides it into
two segments with
equal measures.
7. The midpoint divides a segment into two congruent segments.
If two segments are congruent, then their measures are equal.
Determine whether statement (3) follows from statements (1) and (2) by the Law
of Detachment or the Law of Syllogism. If it does, state which law was used. If it
does not, write invalid. 8. valid; Law of Syllogism
8. (1) If Molly arrives at school at 7:30 A.M., she will get help in math.
(2) If Molly gets help in math, then she will pass her math test.
(3) If Molly arrives at school at 7:30 A.M., then she will pass her math test.
9. (1) Right angles are congruent.
(2) X Y
(3) X and Y are right angles. invalid
3 Practice/Apply
Application
Study Notebook
Have students—
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 2.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
INSURANCE For Exercises 10 and 11, use the following information.
An insurance company advertised the following monthly rates for life insurance.
If you are a:
Coverage
Coverage
Female, age 35
\$14.35
\$19.00
Male, age 35
\$16.50
\$21.63
Female, age 45
\$21.63
\$25.85
Male, age 45
\$23.75
\$28.90
10. If Ann is 35 years old and she wants to purchase \$30,000 of insurance from this
company, then what is her premium? \$14.35
11. If Terry paid \$21.63 for life insurance, can you conclude that Terry is 35?
Explain. No; Terry could be a man or a woman. She could be 45 and
have purchased \$30,000 of life insurance.
84 Chapter 2 Reasoning and Proof
FIND THE ERROR
Explain that if
you have p → q and
p → r, you cannot assume q → r,
as Suzanne tries to do. Tell
students that the same hypothesis
can imply two different
conclusions, but the conclusions
are not related to one another.
84
Chapter 2 Reasoning and Proof
1. Sample answer: a: If it is rainy, the game will be cancelled. b: It
is rainy. c: The game will be cancelled.
2. Transitive Property of Equality:
a b and b c implies a c. Law of Syllogism: a implies b
and b implies c implies a implies c. Each statement establishes
a relationship between a and c through their relationships to b.
3. Lakeisha; if you are dizzy, that does not necessarily mean that
you are seasick and thus have an upset stomach.
Practice and Apply
For
Exercises
See
Examples
12–19
20–23
24–29
1
2
3
Extra Practice
See page 757.
13. Valid; since 5 and
7 are odd, the Law of
Detachment indicates
that their sum is even.
14. Valid; since 11
and 23 are odd, the
Law of Detachment
indicates that their
sum is even.
16. Valid; A, B, and C
are noncollinear, and
by definition three
noncollinear points
determine a plane.
17. Invalid; E, F, and
G are not necessarily
noncollinear.
18. Invalid; the
hypothesis is false as
there are only two
points.
19. Valid; the vertices
of a triangle are noncollinear, and
therefore determine
a plane.
21. If the measure of
an angle is less than
90, then it is not
obtuse.
22. If X is the midpoint of YZ, then
YX XZ.
For Exercises 12–19, determine whether the stated conclusion is valid based on the
given information. If not, write invalid. Explain your reasoning.
If two numbers are odd, then their sum is even.
12. Given: The sum of two numbers is 22.
Conclusion: The two numbers are odd. invalid; 10 12 22
13. Given: The numbers are 5 and 7.
Conclusion: The sum is even.
14. Given: 11 and 23 are added together.
Conclusion: The sum of 11 and 23 is even.
15. Given: The numbers are 2 and 6.
Conclusion: The sum is odd. Invalid; the sum is even.
If three points are noncollinear, then they determine a plane.
16. Given: A, B, and C are noncollinear.
Conclusion: A, B, and C determine a plane.
Organization by Objective
• Law of Detachment: 12–19
• Law of Syllogism: 20–29
Odd/Even Assignments
Exercises 12–29 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Internet or other research
materials.
Assignment Guide
17. Given: E, F, and G lie in plane M.
Conclusion: E, F, and G are noncollinear.
Basic: 13–31 odd, 32–58
Average: 13–31 odd, 32–58
34–55 (optional: 56–58)
18. Given: P and Q lie on a line.
Conclusion: P and Q determine a plane.
19. Given: XYZ
Conclusion: X, Y, and Z determine a plane.
Use the Law of Syllogism to determine whether a valid conclusion can be reached
from each set of statements. If a valid conclusion is possible, write it. If not, write
no conclusion.
20. If you spend money on it, then it is a business.
If you spend money on it, then it is fun. no conclusion
21. If the measure of an angle is less than 90, then it is acute.
If an angle is acute, then it is not obtuse.
22. If X is the midpoint of segment YZ, then YX XZ.
If the measures of two segments are equal, then they are congruent.
23. If two lines intersect to form a right angle, then they are perpendicular.
Lines ᐉ and m are perpendicular. no conclusion
Determine whether statement (3) follows from statements (1) and (2) by the Law
of Detachment or the Law of Syllogism. If it does, state which law was used. If it
does not, write invalid.
24. (1) In-line skaters live dangerously.
(2) If you live dangerously, then you like to dance.
(3) If you are an in-line skater, then you like to dance. yes; Law of Syllogism
25. (1) If the measure of an angle is greater than 90, then it is obtuse.
(2) mABC > 90
(3) ABC is obtuse. yes; Law of Detachment
26. (1) Vertical angles are congruent.
(2) 3 4
(3) 3 and 4 are vertical angles. invalid
27. (1) If an angle is obtuse, then it cannot be acute.
(2) A is obtuse.
(3) A cannot be acute. yes; Law of Detachment
Lesson 2-4 Deductive Reasoning 85
Lesson 2-4 Deductive Reasoning 85
NAME ______________________________________________ DATE
Determine whether statement (3) follows from statements (1) and (2) by the Law
of Detachment or the Law of Syllogism. If it does, state which law was used. If it
does not, write invalid.
28. (1) If you drive safely, then you can avoid accidents.
(2) Tika drives safely.
(3) Tika can avoid accidents. yes; Law of Detachment
29. (1) If you are a customer, then you are always right.
(2) If you are a teenager, then you are always right.
(3) If you are a teenager, then you are a customer. invalid
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
2-4
Study
Guide and
p. 75
(shown)
and p. 76
Deductive
Reasoning
Law of Detachment Deductive reasoning is the process of using facts, rules,
definitions, or properties to reach conclusions. One form of deductive reasoning that draws
conclusions from a true conditional p → q and a true statement p is called the Law of
Detachment.
Law of Detachment
If p → q is true and p is true, then q is true.
Symbols
[(p → q)] p] → q
30. then he could
hear the grating noise
of the fish canneries
Example
The statement If two angles are supplementary to the same angle,
then they are congruent is a true conditional. Determine whether each conclusion
is valid based on the given information. Explain your reasoning.
a. Given: A and C are supplementary to B.
Conclusion: A is congruent to C.
The statement A and C are supplementary to B is
the hypothesis of the conditional. Therefore, by the Law
of Detachment, the conclusion is true.
F
E
G
B
H
A
D
C
J
30. LITERATURE John Steinbeck, a Pulitzer Prize winning author, lived in
Monterey, California, for part of his life. In 1945, he published the book, Cannery
Row, about many of his local working-class heroes from Monterey. If you visited
Cannery Row in Monterey during the 1940s, then you could hear the grating
noise of the fish canneries. Write a valid conclusion to the following hypothesis.
If John Steinbeck lived in Monterey in 1941, . . .
b. Given: A is congruent to C.
Conclusion: A and C are supplementary to B.
The statement A is congruent to C is not the hypothesis
of the conditional, so the Law of Detachment cannot be used.
The conclusion is not valid.
Exercises
Determine whether each conclusion is valid based on the true conditional given.
If not, write invalid. Explain your reasoning.
If two angles are complementary to the same angle, then the angles are congruent.
The given statement is the hypothesis of the conditional statement.
Since the conditional is true, the conclusion A C is true.
2. Given: A C
Conclusion: A and C are complements of B.
The given statement is not the hypothesis of the conditional.
Therefore, the conclusion is invalid.
3. Given: E and F are complementary to G.
Conclusion: E and F are vertical angles.
While the given statement is the hypothesis of the conditional statement,
the statement that E and F are vertical angles is not the conclusion of
the conditional. The conclusion is invalid.
Gl
NAME
______________________________________________
DATE
/M
G
Hill
75
Skills
Practice,
2-4
Practice
(Average)
p. 77 and
Practice,
p. 78 (shown)
Deductive Reasoning
____________
Gl PERIOD
G _____
Determine whether the stated conclusion is valid based on the given information.
If not, write invalid. Explain your reasoning.
If a point is the midpoint of a segment, then it divides the segment into two
congruent segments.
Lesson 2-4
1. Given: A and C are complementary to B.
Conclusion: A is congruent to C.
31. SPORTS In the 2002 Winter Olympics, Canadian speed skater Catriona Le May
Doan won her second Olympic title in 500-meter speed skating. Ms. Doan was
in the last heat for the second round of that race. Use the two true conditional
statements to reach a valid conclusion about Ms. Doan’s 2002 competition.
(1) If Catriona Le May Doan skated her second 500 meters in 37.45 seconds,
then she would beat the time of Germany’s Monique Garbrecht-Enfeldt.
(2) If Ms. Doan beat the time of Monique Garbrecht-Enfeldt, then she would
win the race. If Catriona Le May Doan skated her second 500 meters in
Literature
The Pulitzer Prize is
awarded annually for
outstanding contributions
in the fields of journalism,
literature, drama, and
music.
37.45 seconds, then she would win the race.
Online Research
Data Update Use the Internet or another resource
to find the winning times for other Olympic events. Write statements
using these times that can lead to a valid conclusion. Visit
Source: www.pulitzer.org
.
1. Given: R is the midpoint of QS
Conclusion: QR
RS
Valid; since R is the midpoint of Q
S
, the Law of Detachment indicates
into two congruent segments.
that it divides QS
2. Given: AB
BC
Conclusion: B divides A
C
into two congruent segments.
Invalid; the points A, B, and C may not be collinear, and if they are not,
C
.
then B will not be the midpoint of A
Use the Law of Syllogism to determine whether a valid conclusion can be reached
from each set of statements. If a valid conclusion is possible, write it.
3. If two angles form a linear pair, then the two angles are supplementary.
If two angles are supplementary, then the sum of their measures is 180.
If two angles form a linear pair, then the sum of their measures is 180.
Stacey assumed that
the conditional statement was true.
32. CRITICAL THINKING An advertisement states that “If you like to ski, then
you’ll love Snow Mountain Resort.” Stacey likes to ski, but when she went to
Snow Mountain Resort, she did not like it very much. If you know that Stacey
saw the ad, explain how her reasoning was flawed.
4. If a hurricane is Category 5, then winds are greater than 155 miles per hour.
If winds are greater than 155 miles per hour, then trees, shrubs, and signs are blown down.
33. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson. See p. 123C.
How does deductive reasoning apply to health?
If a hurricane is Category 5, then trees, shrubs, and signs are blown down.
Determine whether statement (3) follows from statements (1) and (2) by the Law
of Detachment or the Law of Syllogism. If it does, state which law was used. If it
does not, write invalid.
5. (1) If a whole number is even, then its square is divisible by 4.
(2) The number I am thinking of is an even whole number.
(3) The square of the number I am thinking of is divisible by 4.
• an explanation of how doctors may use deductive reasoning to prescribe
medicine, and
• an example of a doctor’s uses of deductive reasoning to diagnose an illness,
such as strep throat or chickenpox.
yes; Law of Detachment
6. (1) If the football team wins its homecoming game, then Conrad will attend the school
dance the following Friday.
(2) Conrad attends the school dance on Friday.
(3) The football team won the homecoming game.
invalid
7. BIOLOGY If an organism is a parasite, then it survives by living on or in a host
organism. If a parasite lives in or on a host organism, then it harms its host. What
conclusion can you draw if a virus is a parasite?
If a virus is a parasite, then it harms its host.
Gl
NAME
______________________________________________
DATE
/M
G
Hill
78
____________
Gl PERIOD
G _____
Standardized
Test Practice
2-4
to Learn
Learn Mathematics
Mathematics,
p. 79
Deductive Reasoning
Pre-Activity
ELL
How does deductive reasoning apply to health?
Read the introduction to Lesson 2-4 at the top of page 82 in your textbook.
Suppose a doctor wants to use the dose chart in your textbook to prescribe
an antibiotic, but the only scale in her office gives weights in pounds. How
can she use the fact that 1 kilogram is about 2.2 pounds to determine the
correct dose for a patient? Sample answer: The doctor can divide
the patient’s weight in pounds by 2.2 to find the equivalent
mass in kilograms. She can then use the dose chart.
If s, t, and u are three statements, match each description from the list on the left
with a symbolic statement from the list on the right.
1. negation of t e
a. s u
2. conjunction of s and u g
b. [(s → t) s] → t
3. converse of s → t h
c. s → u
4. disjunction of s and u a
d. u → s
5. Law of Detachment b
e. t
6. contrapositive of s → t j
f. [(u → t) (t → s)] → (u → s)
7. inverse of s → u c
g. s u
8. contrapositive of s → u d
h. t → s
9. Law of Syllogism f
i. t
10. negation of t i
j. t → s
11. Determine whether statement (3) follows from statements (1) and (2) by the Law of
Detachment or the Law of Syllogism. If it does, state which law was used. If it does not,
write invalid.
a. (1) Every square is a parallelogram.
(2) Every parallelogram is a polygon.
(3) Every square is a polygon. yes; Law of Syllogism
b. (1)If two lines that lie in the same plane do not intersect, they are parallel.
(2) Lines and m lie in plane U and do not intersect.
(3) Lines and m are parallel. yes; Law of Detachment
c. (1) Perpendicular lines intersect to form four right angles.
(2) A, B, C, and D are four right angles.
(3) A, B, C, and D are formed by intersecting perpendicular lines. invalid
Helping You Remember
12. A good way to remember something is to explain it to someone else. Suppose that a
classmate is having trouble remembering what the Law of Detachment means?
Sample answer: The word detach means to take something off of another
thing. The Law of Detachment says that when a conditional and its
hypothesis are both true, you can detach the conclusion and feel
confident that it too is a true statement.
86
Chapter 2 Reasoning and Proof
86
34. Based on the following statements, which statement must be true? C
I If Yasahiro is an athlete and he gets paid, then he is a professional athlete.
II Yasahiro is not a professional athlete.
III Yasahiro is an athlete.
A Yasahiro is an athlete and he gets paid.
B Yasahiro is a professional athlete or he gets paid.
C Yasahiro does not get paid.
D Yasahiro is not an athlete.
Chapter 2 Reasoning and Proof
AP/Wide World Photos
NAME ______________________________________________ DATE
2-4
Enrichment
Enrichment,
____________ PERIOD _____
p. 80
Valid and Faulty Arguments
Consider the statements at the right.
What conclusions can you make?
(1) Boots is a cat.
(2) Boots is purring.
(3) A cat purrs if it is happy.
From statements 1 and 3, it is correct to conclude that Boots
purrs if it is happy. However, it is faulty to conclude from only
statements 2 and 3 that Boots is happy. The if-then form of
statement 3 is If a cat is happy, then it purrs.
Advertisers often use faulty logic in subtle ways to help sell
their products. By studying the arguments, you can decide
whether the argument is valid or faulty.
Decide if each argument is valid or faulty.
1. (1) If you buy Tuff Cote luggage, it
will survive airline travel.
(2) Justin buys Tuff Cote luggage.
C
l
’ l
ll
2. (1) If you buy Tuff Cote luggage, it
will survive airline travel.
(2) Justin’s luggage survived airline travel.
C
l
h
ff C
35. ALGEBRA At a restaurant, a diner uses a coupon for 15% off the cost of one
meal. If the diner orders a meal regularly priced at \$16 and leaves a tip of 20%
of the discounted meal, how much does she pay in total? B
A \$15.64
B \$16.32
C \$16.80
D \$18.72
Mixed Review
36. If you try Casa
Fiesta, then you’re
looking for a fast,
37. They are a fast,
easy way to add fun to
38. No; the conclusion
is implied.
KHJ and DHG
JHK and DHK
linear pair
ADVERTISING For Exercises 36–38, use the following information. (Lesson 2-3)
Advertising writers frequently use if-then statements to relay a message and
promote their product. An ad for a type of Mexican food reads, If you’re looking
36. Write the converse of the conditional.
37. What do you think the advertiser wants people to conclude about
Casa Fiesta products?
Construct a truth table for each compound statement. (Lesson 2-2) 39–42. See p. 123C.
39. q r
40. ~p r
41. p (q r)
42. p (~q r)
For Exercises 43–47, refer to the figure at the right. (Lesson 1-5)
43. Which angle is complementary to FDG? HDG
44. Name a pair of vertical angles.
45. Name a pair of angles that are noncongruent
and supplementary.
46. Identify FDH and CDH as congruent, adjacent,
vertical, complementary, supplementary, and/or
a linear pair.
C
C
K? Explain. Yes, slashes
47. Can you assume that D
on the segments indicate that they are congruent.
F
G
H
K
J
Use the Pythagorean Theorem to find the distance between each pair of points.
(Lesson 1-3)
49. C(4, 2), D(2, 6) 10
48. A(1, 5), B(2, 9) 5
50. F(7, 4), G(1, 0)
7.2
52
51. M(5, 0), N(4, 7)
11.4
130
For Exercises 52–55, draw and label a figure for each relationship.
52. FG lies in plane M and contains point H. 52–55. See margin.
53. Lines r and s intersect at point W.
54. Line contains P and Q, but does not contain R.
55. Planes A and B intersect in line n.
the Next Lesson
56–58. See margin.
Quiz (Lessons 2-3 and 2-4) is
available on p. 119 of the Chapter 2
Resource Masters.
Mid-Chapter Test (Lessons 2-1
through 2-4) is available on
p. 121 of the Chapter 2 Resource
Masters.
5
6
M
1
C
N
2
B
www.geometryonline.com/self_check_quiz
Lesson 2-4 Deductive Reasoning 87
52.
M
H
Prerequisite Skill Students will
paragraph proofs in Lesson 2-5.
They will extract information from
figures to test postulates and
write proofs. Use Exercises 56–58
familiarity with information
from figures.
Assessment Options
A
F
Modeling Students can use
shapes of paper to model the Law
of Detachment and the Law of
Syllogism. For example, you
could provide students with two
yellow squares of laminated
construction paper marked with p,
two blue triangles marked with
q, and two red circles marked
with r. Students can physically
arrange the shapes to represent
the symbolic relationships of the
two laws. They could also secure
these shapes in their study
notebooks for a colorful
reminder of these laws.
(Lesson 1-1)
PREREQUISITE SKILL Write what you can assume about the segments or angles
listed for each figure. (To review information from figures, see Lesson 1-5.)
56. A
M, C
M, C
N, B
N
57. 1, 2
58. 4, 5, 6
4
Open-Ended Assessment
Lesson 2-5
D
C
4 Assess
53.
r
W
G
s
54.
Q
P
R
55.
A
n
B
M
AM
, CN
B
N
, AM CM, CN BN, M is
midpoint of A
C
, N is midpoint
of B
C
.
57. Sample answer: 1 and 2 are
complementary, m1 m2 90.
58. Sample answer: 4 and 5 are
supplementary, m4 m5 180, 5 and 6 are supplementary, m5 m6 180,
4 6, m4 m6.
Lesson 2-4 Deductive Reasoning 87
Geometry
Activity
A Follow-Up of Lesson 2-4
A Follow-Up of Lesson 2-4
Matrix Logic
Getting Started
Deductive reasoning can be used in problem-solving situations. One method of
solving problems uses a table. This method is called matrix logic .
You could provide students with
a sheet of three blank logic tables
in which they would fill in the
titles of the rows and columns,
or ask students to use a ruler to
draw the tables as they go along.
Example
GEOLOGY On a recent test, Rashaun
was given five different mineral samples
to identify, along with the chart at right.
Rashaun observed the following.
• Sample C is brown.
• Samples B and E are harder
than glass.
• Samples D and E are red.
Objective Apply deductive
reasoning by using matrix logic
tables to solve problems.
Color
Hardness
Biotite
Mineral
brown or black
softer than glass
Halite
white
softer than glass
Hematite
red
softer than glass
Feldspar
white, pink, or
green
harder than glass
Jaspar
red
harder than glass
Identify each of the samples.
Teach
Make a table to organize the information.
Mark each false condition with an ⫻ and each
true condition with a ⻫. The first observation
is that Sample C is brown. Only one of the
minerals, biotite, is brown, so place a check in
the box that corresponds to biotite and
Sample C. Then place an ⫻ in each of the
other boxes in the same column and row.
• When students are considering
the second observation, tell
them they can think in terms of
boxes to mark out or boxes to
leave open. The only two
minerals that are harder than
glass are feldspar and jasper,
so these boxes are left open in
the columns B and E. Then
they can place an X in the rest
of the boxes in these columns.
• For Exercise 2, advise students
to draw a model with six
apartments. After marking all
obvious information, students
can use the model to place
each person in the correct
apartment.
The second observation is that Samples B and
E are harder than glass. Place an ⫻ in each
box for minerals that are softer than glass.
The third observation is that Samples D and
E are red. Mark the boxes accordingly. Notice
that Sample E has an ⫻ in all but one box.
Place a check mark in the remaining box, and
an ⫻ in all other boxes in that row.
deductive reasoning in problemsolving situations.
88 Chapter 2 Reasoning and Proof
B
C
D
E
⫻
⫻
⫻
⫻
⻫
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
Halite
Hematite
Feldspar
Jaspar
A
B
C
D
E
⫻
⫻
⫻
⫻
⻫
⫻
⫻
⫻
⫻
⫻
⫻
Halite
Hematite
Feldspar
Jaspar
⻫
⫻
⫻
⫻
⻫
⫻
⫻
⫻
⫻
⫻
2. Six friends live in consecutive apartments on the same side of their apartment building.
Anita lives in apartment C. Kelli’s apartment is just past Scott’s. Anita’s closest
neighbors are Eric and Ava. Scott’s apartment is not A through D. Eric’s apartment is
before Ava’s. If Roberto lives in one of the apartments, who lives in which apartment?
A, Roberto; B, Eric; C, Anita; D, Ava; E, Scott; F, Kelli
Chapter 2 Reasoning and Proof
Teaching Geometry with
Manipulatives
• p. 46 (student recording sheet)
⻫
Biotite
1. Nate, John, and Nick just began after-school jobs. One works at a veterinarian’s office,
one at a computer store, and one at a restaurant. Nate buys computer games on the
way to work. Nick is allergic to cat hair. John receives free meals at his job. Who works
at which job? Nate, veterinarian’s office; John, restaurant; Nick, computer store
88
⫻
⫻
Sample
Exercises
Resource Manager
Study Notebook
A
Biotite
Then complete the table. Sample A is Halite, Sample B is Feldspar, Sample C is Biotite,
Sample D is Hematite, and Sample E is Jaspar.
Assess
In Exercises 1 and 2 students
practice their thinking and
deductive reasoning skills by
working more logic tables.
Sample
⻫
⫻
⫻
⻫
Postulates and
Paragraph Proofs
Lesson
Notes
• Identify and use basic postulates about points, lines, and planes.
1 Focus
• Write paragraph proofs.
were postulates used by the founding
fathers of the United States?
Vocabulary
•
•
•
•
•
•
postulate
axiom
theorem
proof
paragraph proof
informal proof
5-Minute Check
Transparency 2-5 Use as a
quiz or review of Lesson 2-4.
U.S. Supreme Court Justice William Douglas stated
“The First Amendment makes confidence in the
common sense of our people and in the maturity of
their judgment the great postulate of our democracy.”
The writers of the constitution assumed that citizens
would act and speak with common sense and maturity.
Some statements in geometry also must be assumed or
accepted as true.
Mathematical Background notes
are available for this lesson on
p. 60D.
POINTS, LINES, AND PLANES In geometry, a postulate, or axiom , is a
statement that describes a fundamental relationship between the basic terms of
geometry. Postulates are accepted as true. The basic ideas about points, lines, and
planes can be stated as postulates.
Postulates
2.1
2.2
Through any two points, there is exactly one line.
Through any three points not on the same line, there is exactly one plane.
Example 1 Points and Lines
COMPUTERS Jessica is setting up a network for her father’s business. There are
five computers in his office. Each computer needs to be connected to every other
computer. How many connections does Jessica need to make?
Study Tip
Explore
There are five computers, and each is connected to four others.
Drawing Diagrams
Plan
Draw a diagram to illustrate the solution.
Solve
Let noncollinear points A, B, C, D, and E represent
the five computers. Connect each point with every
other point. Then, count the number of segments.
When listing segments,
draw all of the segments
from that vertex. Then
move on to the other
vertices until all possible
segments have been
drawn.
Between every two points there is exactly one
segment. So, the connection between computer A
and computer B is the same as the connection
between computer B and computer A. For the five
points, ten segments can be drawn.
were postulates used
by the founding fathers
of the United States?
• How would you interpret the
words of William Douglas?
assumes that people have the
maturity to handle the
responsibilities of democracy.
• Do you think the founding
fathers set up a test to find out
whether or not people do have
common sense and maturity?
that people did.
2 Teach
A
E
B
POINTS, LINES, AND
PLANES
In-Class Example
D
C
Examine A
B
, A
C
, A
D
, A
E
, B
C
, B
D
, BE
, C
D
, C
E
, and D
E
each represent a
connection between two computers. So there will be ten connections
among the five computers.
Power
Point®
1 SNOW CRYSTALS Some snow
crystals are shaped like
regular hexagons. How many
lines must be drawn to
interconnect all vertices of a
hexagonal snow crystal? 15
Lesson 2-5 Postulates and Paragraph Proofs 89
Jeff Hunter/Getty Images
Resource Manager
Workbook and Reproducible Masters
Chapter 2 Resource Masters
• Study Guide and Intervention, pp. 81–82
• Skills Practice, p. 83
• Practice, p. 84
• Reading to Learn Mathematics, p. 85
• Enrichment, p. 86
Teaching Geometry With Manipulatives
Masters, p. 8
Transparencies
5-Minute Check Transparency 2-5
Technology
Interactive Chalkboard
Lesson x-x Lesson Title 89
In-Class Example
There are other postulates that are based on relationships among points, lines,
and planes.
Power
Point®
Teaching Tip
Tell students that
most postulates are very obvious
and make very good sense, but
they do not have a formal proof
behind them. Nonetheless,
students are to accept them as
true and use them to prove
other statements and theorems.
Postulates
2 Determine whether each
2.3
2.4
2.5
A line contains at least two points.
2.6
2.7
If two lines intersect, then their intersection is exactly one point.
A plane contains at least three points not on the same line.
If two points lie in a plane, then the entire line containing those points lies
in that plane.
If two planes intersect, then their intersection is a line.
statement is always, sometimes,
or never true. Explain.
and EF
a. If plane T contains EF
contains point G, then plane
T contains point G. Always;
Postulate 2.5 states that if two
points lie in a plane, then the
entire line containing those
points lies in the plane.
b. For XY, if X lies in plane Q and
Y lies in plane R , then
plane Q intersects plane R .
Sometimes; planes Q and R can
can intersect
be parallel, and XY
both planes.
c. GH contains three
noncollinear points. Never;
noncollinear points do not lie on
the same line by definition.
Building on Prior
Knowledge
Students learned basic principles
about points, lines, and planes in
Chapter 1. In this lesson, they will
revisit those concepts in the form
of postulates that they can use to
write informal proofs and
paragraph proofs.
1. Deductive reasoning is used to
support claims that are made in a
proof.
2.
Example 2 Use Postulates
Determine whether each statement is always, sometimes, or never true. Explain.
a. If points A, B, and C lie in plane M, then they are collinear.
Sometimes; A, B, and C do not necessarily have to be collinear to lie in plane M.
b. There is exactly one plane that contains noncollinear points P, Q, and R.
Always; Postulate 2.2 states that through any three noncollinear points, there is
exactly one plane.
c. There are at least two lines through points M and N.
Never; Postulate 2.1 states that through any two points, there is exactly one line.
PARAGRAPH PROOFS Undefined terms, definitions, postulates, and algebraic
properties of equality are used to prove that other statements or conjectures are true.
Once a statement or conjecture has been shown to be true, it is called a theorem, and
it can be used like a definition or postulate to justify that other statements are true.
Study Tip
Proofs
Before writing a proof,
you should have a plan.
One strategy is to work
you want to prove, and
work backward step by
step until you reach the
given information.
You will study and use various methods to verify or prove statements and
conjectures in geometry. A proof is a logical argument in which each statement you
make is supported by a statement that is accepted as true. One type of proof is
called a paragraph proof or informal proof . In this type of proof, you write a
paragraph to explain why a conjecture for a given situation is true.
Proofs
Five essential parts of a good proof:
• State the theorem or conjecture to be proven.
• List the given information.
• If possible, draw a diagram to illustrate the given information.
• State what is to be proved.
• Develop a system of deductive reasoning.
In Lesson 1-2, you learned the relationship between segments formed by the
midpoint of a segment. This statement can be proven, and the result stated as a
theorem.
90 Chapter 2 Reasoning and Proof
Differentiated Instruction
3. postulates, theorems, algebraic
properties, definitions
90
Chapter 2 Reasoning and Proof
Intrapersonal Tell students to read quietly over the postulates and
examples in this lesson and note the differences in the postulate
statements and the statements they are to write proofs for. Advise
students to go through the text and their study notebooks to compile a
list of useful information they could use to write the proofs in this lesson.
Example 3 Write a Paragraph Proof
PARAGRAPH PROOFS
Given that M is the midpoint of PQ
, write a paragraph proof to show that
—–
–—
PM MQ.
.
Given: M is the midpoint of PQ
Q
Prove:
M
MQ
.
P
In-Class Example
3 Given CD,
AC intersecting M
From the definition of midpoint of a segment,
PM MQ. This means that PM
and M
Q
have
the same measure. By the definition of congruence,
if two segments have the same measure, then they
MQ
.
are congruent. Thus, PM
write a paragraph proof to
show that A, C, and D
determine a plane.
and CD
must intersect at C
AC
because if two lines intersect,
then their intersection is exactly
and
one point. Point A is on AC
point D is on CD . Therefore
points A and D are not collinear.
Therefore ACD is a plane as it
contains three points not on the
same line.
P
Once a conjecture has been proven true, it can be stated as a theorem and used in
other proofs. The conjecture in Example 3 is known as the Midpoint Theorem.
Theorem 2.8
Midpoint Theorem If M is the midpoint of ,
AB then AM
MB
.
Concept Check
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4–5, 11
6
7–10
1
2
3
7. definition of
collinear.
8. Through any three
points not on the same
line, there is exactly
one plane.
9. Through any two
points, there is exactly
one line.
Application
3 Practice/Apply
1. Explain how deductive reasoning is used in a proof. 1–3. See margin.
2. OPEN ENDED Draw figures to illustrate Postulates 2.6 and 2.7.
3. List the types of reasons that can be used for justification in a proof.
Determine the number of segments that can be drawn connecting each pair
of points.
6
5.
15
4.
Study Notebook
Have students—
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 2.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
6. Determine whether the following
statement is always, sometimes, or never
true. Explain. See p. 123C.
The intersection of three planes is two lines.
are
In the figure, BD and BR
BD . State the
in plane P, and W is on postulate or definition that can be used to
show each statement is true.
7. B, D, and W are collinear.
8. E, B, and R are coplanar.
9. R and W are collinear.
E
R
D
B
P
10. PROOF
In the figure at the right, P is the
R
and ST
, and Q
R
ST
.
midpoint of Q
Write a paragraph proof to show that
PQ PT. See p. 123C.
W
Q
T
Organization by Objective
• Points, Lines, and Planes:
12–21
• Paragraph Proofs: 22–28
P
S
R
11. DANCING Six students are participating in a dance to celebrate the opening of
a new community center. The students, each connected to each of the other
students with wide colored ribbons, will move in a circular motion. How many
ribbons are needed? 15 ribbons
www.geometryonline.com/extra_examples
Power
Point®
Odd/Even Assignments
Exercises 12–27 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Lesson 2-5 Postulates and Paragraph Proofs 91
Unlocking Misconceptions
Assignment Guide
Writing Proofs Explain to students that a common mistake in writing
proofs is skipping a step or assuming a step that should be included in
the proof. Sometimes, the missed step can be quite obvious, but it still
has to be included. Tell students to make a habit of listing each piece of
information with a separate explanation for each and to avoid using two
reasons or postulates for the same statement when they are writing
proofs.
Basic: 13–19 odd, 23–31 odd,
33–48
Average: 13–31 odd, 33–48
(optional: 43–48)
Lesson 2-5 Postulates and Paragraph Proofs 91
★ indicates increased difficulty
NAME ______________________________________________ DATE
____________ PERIOD _____
Practice and Apply
Study
Guide
andIntervention
Intervention,
2-5
Study
Guide and
p. 81
(shown)
and Proofs
p. 82
Postulates
and Paragraph
Determine the number of segments that can be drawn connecting each pair
of points.
Points, Lines, and Planes
In geometry, a postulate is a statement that is accepted as
true. Postulates describe fundamental relationships in geometry.
Postulate:
Postulate:
Postulate:
Postulate:
Postulate:
Postulate:
Postulate:
Through any two points, there is exactly one line.
Through any three points not on the same line, there is exactly one plane.
A line contains at least two points.
A plane contains at least three points not on the same line.
If two points lie in a plane, then the line containing those points lies in the plane.
If two lines intersect, then their intersection is exactly one point.
If two planes intersect, then their intersection is a line.
For
Exercises
See
Examples
12–15
16–21
22–28
1
2
3
Example
Determine whether each statement is always,
sometimes, or never true.
a. There is exactly one plane that contains points A, B, and C.
Sometimes; if A, B, and C are collinear, they are contained in many planes. If they are
noncollinear, then they are contained in exactly one plane.
6
12.
10
13.
Extra Practice
See page 757.
b. Points E and F are contained in exactly one line.
Always; the first postulate states that there is exactly one line through any two points.
15
14.
c. Two lines intersect in two distinct points M and N.
Never; the intersection of two lines is one point.
21
15.
Exercises
Use postulates to determine whether each statement is always, sometimes, or
never true.
1. A line contains exactly one point. never
2. Noncollinear points R, S, and T are contained in exactly one plane. always
3. Any two lines and
Determine whether the following statements are always, sometimes, or never true.
Explain. 16–21. See p. 123C.
16. Three points determine a plane.
m intersect. sometimes
4. If points G and H are contained in plane M, then G
H
is perpendicular to plane M. never
5. Planes
R and S intersect in point T. never
6. If points A, B, and C are noncollinear, then segments A
B
, BC
, and C
A
are contained in
exactly one plane. always
In the figure, A
C
and D
E
are in plane Q and A
C
|| D
E
.
State the postulate that can be used to show each
statement is true.
any three points not on the same line, there is
exactly one plane.
Q
C
B
Lesson 2-5
A
E
D
G
8. BE lies in plane
Q. If two points lie in a plane, then
the line containing those points lies in the plane.
NAME
______________________________________________
DATE
/M
G
Hill
81
Gl
17. Points G and H are in plane X. Any point collinear with G and H is in
plane X .
F
7. Exactly one plane contains points F, B, and E. Through
Skills
Practice,
p. 83 and
2-5
Practice
(Average)
Practice,
(shown)
Postulates p.
and 84
Paragraph
Proofs
18. The intersection of two planes can be a point.
19. Points S, T, and U determine three lines.
____________
Gl PERIOD
G _____
★ 20. Points A and B lie in at least one plane.
★ 21. If line lies in plane P and line m lies in plane Q , then lines and m lie
in plane R.
Determine the number of line segments that can be drawn connecting each pair
of points.
21
1.
2.
28
In the figure at the right, AC and BD lie
BY and CX lie in plane K.
in plane J, and State the postulate that can be used to show
each statement is true. 22–27. See p. 123C.
22. C and D are collinear.
23. XB lies in plane K.
Determine whether the following statements are always, sometimes, or never true.
Explain.
3. The intersection of two planes contains at least two points.
Always; the intersection of two planes is a line, and a line contains at
least two points.
4. If three planes have a point in common, then they have a whole line in common.
Sometimes; they might have only that single point in common.
lie in plane A . State the postulate
In the figure, line m and TQ
that can be used to show that each statement is true.
S
24. Points A, C, and X are coplanar.
25. AD lies in plane J.
m A
T
m lie in the same plane.
Postulate 2.5: If two points lie in a plane, then the
entire line containing those points lies in that plane.
5. L, T, and line
Q
L
m and S
T
intersect at T.
Postulate 2.6: If two lines intersect, then their intersection is exactly one
point.
6. Line
7. In the figure, E is the midpoint of A
B
and C
D
, and AB CD. Write a
paragraph proof to prove that A
E
ED
.
C
A
E
Given: E is the midpoint of A
B
and CD
AB CD
D
Prove: AE
ED
Proof: Since E is the midpoint of A
B
and C
D
, we know by the Midpoint
Theorem, that A
E
EB
and C
E
ED
. By the definition of congruent
B
1
1
segments, AE EB AB and CE ED CD. Since AB CD,
2
2
1
1
AB CD by the Multiplication Property. So AE ED, and by the
2
2
definition of congruent segments, A
E
ED
.
Detective
27. Points Y, D, and C are coplanar.
A police detective gathers
facts and collects evidence
for use in criminal cases.
The facts and evidence are
used together to prove a
suspect’s guilt in court.
28. PROOF
Point C is the midpoint of A
B
and B is the midpoint of C
D
. Prove that
C
BD
. See margin.
A
the plane that contains A, B, and C and the plane that contains B, C,
and D
NAME
______________________________________________
DATE
/M
G
Hill
84
____________
Gl PERIOD
G _____
2-5
to Learn
Learn MathematicsELL
Mathematics,
p. 85
Postulates and Paragraph Proofs
Pre-Activity
How are postulates used by the founding fathers of the United States?
Read the introduction to Lesson 2-5 at the top of page 89 in your textbook.
Postulates are often described as statements that are so basic and so clearly
correct that people will be willing to accept them as true without asking for
evidence or proof. Give a statement about numbers that you think most
people would accept as true without evidence. Sample answer: Every
Online Research
a career as a detective,
visit:
www.geometryonline.
com/careers
1. Determine whether each of the following is a correct or incorrect statement of a
geometric postulate. If the statement is incorrect, replace the underlined words to make
the statement correct.
incorrect;
a. A plane contains at least two points that do not lie on the same line. three points
d. A line contains at least one point . incorrect; two points
incorrect;
e. If two lines are parallel , then their intersection is exactly one point. intersect
f. Through any two points, there is at most one line. incorrect; exactly
92
Chapter 2 Reasoning and Proof
NAME ______________________________________________ DATE
a. If two planes intersect, their intersection is a line. always
c. There is exactly one plane that contains three collinear points. never; Sample
answer: There are infinitely many planes if the three points are
collinear, but only one plane if the points are noncollinear.
d. If two lines intersect, their intersection is one point. always
3. Use the walls, floor, and ceiling of your classroom to describe a model for each of the
following geometric situations.
2-5
Enrichment
Enrichment,
____________ PERIOD _____
p. 86
Logic Problems
The following problems can be solved by eliminating possibilities.
It may be helpful to use charts such as the one shown in the first
problem. Mark an X in the chart to eliminate a possible answer.
a. two planes that intersect in a line Sample answer: two adjacent walls that
intersect at an edge of both walls in the corner of the room
b. two planes that do not intersect Sample answer: the ceiling and the floor (or
two opposite walls)
c. three planes that intersect in a point Sample answer: the floor (or ceiling)
and two adjacent walls that intersect at a corner of the floor (or ceiling)
Helping You Remember
4. A good way to remember a new mathematical term is to relate it to a word you already
know. Explain how the idea of a mathematical theorem is related to the idea of a scientific
theory. Sample answer: Scientists do experiments to prove theories;
mathematicians use deductive reasoning to prove theorems. Both
processes involve using evidence to show that certain statements are true.
Solve each problem.
1. Nancy, Olivia, Mario, and Kenji each have
one piece of fruit in their school lunch.
They have a peach, an orange, a banana,
and an apple. Mario does not have a
peach or a banana. Olivia and Mario just
came from class with the student who has
an apple. Kenji and Nancy are sitting
next to the student who has a banana.
Nancy does not have a peach. Which
student has each piece of fruit?
Peach
O
92
Chapter 2 Reasoning and Proof
X
arguments, which is a speech that uses deductive reasoning, in court cases.
Spencer Grant/PhotoEdit
2. Determine whether each statement is always, sometimes, or never true. If the statement
is not always true, explain why.
b. The midpoint of a segment divides the segment into two congruent segments. always
J
29. MODELS Faith’s teacher asked her to make a figure showing the number of
lines and planes formed from four points that are noncollinear and noncoplanar.
Faith decided to make a mobile of straws, pipe cleaners, and colored sheets of
tissue paper. She plans to glue the paper to the straws and connect the straws
together to form a group of connected planes. How many planes and lines will
she have? She will have 4 different planes and 6 lines.
incorrect;
c. Through any four points not on the same line, there is exactly one plane. three points
D
C
B
30. CAREERS Many professions use deductive reasoning and paragraph proofs.
For example, a police officer uses deductive reasoning investigating a traffic
accident and then writes the findings in a report. List a profession, and describe
how it can use paragraph proofs. Sample answer: Lawyers make final
number is equal to itself.
b. If two planes intersect, then the intersection is a line. correct
A
26. X and Y are collinear.
8. LOGIC Points A, B, and C are not collinear. Points B, C, and D are not collinear. Points
A, B, C, and D are not coplanar. Describe two planes that intersect in line BC.
Gl
K
Y
Nancy
Olivia
Mario
X
X
X
X
X
Kenji
X
2. Victor, Leon, Kasha, and Sheri each play
one instrument. They play the viola,
clarinet, trumpet, and flute. Sheri does
not play the flute. Kasha lives near the
student who plays flute and the one
who plays trumpet. Leon does not play
a brass or wind instrument. Which
student plays each instrument?
Victor—flute,
Leon—viola,
Kasha—clarinet,
Sheri—trumpet
31. CRITICAL THINKING You know that three noncollinear points lie in a single
plane. In Exercise 29, you found the number of planes defined by four
noncollinear points. What are the least and greatest number of planes defined
by five noncollinear points? one, ten
4 Assess
Open-Ended Assessment
32. WRITING IN MATH
Answer the question that was posed at the beginning of the
lesson. See margin.
How are postulates used in literature?
• an example of a postulate in historic United States’ documents, and
• an example of a postulate in mathematics.
Standardized
Test Practice
Lesson 2-6
33. Which statement cannot be true? C
A A plane can be determined using three noncollinear points.
B Two lines intersect at exactly one point.
C At least two lines can contain the same two points.
D A midpoint divides a segment into two congruent segments.
Prerequisite Skill Students will
Lesson 2-6. Use Exercises 43–48
familiarity with solving equations.
34. ALGEBRA For all values of x, (8x 4 2x2 3x 5) (2x 4 x 3 3x 5) A
A
6x 4 x 3 2x2 10.
B
6x 4
3x2
6x 10.
C
6x 4 x 3 2x2 6x.
D
6x 4 3x 2.
Mixed Review
35. Determine whether statement (3) follows from statements (1) and (2) by the Law
of Detachment or the Law of Syllogism. If it does, state which law was used. If it
does not, write invalid. (Lesson 2-4) yes; Law of Detachment
(1) Part-time jobs require 20 hours of work per week.
(2) Jamie has a part-time job.
(3) Jamie works 20 hours per week.
Write the converse, inverse, and contrapositive of each conditional statement.
Determine whether each related conditional is true or false. If a statement is false,
find a counterexample. (Lesson 2-3) 36–37. See margin.
37. If ABC is a right triangle, one of its angle measures is greater than 90.
38. BIOLOGY Use a Venn diagram to illustrate the following statement.
If an animal is a butterfly, then it is an arthropod. (Lesson 2-2) See p. 123C.
Use the Distance Formula to find the distance between each pair of points.
(Lesson 1-3)
the Next Lesson
Speaking Have students choose
a paragraph proof they wrote
and explain each step aloud.
4.1
17
P(8, 2), Q(1, 3) 106
10.3
39. D(3, 3), F(4, 1)
40. M(0, 2), N(5, 5)
41.
42. R(5, 12), S(2, 1)
5.8
34
13.0
170
PREREQUISITE SKILL Solve each equation.
(To review solving equations, see pages 737 and 738.)
y
6
43. m 17 8 25
44. 3y 57 19
45. 12 14 12
46. t 3 27 ⴚ24
47. 8n 39 41 10
48. 6x 33 0 www.geometryonline.com/self_check_quiz
11
2
Lesson 2-5 Postulates and Paragraph Proofs 93
28. Given: C is the midpoint of AB
.
B is the midpoint of C
D
.
Prove: AC
BD
Proof: We are given that C is the midpoint of AB
, and B is the
midpoint of CD
. By the definition of midpoint AC
CB
and C
B
BD
.
Using the definition of congruent segments, AC CB, and
CB BD. AC BD by the Transitive Property of Equality.
Thus, A
C
BD
by the definition of congruent segments.
32. Sample answer: The forms and
structures of different types of
writing are accepted as valid,
such as the structure of a poem.
following.
• The Declaration of
Independence, “We hold these
truths to be self-evident, …”
• Through any two points, there is
exactly one line.
36. Converse: If you have a computer,
Internet at your house. False; you
can have a computer and not have
Internet at your house, then you
do not have a computer. False; it
the Internet and still have a
computer. Contrapositive: If you
do not have a computer, then you
at your house. False; you could
have Internet access through your
television or wireless phone.
37. Converse: If ABC has an angle
with measure greater than 90, then
ABC is a right triangle. False; the
triangle would be obtuse. Inverse:
If ABC is not a right triangle,
none of its angle measures are
greater than 90. False; it could be
an obtuse triangle. Contrapositive:
If ABC does not have an angle
measure greater than 90, ABC is
not a right triangle. False; mABC
could still be 90 and ABC be a
right triangle.
Lesson 2-5 Postulates and Paragraph Proofs 93
Lesson
Notes
1 Focus
5-Minute Check
Transparency 2-6 Use as a
quiz or review of Lesson 2-5.
Algebraic Proof
• Use algebra to write two-column proofs.
• Use properties of equality in geometry proofs.
• deductive argument
• two-column proof
• formal proof
Mathematical Background notes
are available for this lesson on
p. 60D.
is mathematical
evidence similar to
evidence in law?
• In math, what one thing do you
need to prove a statement is
false? What evidence can a
lawyer use to prove that
someone is innocent (or has
been falsely accused)?
an alibi
• How does the use of evidence in
law differ from its use in math?
evidence to sway opinions,
sometimes without knowing the
truth. A mathematician presents
evidence to prove factual
statements.
is mathematical evidence
similar to evidence in law?
Vocabulary
Lawyers develop their cases using logical
arguments based on evidence to lead a jury to
a conclusion favorable to their case. At the end
of a trial, a lawyer will make closing remarks
summarizing the evidence and testimony
that they feel proves their case. These closing
arguments are similar to a proof in mathematics.
ALGEBRAIC PROOF Algebra is a system with sets of numbers, operations,
and properties that allow you to perform algebraic operations.
Properties of Equality for Real Numbers
Study Tip
Commutative
and Associative
Properties
Throughout this text,
we shall assume the
Commutative and
Associative Properties
multiplication.
Reflexive Property
For every number a, a a.
Symmetric Property
For all numbers a and b, if a b, then b a.
Transitive Property
For all numbers a, b, and c, if a b and b c, then a c.
Subtraction Properties
For all numbers a, b, and c, if a b, then a c b c
and a c b c.
Multiplication and
Division Properties
For all numbers a, b, and c, if a b, then a c b c
a
b
and if c 0, c c.
Substitution Property
For all numbers a and b, if a b, then a may be replaced
by b in any equation or expression
Distributive Property
For all numbers a, b, and c, a(b c) ab ac.
The properties of equality can be used to justify each step when solving an equation.
A group of algebraic steps used to solve problems form a deductive argument .
Example 1 Verify Algebraic Relationships
Solve 3(x 2) 42.
Algebraic Steps
3(x 2) 42
3x 6 42
3x 6 6 42 6
3x 48
3x
48
3
3
x 16
94
Properties
Original equation
Distributive Property
Substitution Property
Division Property
Substitution Property
Chapter 2 Reasoning and Proof
Bob Daemmrich/The Image Works
Resource Manager
Workbook and Reproducible Masters
Chapter 2 Resource Masters
• Study Guide and Intervention, pp. 87–88
• Skills Practice, p. 89
• Practice, p. 90
• Reading to Learn Mathematics, p. 91
• Enrichment, p. 92
• Assessment, p. 120
Prerequisite Skills Workbook, pp. 41–44,
83–86, 93–94
Teaching Geometry With Manipulatives
Masters, p. 8
Transparencies
5-Minute Check Transparency 2-6
Technology
Interactive Chalkboard
Multimedia Applications: Virtual Activities
Example 1 is a proof of the conditional statement If 5x 3(x 2) 42, then x 6.
Notice that the column on the left is a step-by-step process that leads to a solution.
The column on the right contains the reason for each statement.
In geometry, a similar format is used to prove conjectures and theorems. A
two-column proof , or formal proof , contains statements and reasons organized in
two columns. In a two-column proof, each step is called a statement, and the
properties that justify each step are called reasons.
Write a Two-Column Proof
Example 2
a. If 3x 1, then x 2
5
3
Study Tip
Mental Math
you to do so, some steps
may be eliminated by
performing mental
calculations. For example,
in part a of Example 2,
statements 4 and 6 could
be omitted. Then the
reason for statements 5
Property and Division
Property for statement 7.
Reasons
5
1. 3 x 1
3
5
2. 3x 3 1
3
1. Given
3. 3x 5 1
4. 3x 5 5 1 5
5. 3x 6
3. Substitution
5. Substitution
3x
6
6. 6. Division Property
7. x 2
7. Substitution
3
b. Given:
2. Distributive Property
3
3.
4.
5.
6.
7.
7 2n 8 n
7 2n n 8 n n
7n8
7n787
n 1
1. Given
2. Multiplication Property
3.
4.
5.
6.
7.
Distributive Property
Substitution
Subtraction Property
Substitution
1
n
8. 8. Division Property
9. n 1
9. Substitution
1
1
Power
Point®
2(5 3a) 4(a 7) 92.
2(5 3a) 4(a 7) 92
(Original eqn.)
10 6a 4a 28 92
(Distr. Prop.)
18 10a 92 (Subst. Prop.)
18 10a 18 92 18
10a 110 (Subst. Prop.)
110
10a
(Div. Prop.)
10
10
Teaching Tip
Reasons
7
1
1. n 4 n
2
2
7
1
2. 2 n 2 4 n
2
2
In-Class Examples
a 11 (Subst. Prop.)
7
1
n 4 n
2
2
Prove: n 1
Proof:
Statements
ALGEBRAIC PROOFS
1 Solve
Write a two-column proof.
Statements
2 Teach
Explain that
since students may have
different preferences when
solving algebraic equations,
their proofs might vary slightly
from the examples. For
example, one student may
distribute a variable first, while
subtraction. Assure students that
as long as they use properties of
equality appropriately, their
proofs will be correct.
2 Write a two-column proof for
each of the following.
7d 3
4
a. If 6, then d 3.
GEOMETRIC PROOF Since geometry also uses variables, numbers, and
operations, many of the properties of equality used in algebra are also true in
geometry. For example, segment measures and angle measures are real numbers,
so properties from algebra can be used to discuss their relationships. Some
examples of these applications are shown below.
Property
Segments
Angles
Reflexive
AB AB
m1 m1
Symmetric
If AB CD, then CD AB.
If m1 m2, then m2 m1.
Transitive
If AB CD and CD EF,
then AB EF.
If m1 m2 and m2 m3,
then m1 m3.
www.geometryonline.com/extra_examples
Lesson 2-6 Algebraic Proof 95
Statements (Reasons)
7d 3
4
7d 3
2. 4 4(6) (Mult. Prop.)
4
1. 6 (Given)
3. 7d 3 24 (Substitution)
4. 7d 3 3 24 3
(Subtr. Prop.)
5. 7d 21 (Substitution)
7d
7
21
7
6. (Div. Prop.)
7. d 3 (Substitution)
(continued on the next page)
Lesson 2-6 Algebraic Proofs 95
Standardized Example
Test Practice
9
p
11
b. If 3p , then
5
10
10
p
10
p
9
11
2. 10 3p 10 10
5
10
11
10
If AB
CD
EF
, and C
D
, then which
of the following is a valid conclusion?
1. 3p (Given)
(Mult. Prop.)
3. 30p 18 11 p
(Distr. Prop.)
4. 30p p 18 11 p p
(Subtr. Prop.)
5. 29p 18 11 (Substitution)
6. 29p 18 18 11 18
7. 29p 29 (Substitution)
29p
29
Test-Taking Tip
More than one statement
may be correct. Work
through each problem
completely before
29
29
9. p 1 (Substitution)
GEOMETRIC PROOFS
Power
Point®
ST
, then which of the
RP
following is a valid
conclusion? B
I. GH JK RP
II. PR TS
III. GH JK ST RP
A I only
B I and II
C I and III D I, II, and III
Chapter 2 Reasoning and Proof
B
D
A
I only
B
I and II
C
I and III
D
I, II, and III
E
F
Determine whether the statements are true based on the given information.
In Example 3, each conclusion was justified using a definition or property. This
process is used in geometry to verify and prove statements.
Example 4 Geometric Proof
TIME On a clock, the angle formed by the hands at 2:00 is a 60 angle. If the
angle formed at 2:00 is congruent to the angle formed at 10:00, prove that the
angle at 10:00 is a 60 angle.
Given:
4 SEA LIFE A starfish has five
96
C
I AB CD and CD EF
II AB
EF
III AB EF
Because Statements I, II, and III are true, choice D is correct.
3 If GH JK ST and
arms. If the length of arm 1 is
22 cm, and arm 1 is congruent
to arm 2, and arm 2 is
congruent to arm 3, prove that
arm 3 has length 22 cm. We
are given arm 1 arm 2 and
arm 2 arm 3, so by the
definition of congruence, the
measure of arm 1 the measure
of arm 2 and the measure of
arm 2 the measure of arm 3.
By the Transitive Property of
Equality, we know that the
measure of arm 1 the measure
of arm 3. We can then substitute
22 cm for the measure of arm 1
to prove that the measure of arm
3 is 22 cm.
A
Solve the Test Item
Statement I:
Examine the given information, AB
CD
EF
and C
D
. From the definition of
CD
EF
congruent segments, if A
B
and C
D
, then AB CD and CD EF. Thus,
Statement I is true.
Statement II:
B
E
F.
By the definition of congruent segments, if AB EF, then A
Statement II is true also.
Statement III:
If AB CD and CD EF, then AB EF by the Transitive Property.
Thus, Statement III is true.
8. (Div. Prop.)
In-Class Examples
Justify Geometric Relationships
Multiple-Choice Test Item
p 1.
Statements (Reasons)
9
5
3
Prove:
m2 60
2 10
m10 60
Proof:
Statements
1. m2 60
2 10
2. m2 m10
3. 60 m10
4. m10 60
96
Reasons
1. Given
2. Definition of congruent angles
3. Substitution
4. Symmetric Property
Chapter 2 Reasoning and Proof
Aaron Haupt
Differentiated Instruction
Interpersonal Let groups of students work one or two selected
problems from Exercises 24–29 on p. 98. Stipulate that each group
member should contribute at least one step of the proof. Encourage
groups to brainstorm beforehand to determine the properties they will
use and the order they will use them in. Allow the groups to check and
compare their proofs when they are done to see if any two groups
found different ways to prove the same statement.
Concept Check
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4–7
8
9, 10, 12
11
1
3
2
4
1. OPEN ENDED Write a statement that illustrates the Substitution Property
of Equality. 1–2. See margin.
2. Describe the parts of a two-column proof.
3. State the part of a conditional that is related to the Given statement of a proof.
What part is related to the Prove statement? hypothesis; conclusion
State the property that justifies each statement.
5
2
4. If 2x 5, then x Division Property
x
2
5. If 7, then x 14. Multiplication Property
6. If x 5 and b 5, then x b. Substitution Property
7. If XY AB WZ AB, then XY WZ. Addition Property
x
2
8. Solve 4x 7 11. List the property that justifies each step. See margin.
9. Complete the following proof.
2
3
Given: 5 x 1
Prove: x 6
3 Practice/Apply
Study Notebook
Have students—
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 2.
• include a sample algebraic proof
and a sample geometric proof.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
Proof:
Statements
a. ? 5 2x 1
3
2
b. 35 x 3(1)
3
c. 15 2x 3
d. ? 2x 12
e. x 6
Reasons
a. Given
b.
Mult. Prop.
?
c. ? Dist. Prop.
d. Subtraction Prop.
e. ? Div. Prop.
Organization by Objective
• Algebraic Proofs: 14–21
• Geometric Proofs: 22–29
Write a two-column proof. 10–12. See pp. 123C–123D.
10. Prove that if 25 7(y 3) 5y, then 2 y.
11. If rectangle ABCD has side lengths AD 3 and AB 10, then AC BD.
12. The Pythagorean Theorem states that in a right triangle ABC, c2 a2 b2.
PROOF
2 b2.
Prove that a c
Standardized
Test Practice
13. ALGEBRA If 8 x = 12, then 4 x =
A 28
B 24
? . C
C 0
D
4
Practice and Apply
For
Exercises
See
Examples
15, 16, 20
14, 17–19, 21
22–27
28, 29
1
2
3
4
Extra Practice
See page 757.
State the property that justifies each statement.
14. If mA mB and mB mC, mA mC. Trans. Prop.
15. If HJ 5 20, then HJ 15. Subt. Prop.
16. If XY 20 YW and XY 20 DT, then YW DT. Substitution
17. If m1 m2 90 and m2 m3, then m1 m3 90. Substitution
1
2
1
2
18. If AB EF, then AB EF. Div. or Mult. Prop.
Odd/Even Assignments
Exercises 14–29 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Assignment Guide
Basic: 15–31 odd, 32–33, 35–51
Average: 15–31 odd, 32–33,
35–51
(optional: 49–51)
All: Quiz 2 (1–5)
x
19. AB AB Reflexive Property
Lesson 2-6 Algebraic Proof 97
1. Sample answer: If x 2 and
x y 6, then 2 y 6.
2. given and prove statements and
two columns, one of statements
and one of reasons
8. Given: 4x 7 11
2
Prove: x 4
Proof:
Statements (Reasons)
x
2
1. 4x 7 11 (Given)
2x
2. 2 4x 7 2(11)
(Mult. Prop.)
3. x 8x 14 22 (Dist.Prop.)
4. 9x 14 22 (Substitution)
6. x 4 (Div. Prop.)
Lesson 2-6 Algebraic Proofs 97
20. If 2x 5, which property can be used to support the statement 2x 3 5?
32. Given: Ek hf W
21. Which property allows you to state m4 m5, if m4 35 and m5 35?
3
2
Dist. Prop.
Ek W
h
Substitution
Prove: f 1
2
1
2
22. If AB CD, which property can be used to justify the statement AB CD?
Proof:
Statements (Reasons)
1. Ek hf W (Given)
2. Ek W hf (Subt. Prop.)
Div. or Mult. Prop.
23. Which property could be used to support the statement EF JK, given that
EF GH and GH JK? Transitive Prop.
Complete each proof.
Ek W
h
Ek W
4. f (Sym. Prop.)
h
24. Given:
3. f (Div. Prop.)
3x 5
7
2
x3
Prove:
Proof:
Statements
evidence and testimony as
reasons for justifying statements
and actions. All of the evidence
to prove a lawyer’s case, much as
in a proof in mathematics.
following.
• Evidence is used to verify facts
from witnesses or materials.
• Postulates, theorems,
definitions, and properties can
be used to justify statements
Reasons
3x 5
a. 7
2
b.
c.
d.
e.
a.
3x 5
? 2 2(7)
2
3x 5 14
3x 9
? x3
?
Given
b. Mult. Prop.
? Substitution
c.
d. ? Subt. Prop.
e. Div. Prop.
1
25. Given: 2x 7 x 2
3
Prove: x 3
Proof:
Statements
a. ? 2x 7 1x 2
3
1
b. ? 3(2x 7) 3 x 2
3
c. 6x 21 x 6
d. ? 5x 21 6
e. 5x 15
? x3
f.
A gymnast exhibits kinetic
energy when performing
on the balance beam. The
movements and flips show
the energy that is being
displayed while the
gymnast is moving.
a. Given
b. Mult. Prop.
? Dist. Prop.
c.
d. Subt. Prop.
f. Div. Prop.
Write a two-column proof. 26–31. See p. 123D.
PROOF
Physics
Reasons
1
2
7
2
26. If 4 a a, then a 1.
1
2
3
2
13
4
27. If 2y 8, then y .
2
3
28. If m 9, then m 18.
29. If 5 z 1, then z 6.
30. If XZ ZY, XZ 4x 1,
and ZY 6x 13, then x 7.
31. If mACB mABC,
then mXCA mYBA.
X
4x 1
Z
A
6x 13
X
C
B
Y
Y
32. PHYSICS Kinetic energy is the energy of motion. The formula for kinetic
energy is Ek h f W, where h represents Planck’s Constant, f represents the
frequency of its photon, and W represents the work function of the material
being used. Solve this formula for f and justify each step. See margin.
98
Chapter 2 Reasoning and Proof
Duomo/CORBIS
98
Chapter 2 Reasoning and Proof
33. GARDENING Areas in the southwest
and southeast have cool but mild winters.
In these areas, many people plant pansies
in October so that they have flowers
outside year-round. In the arrangement of
pansies shown, the walkway divides the
two sections of pansies into four beds that
are the same size. If mACB mDCE,
what could you conclude about the
relationship among ACB, DCE, ECF,
and ACG? All of the angle measures
A
G
B
NAME ______________________________________________ DATE
The following properties of algebra can be used to justify the steps
when solving an algebraic equation.
C
Property
Statement
Reflexive
For every number a, a a.
Symmetric
For all numbers a and b, if a b then b a.
Transitive
For all numbers a, b, and c, if a b and b c then a c.
For all numbers a, b, and c, if a b then a c b c and a c b c.
Multiplication and Division
For all numbers a, b, and c, if a b then a c b c, and if c 0 then .
Substitution
For all numbers a and b, if a b then a may be replaced by b in any equation
or expression.
D
E
Example
Solve 6x 2(x 1) 30.
Algebraic Steps
6x 2(x 1) 30
6x 2x 2 30
8x 2 30
8x 2 2 30 2
8x 32
CRITICAL THINKING For Exercises 34 and 35, use the following information.
Below is a family tree of the Gibbs family. Clara, Carol, Cynthia, and Cheryl are all
daughters of Lucy. Because they are sisters, they have a transitive and symmetric
relationship. That is, Clara is a sister of Carol, Carol is a sister of Cynthia, so Clara
is a sister of Cynthia.
Properties
Given
Distributive Property
Substitution
Substitution
8x
32
8
8
Division Property
x4
Substitution
Exercises
Complete each proof.
4x 6
2. Given: 4x 8 x 2
Prove: x 2
1. Given: 9
2
Prove: x 3
Statements
4x 6
2
4x 6
b. 2 2(9)
2
a. 9
Lucy
Cynthia
Carol
Chris
Diane
Kevin
Dierdre
Reasons
Statements
Reasons
a. Given
a. 4x 8 x 2
a. Given
b. Mult. Prop.
b. 4x 8 x x2x
b. Subtr. Prop.
c. 3x 8 2
c. Substitution
c. 4x 6 18
c. Subst.
d. 4x 6 6 18 6
d. Subtr. Prop.
e. 4x 12
e. Substitution
d. 3x 8 8 f. f. Div. Prop.
28
e. 3x 6
g. x 3
g. Substitution
f. 12
4
4x
4
Cheryl
Gl
Michael
3x
3
d. Subtr. Prop.
e. Substitution
6
3
f. Div. Prop.
g. x 2
NAME
______________________________________________
DATE
/M
G
Hill
87
Skills
Practice,
2-6
Practice
(Average)
Steven
b
c
For all numbers a, b, and c, a(b c) ab ac.
Distributive
F
a
c
Lesson 2-6
p. 87
(shown)
and p. 88
Algebraic
Proof
Algebraic Proof
would be equal.
Clara
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
2-6
Study
Guide and
p. 89 and
Practice,
p. 90 (shown)
Algebraic Proof
g. Substitution
____________
Gl PERIOD
G _____
PROOF Write a two-column proof.
Cyle
Standardized
Test Practice
Allycia
Maria
1. If mABC mCBD 90, mABC 3x 5,
x1
2
34. What other relationships in a family have reflexive, symmetric, or transitive
relationships? Explain why. Remember that the child or children of each person
are listed beneath that person’s name. Consider relationships such as first
cousin, ancestor or descendent, aunt or uncle, sibling, or any other relationship.
1. mABC mCBD 90
mABC 3x 5
x1
mCBD 35. Construct your family tree on one or both sides of your family and identify the
reflexive, symmetric, or transitive relationships. See students’ work.
2
x1
2
2. 3x 5 90
36. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson. See margin.
How is mathematical evidence similar to evidence in law?
• a description of how evidence is used to influence jurors’ conclusions
in court, and
• a description of the evidence used to make conclusions in mathematics.
37. In PQR, mP m Q and mR 2(mQ).
Find mP if mP mQ + mR 180. B
A
C
30
B
60
D
D
and mCBD , then x 27.
Given: mABC mCBD 90
mABC 3x 5
x1
m CBD 2
Prove: x 27
Proof:
Statements
x1
2
C
B
A
Reasons
1. Given
2. Substitution Property
3. (2)(3x 5) (2) (2)90
3. Multiplication Property
4.
5.
6.
7.
10 x 1 180
9 180
9 9 180 9
189
4.
5.
6.
7.
8. 189
7
8. Division Property
9. x 27
9. Substitution Property
6x
7x
7x
7x
7x
7
Substitution Property
Substitution Property
Substitution Property
2. FINANCE The formula for simple interest is I prt, where I is interest, p is principal,
r is rate, and t is time. Solve the formula for r and justify each step.
Given: I prt
I
Prove: r pt
Proof:
Statements
1. I prt
prt
I
2. pt
pt
I
3. r
pt
I
4. r pt
Reasons
1. Given
2. Division Property
3. Substitution Property
4. Symmetric Property
P
Gl
NAME
______________________________________________
DATE
/M
G
Hill
90
____________
Gl PERIOD
G _____
2-6
to Learn
Learn Mathematics
Mathematics,
p. 91
Algebraic Proof
45
Pre-Activity
90
R
ELL
How is mathematical evidence similar to evidence in law?
Read the introduction to Lesson 2-6 at the top of page 94 in your textbook.
Q
What are some of the things that lawyers might use in presenting their
closing arguments to a trial jury in addition to evidence gathered prior to
the trial and testimony heard during the trial? Sample answer: They
might tell the jury about laws related to the case, court
rulings, and precedents set by earlier trials.
38. ALGEBRA If 4 more than x is 5 less than y, what is x in terms of y? B
A
y1
B
y9
y9
C
www.geometryonline.com/self_check_quiz
D
y5
Lesson 2-6 Algebraic Proof 99
1. Name the property illustrated by each statement.
a. If a 4.75 and 4.75 b, then a b. Transitive Property of Equality
b. If x y, then x 8 y 8. Addition Property of Equality
c. 5(12 19) 5 12 5 19 Distributive Property
Substitution Property
d. If x 5, then x may be replaced with 5 in any equation or expression. of Equality
e. If x y, then 8x 8y. Multiplication Property of Equality
f. If x 23.45, then 23.45 x. Symmetric Property of Equality
7
g. If 5x 7, then x . Division Property of Equality
5
h. If x 12, then x 3 9. Subtraction Property of Equality
NAME ______________________________________________ DATE
2-6
Enrichment
Enrichment,
____________ PERIOD _____
p. 92
2. Give the reason for each statement in the following two-column proof.
Given: 5(n 3) 4(2n 7) 14
Prove: n 9
Statements
Reasons
1. 5(n 3) 4(2n 7) 14
1. Given
2. 5n 15 8n 28 14
2. Distributive Property
3. 5n 15 8n 42
3. Substitution Property
Symmetric, Reflexive, and Transitive Properties
4. 5n 15 15 8n 42 15
Equality has three important properties.
5. 5n 8n 27
Reflexive
Symmetric
Transitive
aa
If a b, then b a.
If a b and b c, then a c.
Other relations have some of the same properties. Consider the
relation “is next to” for objects labeled X, Y, and Z. Which of the
properties listed above are true for this relation?
X is next to X. False
If X is next to Y, then Y is next to X. True
If X is next to Y and Y is next to Z, then X is next to Z. False
Only the symmetric property is true for the relation “is next to.”
For each relation, state which properties (symmetric, reflexive,
transitive) are true.
5. Substitution Property
6. 5n 8n 8n 27 8n
6. Subtraction Property
7. 3n 27
7. Substitution Property
3n
3
27
3
8. 8. Division Property
9. n 9
9. Substitution Property
Helping You Remember
3. A good way to remember mathematical terms is to relate them to words you already know.
Give an everyday word that is related in meaning to the mathematical term reflexive and
explain how this word can help you to remember the Reflexive Property and to distinguish
it from the Symmetric and Transitive Properties. Sample answer: Reflection: If you
look at your reflection, you see yourself. The Reflexive Property says that
every number is equal to itself. The Reflexive Property involves only one
number, while the Symmetric and Transitive Properties each involve two
or three numbers.
Lesson 2-6 Algebraic Proofs 99
Lesson 2-6
are: Michael has a
symmetric relationship
of first cousin with
Chris, Kevin, Diane,
Dierdre, and Steven.
Diane, Dierdre,
and Steve have a
symmetric and
transitive relationship
of sibling. Any direct
line from bottom to
top has a transitive
descendent
relationship.
Ryan
4 Assess
Mixed Review
Open-Ended Assessment
Writing Select some statements
to prove and write them on the
board. Have different volunteers
come up to the board and write
one statement and reason to
students have proven the
original statement.
Lesson 2-7
Prerequisite Skill Students will
relationships in Lesson 2-7. They
and will use segment measures
to prove segment congruence.
Use Exercises 51–53 to determine
segment measures.
(Lesson 2-5)
40. Valid; since 24 is
divisible by 6, the Law
of Detachment says it
is divisible by 3.
41. Invalid; 27 6 4.5, which is not an
integer.
42. Valid; since 85 is
not divisible by 3, the
contrapositive of the
statement and the Law
of Detachment say
that 85 is not divisible
by 6.
the Next Lesson
Assessment Options
Practice Quiz 2 The quiz
provides students with a brief
review of the concepts and skills
in Lessons 2-4 through 2-6.
Lesson numbers are given to the
right of the exercises or
instruction lines so students can
review concepts not yet
mastered.
Quiz (Lessons 2-5 and 2-6) is
available on p. 120 of the Chapter 2
Resource Masters.
43. If people are happy, then they
rarely correct their faults.
44. If you don’t know where you are
going, then you will probably end
up somewhere else.
45. If a person is a champion, then
the person is afraid of losing.
46. If we would have new knowledge,
then we must get a whole new
world of questions.
100
Chapter 2 Reasoning and Proof
39. CONSTRUCTION There are four buildings on the Medfield High School
Campus, no three of which stand in a straight line. How many sidewalks need
to be built so that each building is directly connected to every other building?
6
Determine whether the stated conclusion is valid based on the given information.
If not, write invalid. Explain your reasoning. A number is divisible by 3 if it is
divisible by 6. (Lesson 2-4)
40. Given: 24 is divisible by 6.
Conclusion: 24 is divisible by 3.
41. Given: 27 is divisible by 3.
Conclusion: 27 is divisible by 6.
42. Given: 85 is not divisible by 3.
Conclusion: 85 is not divisible by 6.
Write each statement in if-then form. (Lesson 2-3) 43–46. See margin.
43. “Happy people rarely correct their faults.” (La Rochefoucauld)
44. “If you don’t know where you are going, you will probably end up
somewhere else.” (Laurence Peters)
45. “A champion is afraid of losing.” (Billie Jean King)
46. “If we would have new knowledge, we must get a whole new world of
questions.” (Susanne K. Langer)
Find the precision for each measurement. (Lesson 1-2)
47. 13 feet
48. 5.9 meters
49. 74 inches
1 ft
0.05
m
0.5 in.
2
PREREQUISITE SKILL Find the measure of each segment.
50. 3.1 kilometers
0.05 km
(To review segment measures, see Lesson 1-2.)
51. K
L
11
52. Q
S 28
25
51
K
L
P
Q
P ractice Quiz 2
9
38
23
14
J
53. W
Z 47
W
R S
X
Y Z
Lessons 2-4 through 2-6
1. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the
Law of Syllogism. If it does, state which law was used. If it does not, write invalid. (Lesson 2-4)
(1) If n is an integer, then n is a real number.
(2) n is a real number.
(3) n is an integer. invalid
In the figure at the right, A, B, and C are collinear. Points A, B, C,
and D lie in plane N. State the postulate or theorem that can be
used to show each statement is true. (Lesson 2-5) 2–4. See margin.
2. A, B, and D determine plane N.
AC at B.
3. BE intersects 4. lies in plane N.
5.
PROOF
E
A
D
If 2(n 3) 5 3(n 1), prove that n 2. (Lesson 2-6) See p. 123E.
100 Chapter 2 Reasoning and Proof
Practice Quiz 2
2. Through any three points not on the same line,
there is exactly one plane.
3. If two lines intersect, then their intersection is
exactly one point.
4. If two points lie in a plane, then the entire line
containing those points lies in that plane.
N
B
C
Proving Segment Relationships
• Write proofs involving segment addition.
1 Focus
• Write proofs involving segment congruence.
can segment relationships
be used for travel?
San
Diego
When leaving San Diego, the pilot said
that the flight would be about 360 miles
to Phoenix before continuing on to
Dallas. When the plane left Phoenix,
the pilot said that the flight would be
flying about 1070 miles to Dallas.
5-Minute Check
Transparency 2-7 Use as a
quiz or review of Lesson 2-6.
Phoenix
Dallas
Mathematical Background notes
are available for this lesson on
p. 60D.
1/2 inch = 400 mi.
SEGMENT ADDITION In Lesson 1-2, you measured segments with a ruler by
placing the mark for zero on one endpoint, then finding the distance to the other
endpoint. This illustrates the Ruler Postulate.
Postulate 2.8
Ruler Postulate The points on any line or line
segment can be paired with real numbers so
that, given any two points A and B on a line,
A corresponds to zero, and B corresponds to
a positive real number.
A
B
0
The Ruler Postulate can be used to further investigate line segments.
Construct a Figure
Segment Measures
• Use The Geometer’s Sketchpad to
C
.
construct A
C
.
• Place point B on A
• Find AB, BC, and AC.
AB = 1.79 cm
BC = 3.21 cm
AC = 5.00 cm
A
Analyze the Model
Lesson
Notes
B
can segment
relationships be used
for travel?
• Use a ruler to measure the
length in millimeters from San
Diego to Phoenix and from
Phoenix to Dallas. about 11 mm;
• Given the pilot’s information,
how many miles will Janelle be
flying from San Diego to
Dallas? 1430
• How are segment lengths
lengths to calculate distance, flight
time and necessary fuel
requirements.
C
1. What is the sum AB BC?
2. Move B. Find AB, BC and AC.
What is the sum of AB BC?
3. Repeat moving B, measuring the segments, and finding the sum AB BC
three times. Record your results. 1–3. See students’ work. The sum AB BC
Make a Conjecture
should always equal AC.
4. What is true about the relationship of AB, BC, and AC ? AB BC AC
5. Is it possible to place B on A
C
so that this relationship is not true? no
Lesson 2-7 Proving Segment Relationships 101
Resource Manager
Workbook and Reproducible Masters
Chapter 2 Resource Masters
• Study Guide and Intervention, pp. 93–94
• Skills Practice, p. 95
• Practice, p. 96
• Reading to Learn Mathematics, p. 97
• Enrichment, p. 98
School-to-Career Masters, p. 4
Prerequisite Skills Workbook, pp. 89–90,
101–104
Teaching Geometry With Manipulatives
Masters, p. 8
Transparencies
5-Minute Check Transparency 2-7
Technology
Interactive Chalkboard
Lesson x-x Lesson Title 101
Study Tip
2 Teach
Betweenness
In-Class Example
Power
Point®
1 Prove the following. Use the
figure from Example 1 in the
Student Edition.
Given: PR QS
Prove: PQ RS
Statements (Reasons)
1. PR QS (Given)
2. PR QR QS QR
(Subtr. Prop.)
3. PR QR PQ;
QS QR RS
4. PQ RS (Substitution)
In general, the definition
of between is that B is
between A and C if A, B,
and C are collinear and
AB BC AC.
Examine the measures AB, BC, and AC in the Geometry Activity. Notice that
wherever B is placed between A and C, AB BC AC. This suggests the following
postulate.
Postulate 2.9
Segment Addition Postulate If B is between
BC
AB
A and C, then AB BC AC.
A
B
If AB BC AC, then B is between A and C.
C
AC
Example 1 Proof With Segment Addition
Prove the following.
Given:
PQ RS
Prove:
PR QS
P
Q
R S
Proof:
Statements
Reasons
1. PQ RS
2. PQ QR QR RS
3. PQ QR PR
QR RS QS
4. PR QS
1. Given
4. Substitution
Teaching Tip
Tell students that
with each new lesson, they are
accumulating more postulates and
theorems that they can use for
writing proofs. Encourage students
to practice using these concepts as
much as possible before moving on
to the next lesson to strengthen
their ability to recall important facts
for proof-writing skills.
SEGMENT CONGRUENCE In Lesson 2-5, you learned that once a theorem is
proved, it can be used in proofs of other theorems. One theorem we can prove is
similar to properties of equality from algebra.
Segment Congruence
Theorem 2.2
Congruence of segments is reflexive, symmetric, and transitive.
Reflexive Property
A
AB
B
Symmetric Property If AB
CD
CD
AB
, then .
Transitive Property
If AB
CD
CD
EF
AB
EF
, and , then .
You will prove the first two properties in Exercises 10 and 24.
Proof
Given:
Transitive Property of Congruence
N
M
N
PQ
P
Q
P
RS
Prove:
RS
M
N
M
Proof:
Method 1
Paragraph Proof
Q
R
PQ
and P
Q
RS
, MN PQ and PQ RS by the
Since MN
definition of congruent segments. By the Transitive Property
RS
by the definition of
of Equality, MN RS. Thus, MN
congruent segments.
102 Chapter 2 Reasoning and Proof
Geometry Software Investigation
Adding Segment Measures Have students repeat the activity for different
lengths of AC
. Students can also construct AC
vertically and at various
diagonals. Tell students that this activity provides several examples to
102
Chapter 2 Reasoning and Proof
S
Two-Column Proof
Method 2
Statements
1.
2.
3.
4.
SEGMENT CONGRUENCE
Reasons
M
N
PQ
, P
Q
R
S
MN PQ, PQ RS
MN RS
N
RS
M
1.
2.
3.
4.
In-Class Example
Given
Definition of congruent segments
Transitive Property
Definition of congruent segments
2 Prove the following.
3 cm
Y
W
Example 2 Proof With Segment Congruence
J
1.
2.
3.
4.
5.
Concept Check
Guided Practice
GUIDED PRACTICE KEY
Exercises
6
4, 5, 7–10
Examples
1
2
K
G
Reasons
1.
2.
3.
4.
5.
Given
Transitive Property
Given
Transitive Property
Symmetric Property
from Cleveland to Chicago is the
same as the distance from
Cleveland to Chicago.
XY
and
X
Y
PQ
, then AB
PQ
.
If A, B, and C are collinear and AB BC AC, then B is between A and C.
Justify each statement with a property of equality or a property of congruence.
Y
XY
Reflexive
4. X
5. If G
H
MN
, then MN
G
H
. Symmetric
6. If AB AC CB, then AB AC CB. Subtraction
Prove:
Y
S T
Q
X
RT
P
S
Q
P
Statements
Reasons
a.
? , ?
b. PQ RS, QS ST
c. PS PQ QS, RT RS ST
d. ? PQ QS RS ST
e.
? PS RT
RT
S
f. P
a. Given
b. ? Def. of segs.
c.
e. Substitution
f.
? Def. of segs.
www.geometryonline.com/extra_examples
P
B
A
R
Proof:
7a. PQ
RS
, QS
ST
1. Choose two cities from a United States road map. Describe the distance between
the cities using the Reflexive Property. See margin.
2. OPEN ENDED Draw three congruent segments, and illustrate the Transitive
Property using these segments. See margin.
3. Describe how to determine whether a point B is between points A and C.
7. Copy and complete the proof.
RS
ST
Q
, Q
S
Given: P
X
Given: WY YZ
Z
XZ
Y
Z
X
WX
Prove: WX
WY
Proof:
Statements (Reasons)
1. WY YZ (Given)
2. WY
Y
Z (Def. of Segs.)
3. YZ
XZ; X
Z WX
(Given)
4. WY
WX
(Trans. Prop.)
5. WX
WY
(Symmetric)
L
H
JK
KL
, K
L
HJ
HJ
JK
J GH
H
GH
JK
H
JK
G
Z
3 cm
The theorems about segment congruence can be used to prove segment
relationships.
Prove the following.
KL
, H
J GH
, KL
HJ
Given: JK
Prove: G
H
JK
Proof:
Statements
Power
Point®
Lesson 2-7 Proving Segment Relationships 103
C Squared Studios/PhotoDisc
(t) file photo
(b)
Differentiated Instruction
Visual/Spatial When students are first examining figures to determine
the steps necessary for writing proofs, encourage them to use their spatial
skills to locate obvious and hidden congruent segments and parts of
to use the given information to mark the figures so they can easily refer to
the relationships in the figures while they are writing their proofs.
Lesson 2-7 Proving Segment Relationships 103
For Exercises 8–10, write a two-column proof. 8–9. See p. 123E.
8. Given: AP
CP
9. Given: H
I TU
P
B
DP
J H
TV
Prove: A
B
CD
Prove: IJ UV
PROOF
3 Practice/Apply
A
D
Study Notebook
P
Have students—
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 2.
• include a sample proof using
segment congruence.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
Odd/Even Assignments
Exercises 12–23 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Assignment Guide
Basic: 13–27 odd, 29–45
Average: 13–27 odd, 29–45
(optional: 40–45)
I
B
C
T
J
U
V
10. Symmetric Property of Congruence (Theorem 2.2) See margin.
Application
11. GEOGRAPHY Aberdeen in South Dakota and Helena, Miles City, and
Missoula, all in Montana, are connected in a straight line by interstate highways.
Missoula is 499 miles from Miles City and 972 miles from Aberdeen. Aberdeen is
473 miles from Miles City and 860 miles from Helena. Between which cities does
Helena lie? Helena is between Missoula and Miles City.
Practice and Apply
For
Exercises
See
Examples
14, 16, 17
12, 13, 15,
18–24
1
2
Extra Practice
Organization by Objective
• Segment Addition: 14, 16, 17
• Segment Congruence: 12, 13,
15, 18–24
H
See page 758.
Justify each statement with a property of equality or a property of congruence.
LM
, then LM
JK
. Symmetric
12. If JK
13. If AB 14 and CD 14, then AB CD. Substitution
14. If W, X, and Y are collinear, in that order, then WY WX XY. Segment
15. If M
N
PQ
and P
Q
RS
, then MN
RS
. Transitive
16. If EF TU and GH VW, then EF GH TU VW. Addition
17. If JK MN JK QR, then MN QR. Subtraction
B
18. Copy and complete the proof.
D
CE
, D
B
EB
Given: A
Prove:
Proof:
Statements
CB CE EB
a.
b.
c.
d.
e.
f.
D
A
Reasons
D
CE
, DB
EB
? A
?
AB CB
A
B
CB
Write a two-column proof. 19 – 20. See p. 123E.
PROOF
19. If X
Y
WZ
and W
Z
AB
,
20. If AB
AC
and P
C
QB
,
AB
.
then AP
AQ
.
then XY
W
C
B
A
P
Y
B
Z
A
104 Chapter 2 Reasoning and Proof
C
D
Proof:
Statements (Reasons)
1. AB
CD
(Given)
2. AB CD (Def. of segs.)
3. CD AB (Symmetric Prop.)
4. CD
AB
(Def. of segs.)
104
Q
X
10. Given: A
B
CD
Prove: C
D
AB
A
Chapter 2 Reasoning and Proof
C
a. Given
b. ? Def. of segs.
c.
e. ? Substitution
f.
? Def. of segs.
B
E
AB
CB
21. Copy and complete the proof.
Y
ZX
Given: W
A is the midpoint of WY
.
.
A is the midpoint of ZX
ZA
Prove: WA
Proof:
Statements
W
NAME ______________________________________________ DATE
p. 93
(shown)
and p. 94
Proving
Segment Relationships
X
A
Y
Reasons
Two basic postulates for working with segments and lengths are
the Ruler Postulate, which establishes number lines, and the Segment Addition Postulate,
which describes what it means for one point to be between two other points.
Ruler Postulate
The points on any line or line segment can be paired with real numbers so that, given any two
points A and B on a line, A corresponds to zero and B corresponds to a positive real number.
Postulate
B is between A and C if and only if AB BC AC.
Example
Write a two-column proof.
Given: Q is the midpoint of P
R
.
.
R is the midpoint of QS
Prove: PR QS
Given
?
a.
b. ? Def. of segs.
c. Definition of midpoint
e. ? Substitution
f.
? Substitution
g. ? Substitution
h. Division Property
i.
? Def. of segs.
For Exercises 22–24, write a two-column proof. 22–24. See p. 123E.
22. If L
M
PN
and X
M
XN
,
23. If AB BC,
PX
.
then AC 2BC.
then LX
S
Statements
Reasons
1. Given
2. Definition of midpoint
3. Given
4. Definition of midpoint
7. Substitution
Exercises
Complete each proof.
1. Given: BC DE
Prove: AB DE AC
Statements
a. BC DE
A
2. Given: Q is between
P and R, R is between
Q and S, PR QS.
Prove: PQ RS
C
B
E
D
Reasons
a. Given
b. AB BC AC b. Seg. Add. Post.
c. AB DE AC
Gl
c. Substitution
Q
Reasons
a. Q is between
P and R.
b.PQ QR PR
c. R is between
Q and S.
d. QR RS QS
e. PR QS
f. PQ QR QR RS
g. PQ QR QR QR RS QR
h. PQ RS
a. Given
NAME
______________________________________________
DATE
/M
G
Hill
93
p. 95 and
Practice,
p. 96Relationships
(shown)
Proving Segment
S
R
P
Statements
Skills
Practice,
2-7
Practice
(Average)
P
c. Given
e. Given
f. Substitution
g. Subtraction
Prop.
h. Substitution
____________
Gl PERIOD
G _____
Complete the following proof.
L
M
A
X
B
1. Given: A
B
DE
B is the midpoint of A
C
.
E is the midpoint of D
F
.
C
EF
Prove: B
Proof:
C
N
A
B
D
C
E
F
Statements
Reasons
DE
a. A
B
a. Given
B is the midpoint of A
C
.
E is the midpoint of D
F
.
24. Reflexive Property of Congruence (Theorem 2.2)
25. DESIGN The front of a building has a
DE
and C is
triangular window. If AB
D
, prove that A
C
CE
.
the midpoint of B
b. AB DE
b. Definition of segments
c. AB BC
c. Definition of Midpoint
DE EF
F
d. AC AB BC
DF DE EF
e. AB BC DE EF
See p. 123E.
A
B
C
D
f. Substitution Property
g. AB BC AB AB EF AB
g. Subtraction Property
h. BC EF
h. Substitution Property
EF
i. B
C
i. Definition of segments
2. TRAVEL Refer to the figure. DeAnne knows that the
Grayson
distance from Grayson to Apex is the same as the distance
G
from Redding to Pine Bluff. Prove that the distance from
Grayson to Redding is equal to the distance from Apex to Pine Bluff.
E
Given: G
A
RP
Prove: G
R
AP
Proof:
Statements
26. LIGHTING The light fixture in Gerrard Hall
of the University of North Carolina is shown at
B
EF
and B
C
DE
, prove that
the right. If A
DF
C
. See p. 123F.
A
B
A
E
D
e. Substitution Property
f. AB BC AB EF
1.
2.
3.
4.
5.
6.
F
C
Gl
A
G
GA
GA
GR
GR
GR
RP
RP
AR AR RP
GA AR, AP AR RP
AP
AP
www.geometryonline.com/self_check_quiz
L
M
N
NAME
______________________________________________
DATE
/M
G
Hill
96
____________
Gl PERIOD
G _____
• Before leaving home, a passenger used a road atlas to determine that the
distance between San Diego and Dallas is about 1350 miles. Why is the
flying distance greater than that? Sample answer: Phoenix is not
T
on a straight line between San Diego and Dallas, so the stop
added to the distance traveled. A nonstop flight would have
been shorter.
P
O
NAME ______________________________________________ DATE
2-7
Enrichment
Enrichment,
____________ PERIOD _____
p. 98
1
2
L
L
L
I
O
N
E
A
N
8
A
J
13
C
Q
O
16
P
U
L
A
N
4
R
R
6
I
E
C
9
G
D
12
O
I
10
S
E
11
G M
A
T
I
C
R
D
E
A
P
17
C
O
E
N
B
T
P
A
Y
7
U
P
S
E
T
R
E
P
14
L
G
5
C
M
T
V
E
V
E
E
N
E
M
R
D
T
I
15
N
R
1. If E is between Y and S, which of the following statements are always true? B, E
A. YS ES YE
B. YS ES YE
C. YE ES
D. YE ES YS
E. SE EY SY
F. E is the midpoint of Y
S
.
2. Give the reason for each statement in the following
two-column proof.
Given: C is the midpoint of B
D
.
D is the midpoint of CE
.
Prove: BD
CE
Statements
Reasons
1. C is the midpoint of B
D
.
Geometry Crossword Puzzle
O
ELL
How can segment relationships be used for travel?
• What is the total distance that the plane will fly to get from San Diego to
Dallas? 1430 mi
S
C
P
Read the introduction to Lesson 2-7 at the top of page 101 in your textbook.
Lesson 2-7 Proving Segment Relationships 105
3
Pine Bluff
R
Given
Definition of segments
Substitution Property
Definition of segments
Mathematics,
p. 97
Proving Segment Relationships
RT
27. CRITICAL THINKING Given that L
N
,
T
QO
, L
Q
NO
, M
P
NO
, S is the
R
midpoint of R
T
, M is the midpoint of L
N
, and P
R
, list three statements that
is the midpoint of QO
you could prove using the postulates, theorems,
and definitions that you have learned. See p. 123F. Q
A
Reasons
1.
2.
3.
4.
5.
6.
2-7
to Learn
Learn Mathematics
Pre-Activity
Apex Redding
L
D
N
I
U
E
A
R
T
G
I
T
N
I
C
N
A
H
N
E
T
C
U
A
N
T
T
X
A
R
T
R
N
18
A
L
N
G
L
A
B
C
D
E
1. Given
2. BC CD
2. Definition of midpoint
3. D is the midpoint of CE
.
3. Given
4. CD DE
4. Definition of midpoint
5. BC DE
5. Transitive Property of Equality
6. BC CD CD DE
7. BC CD BD
CD DE CE
8. BD CE
8. Substitution Property
9. B
D
CE
9. Def. of segments
Helping You Remember
3. One way to keep the names of related postulates straight in your mind is to associate
something in the name of the postulate with the content of the postulate. How can you use
this idea to distinguish between the Ruler Postulate and the Segment Addition Postulate?
Sample answer: There are two words in “Ruler Postulate” and three words
in “Segment Addition Postulate.” The statement of the Ruler Postulate
mentions two points, and the statement of the Segment Addition
Postulate mentions three points.
E
A
Lesson 2-7 Proving Segment Relationships 105
Lesson 2-7
Source: www.pella.com
R
R
.
1. Q is the midpoint of P
2. PQ QR
3. R is the midpoint of Q
S
.
4. QR RS
5. PQ QR QR RS
6. PQ QR PR, QR RS QS
7. PR QS
PROOF
Windows come in many
different shapes and sizes.
Some commonly used
shapes for windows are
circles, rectangles, squares,
triangles, pentagons, and
octagons.
Q
P
Lesson 2-7
Z
a. W
Y
ZX
A is the midpoint of W
Y
.
X
.
A is the midpoint of Z
b. WY ZX
c.
? WA AY, ZA AX
d. WY WA AY, ZX ZA AX
e. WA AY ZA AX
f. WA WA ZA ZA
g. 2WA = 2ZA
h. ? WA ZA
A
ZA
i. W
Design
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
2-7
Study
Guide and
28. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson. See margin.
How can segment relationships be used for travel?
4 Assess
• an explanation of how a passenger can use the distances the pilot announced
to find the total distance from San Diego to Dallas, and
• an explanation of why the Segment Addition Postulate may or may not be
useful when traveling.
Open-Ended Assessment
Modeling Make a
“Reasons/Statements” board
with Velcro in positions where you
could place given information,
statements and reasons. Create
three or four proofs using segment
and write the given information
and each statement and reason on
a separate rectangular piece of
poster board (large enough to read
from the back of the classroom).
Affix Velcro to the back of the
boards so they can be easily
placed on the R/S board. Place
the given information at the top
of the R/S board. Have students
select each statement, match it
with its corresponding reason
and place it in the correct order
on the R/S board.
Standardized
Test Practice
28. Sample answer: You can use
segment addition to find the total
distance between two destinations
by adding the distances of various
should include the following.
• A passenger can add the
distance from San Diego to
Phoenix and the distance from
Phoenix to Dallas to find the
distance from San Diego to
Dallas.
can be useful if you are
traveling in a straight line.
106 Chapter 2 Reasoning and Proof
A
1
2
B
1
C
2
D
2
1
3
14 4
1
12 4
A
B
12 4
C
D
1
2
30. GRID IN A refreshment stand sells a large tub of popcorn for twice the price of
a box of popcorn. If 60 tubs were sold for a total of \$150 and the total popcorn
sales were \$275, how many boxes of popcorn were sold? 100
Mixed Review
State the property that justifies each statement. (Lesson 2-6)
31. If mP mQ 110 and mR 110, then mP mQ mR. Substitution
32. If x(y z) a, then xy xz = a. Dist. Prop.
33. If n 17 39, then n 56. Add. Prop.
34. If cv md and md 15, then cv 15. Trans. Prop.
Determine whether the following statements are always, sometimes, or never true.
Explain. (Lesson 2-5) 35–38. See margin for explanations.
35. A midpoint divides a segment into two noncongruent segments. never
36. Three lines intersect at a single point. sometimes
37. The intersection of two planes forms a line. always
38. Three single points determine three lines. sometimes
Lesson 2-8
Prerequisite Skill In Lesson 2-8,
students will apply properties of
supplementary and
complementary angles to prove
angle relationships. Use
Exercises 4045 to determine
complementary and
supplementary angles.
and Q is
29. If P is the midpoint of BC
D
, what is PQ? B
the midpoint of A
D
39. If the perimeter of rectangle ABCD
is 44 centimeters, find x and the
dimensions of the rectangle.
(Lesson 1-6)
A
(x 6) cm
3; 9 cm by 13 cm
C (2x 7) cm
the Next Lesson
B
PREREQUISITE SKILL Find x.
(To review complementary and supplementary angles, see Lesson 1-5.)
30
40.
2x °
43.
15
41.
42.
x°
22
(3x 2)˚
2x °
x°
4x °
45
x°
45. 25
44. 5
3x °
26x °
10x °
106 Chapter 2 Reasoning and Proof
35. The midpoint of a segment divides it into two congruent segments.
36. If the lines have a common intersection point, then it is a single point.
37. If two planes intersect, they intersect in a line.
38. If the points are noncollinear, then they lie on three distinct lines.
(4x 10)˚ (3x 5)˚
1/16/2003 1:06 PM
T_Maria_Manko 107-114 GEO C2L8-
Lesson
Notes
Proving Angle Relationships
• Write proofs involving supplementary and complementary angles.
1 Focus
• Write proofs involving congruent and right angles.
do scissors illustrate
supplementary angles?
5-Minute Check
Transparency 2-8 Use as a
quiz or review of Lesson 2-7.
Notice that when a pair of scissors is
opened, the angle formed by the two
blades, 1, and the angle formed by a
blade and a handle, 2, are a linear pair.
Likewise, the angle formed by a blade and
a handle, 2, and the angle formed by the
two handles, 3, also forms a linear pair.
2
1
Mathematical Background notes
are available for this lesson on
p. 60D.
3
SUPPLEMENTARY AND COMPLEMENTARY ANGLES Recall that when
you measure angles with a protractor, you position the protractor so that one of the
rays aligns with zero degrees and then determine the position of the second ray.
This illustrates the Protractor Postulate.
Postulate 2.10
80
70
110
100
90
100
80
110
70
12
0
60
13
0
50
14
0
40
30
30
15
0
15
170
10
r°
A
170
10
20
160
160
20
B
In Lesson 2-7, you learned about the Segment Addition Postulate. A similar
relationship exists between the measures of angles.
A
D
B
60
0
12
40
14
0
50
0
13
0
and a number r
Protractor Postulate Given AB
between 0 and 180, there is exactly one ray with
, such
endpoint A, extending on either side of AB
that the measure of the angle formed is r.
do scissors illustrate
supplementary angles?
• In the figure, label 4 vertical
to 2 and name all pairs of
supplementary angles.
1 and 2, 2 and 3, 3 and
4, 4 and 1
• Use a protractor to measure
angles 1 and 2. What is the
sum of these two measures?
• Will the same angles still form
linear pairs if the scissors were
opened wider? narrower?
yes; yes
Postulate 2.11
C
Angle Addition Postulate If R is in the interior
of PQS, then mPQR mRQS mPQS.
If mPQR mRQS mPQS, then R is in the
interior of PQS.
P
R
Q
S
History
The Grand Union flag was
the first flag used by the
colonial United States that
resembles the current flag.
It was made up of thirteen
stripes with the flag of
Great Britain in the corner.
Source: www.usflag.org
HISTORY The Grand Union Flag at the left contains several angles.
If mABD 44 and mABC 88, find mDBC.
mABD mDBC mABC Angle Addition Postulate
44 mDBC 88
mABD 44, mABC 88
mDBC 44
Subtraction Property
Lesson 2-8 Proving Angle Relationships 107
(t)C Squared Studios, (b)file photo
Resource Manager
Workbook and Reproducible Masters
Chapter 2 Resource Masters
• Study Guide and Intervention, pp. 99–100
• Skills Practice, p. 101
• Practice, p. 102
• Reading to Learn Mathematics, p. 103
• Enrichment, p. 104
• Assessment, p. 120
Prerequisite Skills Workbook, pp. 81–82,
85–86
Teaching Geometry With Manipulatives
Masters, pp. 8, 16, 48
Transparencies
5-Minute Check Transparency 2-8
Technology
Interactive Chalkboard
Lesson x-x Lesson Title 107
The Angle Addition Postulate can be used with other angle relationships to
provide additional theorems relating to angles.
2 Teach
SUPPLEMENTARY AND
COMPLEMENTARY ANGLES
In-Class Examples
Power
Point®
Theorems
Study Tip
2.3
Look Back
To review supplementary
and complementary angles,
see Lesson 1-5.
1 TIME At 4 o’ clock, the angle
a linear pair, then they are supplementary
angles.
1
2
m1 m2 180
2.4
between the hour and minute
hands of a clock is 120°. If the
second hand stops where it
bisects the angle between the
hour and minute hands, what
are the measures of the angles
between the minute and
second hands and between
the second and hour hands?
They are both 60° by the
definition of angle bisector and
Supplement Theorem If two angles form
Complement Theorem If the noncommon
sides of two adjacent angles form a right
angle, then the angles are
complementary angles.
1
2
m1 m2 90
You will prove Theorems 2.3 and 2.4 in Exercises 10 and 11.
Example 2 Supplementary Angles
If 1 and 2 form a linear pair and
m 2 67, find m1.
m1 m2 180 Supplement Theorem
1
m1 67 180 m2 67
2
m1 113 Subtraction Property
2 If 1 and 2 form a linear
pair and m2 166, find
m1. 14
CONGRUENT AND RIGHT ANGLES The properties of algebra that applied
to the congruence of segments and the equality of their measures also hold true for
the congruence of angles and the equality of their measures.
Theorem 2.5
Congruence of angles is reflexive, symmetric, and transitive.
Reflexive Property
1 1
Symmetric Property If 1 2, then 2 1.
Transitive Property
If 1 2, and 2 3, then 1 3.
You will prove the Reflexive and Transitive Properties of Angle Congruence in Exercises 26 and 27.
TEACHING TIP
The Symmetric Property
is often assumed in proofs
to condense the number
of steps in a proof. The
rigor of proof is left up
to the teacher, but we
will assume symmetric
property statements in
future chapters.
Proof
Symmetric Property of Congruence
Given: A B
Prove: B A
Paragraph Proof:
A
B
We are given A B. By the definition of congruent angles, mA mB.
Using the Symmetric Property, mB mA. Thus, B A by the definition of
congruent angles.
Algebraic properties can be applied to prove theorems for congruence
relationships involving supplementary and complementary angles.
108 Chapter 2 Reasoning and Proof
Differentiated Instruction
2.3–2.13 aloud for students one by one. After each one, ask students
to discuss how they know the theorem is true and how they might use
the theorem in a proof.
108
Chapter 2 Reasoning and Proof
CONGRUENT AND RIGHT
ANGLES
Theorems
2.6 Angles supplementary to the same
angle or to congruent angles are
congruent.
Abbreviation: suppl. to same or are .
Example:
2.7
3
1
form a linear pair, and m3
m1 180. Prove that 3
and 4 are congruent.
1
2
1
2
3
If m1 m2 90 and
m2 m3 90, then
1 3.
You will prove Theorem 2.6 in Exercise 6.
Proof
Theorem 2.7
Given: 1 and 3 are complementary.
2 and 3 are complementary.
Prove: 1 2
Proof:
Statements
1. 1 and 3 are complementary.
2 and 3 are complementary.
2. m1 m3 90
m2 m3 90
3. m1 m3 m2 m3
4. m3 m3
5. m1 m2
6. 1 2
Power
Point®
3 In the figure, 1 and 4
If m1 m2 180 and
m2 m3 180, then
1 3.
Angles complementary to the same
angle or to congruent angles are
congruent.
Abbreviation: compl. to same or
are .
Example:
In-Class Example
2
2
1
3
Reasons
1. Given
4
3
Statements (Reasons)
1. m3 m1 180; 1 and
4 form a linear pair. (Given)
2. 1 and 4 are supplementary.
(Linear pairs are suppl.)
3. 3 and 1 are supplementary.
(Def. of suppl. )
4. 3 4 ( supplementary
to same are .)
2. Definition of complementary angles
3.
4.
5.
6.
Substitution
Reflective Property
Subtraction Property
Definition of congruent angles
Example 3 Use Supplementary Angles
In the figure, 1 and 2 form a linear pair
and 2 and 3 form a linear pair. Prove
1
that 1 and 3 are congruent.
4 2
Given: 1 and 2 form a linear pair.
3
2 and 3 form a linear pair.
Prove: 1 3
Proof:
Statements
Reasons
1. 1 and 2 form a linear pair.
1. Given
2 and 3 form a linear pair.
2. 1 and 2 are supplementary.
2. Supplement Theorem
2 and 3 are supplementary.
3. 1 3
3. suppl. to same or are .
www.geometryonline.com/extra_examples
Lesson 2-8 Proving Angle Relationships 109
Lesson 2-8 Proving Angle Relationships 109
In-Class Example
Note that in Example 3, 1 and 3 are vertical angles. The conclusion in the
example is a proof for the following theorem.
Power
Point®
Teaching Tip
Tell students to
can be sure to provide the
information requested. For this
example, point out that students
are to find angle measures, not
just the value of the variable;
however, they have to use the
value of the variable to find the
4 If 1 and 2 are vertical
angles and m1 d 32
and m2 175 2d, find
m1 and m2. 37; 37
Study Tip
Theorem 2.8
Look Back
Vertical Angles Theorem If two angles are
To review vertical angles,
see Lesson 1-5.
vertical angles, then they are congruent.
Abbreviation: Vert. are .
2
1
3
4
1 3 and 2 4
Example 4 Vertical Angles
If 1 and 2 are vertical angles and m1 x and m2 228 3x, find
m1 and m2.
Vertical Angles Theorem
1 2
m1 m2
x 228 3x
4x 228
x 57
m1 x
Definition of congruent angles
Substitution
Divide each side by 4.
m2 m1
57
57
The theorems you have learned can be applied to right angles. You can create
right angles and investigate congruent angles by paper folding.
Right Angles
Make a Model
• Fold the paper so that one corner is folded downward.
• Fold along the crease so that the top edge meets the
side edge.
• Unfold the paper and measure each of the angles
formed.
• Repeat the activity three more times.
Analyze the Model 1. The lines are perpendicular.
1. What do you notice about the lines formed?
angles? They are congruent and they form linear pairs.
3. What are the measures of the angles formed? 90
Make a Conjecture 4. They form right angles.
4. What is true about perpendicular lines?
5. What is true about all right angles? They all measure 90 and are congruent.
The following theorems support the conjectures you made in the Geometry Activity.
110 Chapter 2 Reasoning and Proof
Geometry Activity
Materials: paper, protractor
• When students are repeating the activity, tell them to use different folds
from the right and the left sides of the paper each time.
• Ask students what they notice about each pair of vertical angles (they are
congruent and form right angles).
110
Chapter 2 Reasoning and Proof
Theorems
Right Angles
2.9
Perpendicular lines intersect to form four right angles.
2.10 All right angles are congruent.
2.11 Perpendicular lines form congruent adjacent angles.
2.12 If two angles are congruent and supplementary, then each angle
is a right angle.
2.13
Concept Check
If two congruent angles form a linear pair, then they are right angles.
1. FIND THE ERROR Tomas and Jacob wrote equations involving the angle
measures shown.
Tomas
m∆ABE + m∆EBC = m∆ABC
E
F
C
Jacob
m∆ABE + m∆FBC = m∆ABC
B
A
Who is correct? Explain your reasoning. 1–2. See margin.
2. OPEN ENDED Draw three congruent angles. Use these angles to illustrate the
Transitive Property for angle congruence.
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4
5
6
7–11
1
2
3
4
Find the measure of each numbered angle.
3. m1 65
4. 6 and 8 are
complementary.
m8 47
1
5. m11 x 4,
m12 2x 5
2
6
7
11
12
8
m11 59,
m12 121
m2 65
m6 43, m7 90
6. PROOF
Given:
Copy and complete the proof of Theorem 2.6.
1 and 2 are supplementary.
3 and 4 are supplementary.
1
1 4
2 3
Prove:
Proof:
Statements
a. 1 and 2 are supplementary.
3 and 4 are supplementary.
1 4
b. m1 m2 180
m3 m4 180
c. m1 m2 m3 m4
d. m1 m4
e. m2 m3
f. 2 3
2
3
4
Reasons
Given
a.
?
b.
?
Def. of suppl. c.
d.
e.
f.
?
?
?
?
Substitution
Def. of Subtr. Prop.
Def. of Lesson 2-8 Proving Angle Relationships 111
left out the part of ABC
represented by EBF.
2. Sample answer: If 1 2 and
2 3, then 1 3.
3 Practice/Apply
Study Notebook
Have students—
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 2.
• include an example each of a
proof involving supplementary,
complementary, congruent, and
right angles.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
FIND THE ERROR
Explain that
when two angle
must share a common ray.
Students can note that the
, and a combination of these
is BE
letters appears in both angles
that are being added (ABE and
EBC).
Organization by Objective
• Supplementary and
Complementary Angles:
16–18
• Congruent and Right
Angles: 19–39
Odd/Even Assignments
Exercises 20–36 and 42–43 are
structured so that students
practice the same concepts
whether they are assigned
odd or even problems.
Assignment Guide
1
2
3
Basic: 17–41 odd, 42–55
Average: 17–41 odd, 42–55
Lesson 2-8 Proving Angle Relationships 111
Write a two-column proof. See margin.
VX bisects WVY.
bisects XVZ.
VY
7. PROOF
Given:
bisects WVY,
7. Given: VX
VY bisects XVZ.
Prove: WVX YVZ
W
X
V
WVX YVZ
Prove:
Y
Z
W
Determine whether the following statements are always, sometimes, or never true.
8. Two angles that are nonadjacent are ? vertical. sometimes
9. Two angles that are congruent are ? complementary to the same angle.
X
V
Y
sometimes
Z
Proof:
Statements (Reasons)
bisects WVY; VY
bisects
1.VX
XVZ. (Given)
2.WVX XVY (Def. of bisector)
3.XVY YVZ (Def. of bisector)
4.WVX YVZ (Tran. Prop.)
10. Given: Two angles form a
linear pair.
Prove: The angles are
supplementary
1
2
Paragraph Proof: When two angles
form a linear pair, the resulting
angle is a straight angle whose
measure is 180. By definition, two
angles are supplementary if the
sum of their measures is 180. By
m1 m2 180. Thus, if two
angles form a linear pair, then the
angles are supplementary.
11. Given: ABC is a right angle.
Prove: 1 and 2 are
complementary angles.
Application
ALGEBRA For Exercises 12–15, use the
following information.
1 and X are complementary,
2 and X are complementary,
m1 2n 2, and m2 n 32.
12. Find n. 30
14. What is m2? 62
1
2
X
13. Find m1. 62
15. Find mX. 28
Practice and Apply
For
Exercises
See
Examples
16–18
19–24
25–39
1, 2
4
3
Find the measure of each numbered angle.
16. m2 = 67 m1 113 17. m3 = 38 m4 52 18. 7 and 8 are
complementary. 5 8
and m6 = 29.
1
3
2
4
Extra Practice
6
1
5
18. m5 61,
m7 29,
m8 61
19. m9 2x 4,
m10 2x 4
20. m11 4x,
m12 2x 6
10
9
11
22. m15 x,
m16 6x 290
15
16
m15 58,
m16 58
Proof:
Statements (Reasons)
1.ABC is a right angle. (Given)
2.mABC 90 (Def. of rt. )
3.mABC m1 m2
4.90 m1 m2 (Subst.)
5.1 and 2 are complementary
angles. (Def. of comp. )
Chapter 2 Reasoning and Proof
112 Chapter 2 Reasoning and Proof
8
21. m13 2x 94,
m14 7x 49
13
12
14
m9 86, m10 94
2
A
7
See page 758.
m11 124,
m12 56
23. m17 2x 7,
m18 x 30
m13 112,
m14 112
24. m19 100 20x,
m20 20x
m19 140,
m20 40
C
B
112
Write a proof for each theorem. 10–11. See margin.
10. Supplement Theorem
11. Complement Theorem
PROOF
17
18
m17 53,
m18 53
20
19
25. Prove that congruence of angles is reflexive. See p. 123F.
26. Write a proof of the Transitive Property of Angle Congruence. See p. 123F.
NAME ______________________________________________ DATE
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
2-8
Study
Guide and
p. 99
(shown)
and p. 100
Proving
Angle Relationships
Supplementary and Complementary Angles There are two basic postulates for
working with angles. The Protractor Postulate assigns numbers to angle measures, and the
Angle Addition Postulate relates parts of an angle to the whole angle.
Protractor
Postulate
and a number r between 0 and 180, there is exactly one ray
Given AB
, such that the measure
with endpoint A, extending on either side of AB
of the angle formed is r.
Postulate
R is in the interior of PQS if and only if
mPQR mRQS mPQS.
R
Supplement
Theorem
If two angles form a linear pair, then they are supplementary angles.
If 1 and 2 form a linear pair, then m1 m2 180.
Complement
Theorem
Example 1 If 1 and 2 form a
linear pair and m2 115, find m1.
D
R
A
Interstate highways that
run from north to south
are odd-numbered with
the lowest numbers in the
west. East-west interstates
are even-numbered, and
begin in the south.
B
W
1
S
T
m1 m2 90
m1 20 90
m1 70
Suppl. Theorem
Substitution
Subtraction Prop.
Compl. Theorem
Substitution
Subtraction Prop.
1.
2.
T
P
7
Y
X
8
Q
8
R
U
S
m7 5x 5,
m8 x 5
m7 155,
m8 25
7 6
Z
3. A
13
5
V
F
W
m5 5x, m6 4x 6,
m7 10x,
m8 12x 12
H
J
C
m11 110,
m12 110,
m13 70
m5 30, m6 30,
m7 60, m8 60
NAME
______________________________________________
DATE
/M
G
Hill
99
Skills
Practice,
2-8
Practice
(Average)
____________
Gl PERIOD
G _____
p. 101 and
Practice,
p. Relationships
102 (shown)
Proving Angle
Find the measure of each numbered angle.
1. m1 x 10
m2 3x 18
2. m4 2x 5
m5 4x 13
3. m6 7x 24
m7 5x 14
6
3 5
1
T
U
7
2
m1 48,
m2 132
W
S
12
m11 11x,
m12 10x 10
4
C
11
m3 90, m4 31,
m5 59
m6 109,
m7 109
Determine whether the following statements are always, sometimes, or never true.
4. Two angles that are supplementary are complementary.
never
5. Complementary angles are congruent.
sometimes
40. RIVERS Tributaries of rivers sometimes
form a linear pair of angles when they meet
the main river. The Yellowstone River forms
the linear pair 1 and 2 with the Missouri
River. If m1 is 28, find m2. 152
1
6. Write a two-column proof.
Given: 1 and 2 form a linear pair.
2 and 3 are supplementary.
Prove: 1 3
2
Proof:
Statements
1. 1 and 2 form a linear pair.
2 and 3 are supplementary.
2. 1 and 2 are supplementary.
3. 1 3
41. HIGHWAYS Near the city of Hopewell,
Virginia, Route 10 runs perpendicular to
Interstate 95 and Interstate 295. Show that the
angles at the intersections of Route 10 with
Interstate 95 and Interstate 295 are congruent.
95
1
2
2. Supplement Theorem
3. suppl. to the same or are .
Barton
Rd
Tryon
St
Olive Tree Lane
Route 10
NAME
______________________________________________
DATE
/M
G
Hill
102
Gl
____________
Gl PERIOD
G _____
2-8
to Learn
Learn Mathematics
Mathematics,
p. 103
Proving Angle Relationships
Pre-Activity
ELL
How do scissors illustrate supplementary angles?
Read the introduction to Lesson 2-8 at the top of page 107 in your textbook.
Is it possible to open a pair of scissors so that the angles formed by the two
blades, a blade and a handle, and the two handles, are all congruent? If so,
explain how this could happen. Sample answer: Yes; open the
scissors so that the two blades are perpendicular. Then all the
angles will be right angles and will be congruent.
1. Complete each sentence to form a statement that is always true.
2 3
1
a. If two angles form a linear pair, then they are adjacent and supplementary .
4
b. If two angles are complementary to the same angle, then they are
Lesson 2-8 Proving Angle Relationships 113
(l)Richard Pasley/Stock Boston, (r)Sam Abell/National Geographic Image Collection
NAME ______________________________________________ DATE
2-8
Enrichment
Enrichment,
____________ PERIOD _____
.
.
e. If two angles are congruent and supplementary, then each angle is a(n)
right
angle.
Perpendicular lines form congruent adjacent angles.
g. “Every angle is congruent to itself” is a statement of the
of angle congruence.
Reflexive
Property
h. If two congruent angles form a linear pair, then the measure of each angle is
p. 104
90
.
i. If the noncommon sides of two adjacent angles form a right angle, then the angles are
complementary .
2. Determine whether each statement is always, sometimes, or never true.
a. Supplementary angles are congruent. sometimes
b. If two angles form a linear pair, they are complementary. never
c. Two vertical angles are supplementary. sometimes
d. Two adjacent angles form a linear pair. sometimes
e. Two vertical angles form a linear pair. never
f. Complementary angles are congruent. sometimes
g. Two angles that are congruent to the same angle are congruent to each other. always
h. Complementary angles are adjacent angles. sometimes
Bisecting a Hidden Angle
The vertex of BAD at the
right is hidden in a region.
Within the region, you are
not allowed to use a compass.
Can you bisect the angle?
C
B
P
Q
Helping You Remember
A
D
E
congruent
mDBC
c. If D is a point in the interior of ABC, then mABC mABD and a number x between
0
and 180 , there is exactly one ray
d. Given RS
with endpoint R, extended on either side of RS, such that the measure of the angle
formed is x.
f.
42. m1 m4 90;
m1 m2 m3 m4 180
m1 m1 m4 m4 180
2(m1) 2(m4) 180
2(m1 m4) 180
m1 m4 90
Reasons
1. Given
7. STREETS Refer to the figure. Barton Road and Olive Tree Lane
form a right angle at their intersection. Tryon Street forms a 57°
angle with Olive Tree Lane. What is the measure of the acute angle
Tryon Street forms with Barton Road? 33
42. CRITICAL THINKING What conclusion can you
make about the sum of m1 and m4 if m1 m2
and m3 m4? Explain. See margin.
3
295
Because the lines are perpendicular, the
angles formed are right angles. All right
angles are congruent. Therefore, 1 is
congruent to 2.
www.geometryonline.com/self_check_quiz
1 2
Lesson 2-8
Highways
H
Example 2 If 1 and 2 form a
right angle and m2 20, find m1.
P
Y
Z
4
G
Exercises
Gl
R
C
J
3
Find the measure of each numbered angle.
Write a two-column proof. 38–39. See p. 123G.
38. Given: ABD YXZ
39. Given: mRSW mTSU
Prove: CBD WXZ
Prove: mRST mWSU
X
B
F
2
N
m1 m2 180
m1 115 180
m1 65
PROOF
W
2
1
If the noncommon sides of two adjacent angles form a right angle,
then the angles are complementary angles.
⊥ GH
, then m3 m4 90.
If GF
2 1
33.
34.
35.
36.
37.
D
A
M
Use the figure to write a proof of each theorem. 33–37. See p. 123F.
Theorem 2.9
Theorem 2.10
m
1 2
Theorem 2.11
3 4
Theorem 2.12
Theorem 2.13
S
The two postulates can be used to prove the following two theorems.
Q
PROOF
P
Q
Lesson 2-8
Determine whether the following statements are always, sometimes, or never true.
27. Two angles that are complementary ? form a right angle. sometimes
28. Two angles that are vertical are ? nonadjacent. always
29. Two angles that form a right angle are ? complementary. always
30. Two angles that form a linear pair are ? congruent. sometimes
31. Two angles that are supplementary are ? congruent. sometimes
32. Two angles that form a linear pair are ? supplementary. always
3. A good way to remember something is to explain it to someone else. Suppose that a
classmate thinks that two angles can only be vertical angles if one angle lies above the
other. How can you explain to him the meaning of vertical angles, using the word vertex
in your explanation? Sample answer: Two angles are vertical angles if they
share the same vertex and their sides are opposite rays. It doesn’t matter
how the angles are positioned.
1. Use a straightedge to draw lines CE and BD.
Lesson 2-8 Proving Angle Relationships 113
43. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson. See margin.
How do scissors illustrate supplementary angles?
4 Assess
• a description of the relationship among 1, 2, and 3,
• an example of another way that you can tell the relationship between 1
and 3, and
• an explanation of whether this relationship is the same for two angles
complementary to the same angle.
Open-Ended Assessment
Writing Give students a list of
theorems from this chapter. Have
students choose a theorem and
write a proof of it with their
books closed.
Assessment Options
Standardized
Test Practice
Quiz (Lessons 2-7 and 2-8) is
available on p. 120 of the Chapter 2
Resource Masters.
44. The measures of two complementary angles are in the ratio 4:1. What is the
measure of the smaller angle? B
A 15
B 18
C 24
D 36
45. ALGEBRA T is the set of all positive numbers n such that n 50 and n
is an
integer. What is the median of the members of set T? B
A 4
B 16
C 20
D 25
43. Two angles that are
supplementary to the same angle
include the following.
• 1 and 2 are supplementary;
2 and 3 are supplementary.
• 1 and 3 are vertical angles,
and are therefore congruent.
• If two angles are
complementary to the same
angle, then the angles are
congruent.
46. Given: G is between F and H.
H is between F and J.
Prove: FG GJ FH HJ
F
G
H
X
Y
Mixed Review
Write a two-column proof. (Lesson 2-7) 46–47. See margin.
PROOF
46. Given: G is between F and H.
J
H is between G and J.
H
F G
Prove: FG GJ FH HJ
47. Given: X is the midpoint of W
Y
.
Prove:
W
WX YZ XZ
X
48. PHOTOGRAPHY Film is fed through a camera
by gears that catch the perforation in the film. The
distance from the left edge of the film, A, to the right
edge of the image, C, is the same as the distance
from the left edge of the image, B, to the right edge
of the film, D. Show that the two perforated strips
are the same width. (Lesson 2-6) See p. 123G.
Y
Z
A
B
C
D
J
Proof:
Statements (Reasons)
1.G is between F and H; H is
between F and J. (Given)
2.FG GJ FJ, FH HJ FJ
3.FJ FH HJ (Sym. Prop.)
4.FG GJ FH HJ (Transitive
Prop.)
47. Given: X is the midpoint of WY
.
Prove: WX YZ XZ
W
Z
Proof:
Statements (Reasons)
1.X is the midpoint of WY
. (Given)
2.WX XY (Def. of midpoint)
Post.)
4.WX YZ XZ (Substitution)
114 Chapter 2 Reasoning and Proof
For Exercises 49–55, refer to the figure at the right.
I
(Lesson 1-4)
50. PMQ QMN
52. POQ, QON,
NOM, MOP
P
O
Q
Name two angles that have N as a vertex. ONM, MNR
N
M
R
៮
៮Q៬ bisects PMN, name two congruent angles.
If M
Name a point in the interior of LMQ. N or R
L
List all the angles that have O as the vertex.
Does QML appear to be acute, obtuse, right, or
straight? obtuse
and NP
54. Name a pair of opposite rays. Sample answer: NR
55. List all the angles that have M
N
as a side. NML, NMP, NMO, RNM, ONM
49.
50.
51.
52.
53.
114 Chapter 2 Reasoning and Proof
9.
45
X
N
L
O
135
A
10.
11. M
Y
B
Z
Study Guide
and Review
Vocabulary and Concept Check
axiom (p. 89)
biconditional (p. 81)
compound statement (p. 67)
conclusion (p. 75)
conditional statement (p. 75)
conjecture (p. 62)
conjunction (p. 68)
contrapositive (p. 77)
converse (p. 77)
counterexample (p. 63)
deductive argument (p. 94)
deductive reasoning (p. 82)
disjunction (p. 68)
formal proof (p. 95)
hypothesis (p. 75)
if-then statement (p. 75)
inductive reasoning (p. 62)
informal proof (p. 90)
inverse (p. 77)
Law of Detachment (p. 82)
Law of Syllogism (p. 83)
logically equivalent (p. 77)
negation (p. 67)
paragraph proof (p. 90)
Vocabulary and
Concept Check
postulate (p. 89)
proof (p. 90)
related conditionals (p. 77)
statement (p. 67)
theorem (p. 90)
truth table (p. 70)
truth value (p. 67)
two-column proof (p. 95)
• This alphabetical list of
vocabulary terms in Chapter 2
includes a page reference
where each term was
introduced.
• Assessment A vocabulary
test/review for Chapter 2 is
available on p. 118 of the
Chapter 2 Resource Masters.
A complete list of postulates and theorems can be found on pages R1–R8.
Exercises Choose the correct term to complete each sentence.
1. A (counterexample, conjecture ) is an educated guess based on known information.
2. The truth or falsity of a statement is called its (conclusion, truth value ).
3. Two or more statements can be joined to form a (conditional, compound ) statement.
Lesson-by-Lesson
Review
4. A conjunction is a compound statement formed by joining two or more
statements using (or, and ).
5. The phrase immediately following the word if in a conditional statement is
called the ( hypothesis , conclusion).
For each lesson,
• the main ideas are
summarized,
concepts, and
• practice exercises are provided.
6. The (inverse, converse ) is formed by exchanging the hypothesis and the conclusion.
7. (Theorems, Postulates ) are accepted as true without proof.
8. A paragraph proof is a (an) ( informal proof , formal proof ).
Vocabulary
PuzzleMaker
2-1 Inductive Reasoning and Conjecture
See pages
62–66.
Example
Concept Summary
• Conjectures are based on observations and patterns.
• Counterexamples can be used to show that a conjecture is false.
Given that points P, Q, and R are collinear, determine whether the
conjecture that Q is between P and R is true or false. If the conjecture
is false, give a counterexample.
In the figure, R is between P and Q. Since we can find
P
R
a counterexample, the conjecture is false.
Q
Exercises Make a conjecture based on the given information. Draw a figure to
illustrate your conjecture. See Example 2 on page 63. 9–11. See margin for figures.
9. A and B are supplementary. mA mB 180
10. X, Y, and Z are collinear and XY YZ. Y is the midpoint of XZ.
11. In quadrilateral LMNO, LM LO MN NO, and m L 90. LMNO is a square.
www.geometryonline.com/vocabulary_review
Chapter 2 Study Guide and Review 115
TM
Teaching Mathematics
with Foldables.
Have students look through the chapter to make sure they have
included notes and examples in their Foldables for each lesson of
Chapter 2.
Encourage students to refer to their Foldables while completing
the Study Guide and Review and to use them in preparing for the
Chapter Test.
ELL The Vocabulary PuzzleMaker
software improves students’ mathematics
vocabulary using four puzzle formats—
crossword, scramble, word search using a
word list, and word search using clues.
Students can work on a computer screen
or from a printed handout.
MindJogger
Videoquizzes
ELL MindJogger Videoquizzes
provide an alternative review of concepts
presented in this chapter. Students work
in teams in a game show format to gain
points for correct answers. The questions
are presented in three rounds.
Round 1 Concepts (5 questions)
Round 2 Skills (4 questions)
Round 3 Problem Solving (4 questions)
Chapter 2 Study Guide and Review 115
1/16/2003 1:48 PM
Study Guide and Review
12. 1 0 and in a right triangle
with right angle C, a 2 b 2 c 2.
13. In a right triangle with right angle
C, a 2 b 2 c 2 or the sum of the
measures of two supplementary
angles is 180.
14. The sum of the measures of two
supplementary angles is 180 and
1 0.
15. 1 0, and in a right triangle
with right angle C, a2 b2 c2,
or the sum of the measures of two
supplementary angles is 180.
16. In a right triangle with right angle
C, a2 b2 c2, or 1 0 or the
sum of the measures of two
supplementary angles is 180.
17. In a right triangle with right angle
C, a2 b2 c2 and the sum of the
measures of two supplementary
angles is 180, and 1 0.
18. Converse: If an angle is obtuse,
then it measures 120. False; the
measure could be any value
between 90 and 180. Inverse: If
an angle measure does not equal
120, then it is not obtuse. False;
the measure could be any value
other than 120 between 90 and
180. Contrapositive: If an angle is
not obtuse, then its measure does
not equal 120; true.
19. Converse: If a month has 31 days,
then it is March. False; July has
31 days. Inverse: If a month is not
March, then it does not have
31 days. False; July has 31 days.
Contrapositive: If a month does
not have 31 days, then it is not
March; true.
20. Converse: If a point lies on the
y-axis, then its ordered pair has 0
for its x-coordinate; true.
Inverse: If an ordered pair does
not have 0 for its x-coordinate,
then the point does not lie on the
y-axis; true. Contrapositive: If a
point does not lie on the y-axis,
then its ordered pair does not
have 0 for its x-coordinate; true.
116
Chapter 2 Reasoning and Proof
T_Maria_Manko 115-121 GEO C2SGT-
Chapter 2
X Study Guide and Review
• Extra Practice, see pages xxx-xxx.
• Mixed Problem Solving, see page xxx.
2-2 Logic
See pages
67–74.
Example
Concept Summary
• The negation of a statement has the opposite truth value of the original
statement.
• Venn diagrams and truth tables can be used to determine the truth values of
statements.
Use the following statements to write a compound statement for each conjunction.
Then find its truth value.
p: 15
5
q: The measure of a right angle equals 90.
a. p and q
15
5, and the measure of a right angle equals 90.
p and q is false because p is false and q is true.
b. p q
15
5, or the measure of a right angle equals 90.
p q is true because q is true. It does not matter that p is false.
Exercises Use the following statements to write a compound statement for each
conjunction. Then find its truth value. See Examples 1 and 2 on pages 68 and 69.
p: 1 0
q: In a right triangle with right angle C, a2 b2 c 2.
r: The sum of the measures of two supplementary angles is 180.
13. q or r true
14. r p false
12. p and q false
15. p (q r) false
16. q (p r) true
17. (q r) p false
12 – 17. See margin for statements.
2-3 Conditional Statements
See pages
75–80.
Example
Concept Summary
• Conditional statements are written in if-then form.
• Form the converse, inverse, and contrapositive of an if-then statement by using
negations and by exchanging the hypothesis and conclusion.
Identify the hypothesis and conclusion of the statement The intersection of two
planes is a line. Then write the statement in if-then form.
Hypothesis: two planes intersect
Conclusion: their intersection is a line
If two planes intersect, then their intersection is a line.
Exercises Write the converse, inverse, and contrapositive of each conditional
statement. Determine whether each related conditional is true or false. If a
statement is false, find a counterexample. See Example 4 on page 77. 18–20. See margin.
18. If an angle measure equals 120, then the angle is obtuse.
19. If the month is March, then it has 31 days.
20. If an ordered pair for a point has 0 for its x-coordinate, then the point lies on
the y-axis.
116 Chapter 2 Reasoning and Proof
Chapter 2 Study Guide and Review
Study Guide and Review
Determine the truth value of the following statement for each set of conditions.
If the temperature is at most 0°C, then water freezes. See Example 3 on page 76.
21. The temperature is 10°C, and water freezes. true
22. The temperature is 15°C, and water freezes. true
23. The temperature is 2°C, and water does not freeze. false
24. The temperature is 30°C, and water does not freeze. true
2-4 Deductive Reasoning
See pages
82–87.
Example
Concept Summary
• The Law of Detachment and the Law of Syllogism can be used to
determine the truth value of a compound statement.
Use the Law of Syllogism to determine whether a valid conclusion can be
reached from the following statements.
(1) If a body in our solar system is the Sun, then it is a star.
(2) Stars are in constant motion.
p: a body in our solar system is the sun
q: it is a star
r: stars are in constant motion
Statement (1): p → q Statement (2): q → r
Since the given statements are true, use the Law of Syllogism to conclude p → r. That
is, If a body in our solar system is the Sun, then it is in constant motion.
Exercises Determine whether the stated conclusion is valid based on the given
information. If not, write invalid. Explain your reasoning. See Example 1 on page 82.
If two angles are adjacent, then they have a common vertex.
25. Given: 1 and 2 are adjacent angles. Valid; by definition, adjacent angles
Conclusion: 1 and 2 have a common vertex. have a common vertex.
26. Given: 3 and 4 have a common vertex. Invalid; vertical angles also have
Conclusion: 3 and 4 are adjacent angles. a common vertex.
Determine whether statement (3) follows from statements (1) and (2) by the Law
of Detachment or the Law of Syllogism. If it does, state which law was used. If it
does not follow, write invalid. See Example 3 on page 83.
27. (1) If a student attends North High School, then the student has an ID number.
(2) Josh Michael attends North High School.
(3) Josh Michael has an ID number. yes; Law of Detachment
28. (1) If a rectangle has four congruent sides, then it is a square.
(2) A square has diagonals that are perpendicular.
(3) A rectangle has diagonals that are perpendicular. invalid
29. (1) If you like pizza with everything, then you’ll like Cardo’s Pizza. yes; Law
(2) If you like Cardo’s Pizza, then you are a pizza connoisseur. of Syllogism
(3) If you like pizza with everything, then you are a pizza connoisseur.
Chapter 2 Study Guide and Review 117
Chapter 2 Study Guide and Review 117
Study Guide and Review
Chapter 2 Study Guide and Review
2-5 Postulates and Paragraph Proofs
30. Never; the intersection of two
lines is a point.
31. Always; if P is the midpoint of X
Y
,
then X
P
PY
. By definition of
congruent segments, XP PY.
32. Sometimes; if M, X, and Y are
collinear.
33. Sometimes; if the points are
collinear.
34. Always; there is exactly one line
through Q and R. The line lies in
at least one plane.
35. Sometimes; if the right angles
form a linear pair.
36. Always; the Reflexive Property
states that 1 1.
37. Never; adjacent angles must share
a common side, and vertical
angles do not.
38. If M is the midpoint of AB
, then
See pages
89–93.
Example
Concept Summary
• Use undefined terms, definitions, postulates, and theorems to prove
that statements and conjectures are true.
Determine whether the following statement is always, sometimes, or never
true. Explain. Two points determine a line.
According to a postulate relating to points and lines, two points determine a line.
Thus, the statement is always true.
Exercises Determine whether the following statements are always,
sometimes, or never true. Explain. See Example 2 on page 90. 30–37. See margin.
30. The intersection of two lines can be a line.
31. If P is the midpoint of X
Y
, then XP PY.
32. If MX MY, then M is the midpoint of XY.
33. Three points determine a line.
34. Points Q and R lie in at least one plane.
35. If two angles are right angles, they are adjacent.
36. An angle is congruent to itself.
38. PROOF
Write a paragraph proof to prove that
A
Q
if M is the midpoint of A
B
and Q is the midpoint
1
4
of A
M
, then AQ AB. See margin.
1
AM (AB). Since Q is the
2
1
midpoint of AM
, AQ AM or
1 1
1
(AB) AB.
2 2
4
2
2-6 Algebraic Proof
See pages
94–100.
Example
Concept Summary
• The properties of equality used in algebra can be applied to the
measures of segments and angles to verify and prove statements.
5
Given: 2x 6 3 x
3
Prove: x 9
Proof:
Statements
5
1. 2x 6 3 x
3
1. Given
5
2. 3(2x 6) 3 3 x
3
3.
4.
5.
6.
7.
6x 18 9 5x
6x 18 5x 9 5x 5x
x 18 9
x 18 18 9 18
x9
118 Chapter 2 Reasoning and Proof
118
Chapter 2 Reasoning and Proof
Reasons
2. Multiplication Property
3.
4.
5.
6.
7.
Distributive Property
Subtraction Property
Substitution
Subtraction Property
Substitution
M
B
Study Guide and Review
Chapter 2 Study Guide and Review
Exercises State the property that justifies each statement. See Example 1 on page 94.
39. If 3(x 2) 6, then 3x 6 6. Dist. Prop.
1
43. Given: 5 2 x
2
Prove: x 6
Proof:
Statements (Reasons)
40. If 10x 20, then x 2. Div. Prop.
41. If AB 20 45, then AB 25. Subt. Prop.
42. If 3 CD and CD XY, then 3 XY. Transitive Prop.
PROOF
1
2
Write a two-column proof. See Examples 2 and 4 on pages 95 and 96.
1.5 2 x (Given)
1
43. If 5 2 x, then x 6.
2
x 10
44. If x 1 , then x 4.
2
45. If AC = AB, AC = 4x 1, and AB 6x 13, then x 7.
2
46. If MN PQ and PQ RS, then MN RS.
P
x 10
Q
Proof:
Statements
1.
2.
3.
4.
5.
x 10
2
1.x 1 (Given)
T
S
R
1.
2.
3.
4.
5.
Given
Substitution
Substitution
3x
3
9.x 4 (Substitution)
45. Given: AC AB, AC 4x 1,
AB 6x 13
Prove: x 7
A
1
2
51. If 2(XY) AB, then XY (AB). Div. or Mult. Prop.
6x 13
4x 1
52. If AB CD, then AB BC CD BC. Add. Prop.
Chapter 2 Study Guide and Review 119
46. Given: MN PQ, PQ RS
Prove: MN RS
N
P
12
3
8. (Div. Prop.)
Exercises Justify each statement with a property of equality or a property
of congruence. See Example 1 on page 102.
47. PS PS Reflexive Prop.
48. If XY OP, then OP XY. Symmetric Prop.
49. If AB 8 CD 8, then AB CD. Add. Prop.
50. If EF GH and GH LM, then EF LM. Transitive Prop.
M
x 10 2.2(x 1) 2 2
(Mult. Prop.)
3.2x 2 x 10 (Dist. Prop.)
4.2x 2 2 x 10 2
(Subt. Prop.)
5.2x x 12 (Substitution)
6.2x x x 12 x
(Subt. Prop.)
7.3x 12 (Substitution)
Reasons
QT RT, TS TP
QT TS RT TS
QT TS RT TP
QT TS QS, RT TP RP
QS RP
44. Given: x 1 2
Prove: x 4
Proof:
Statements (Reasons)
Concept Summary
• Use properties of equality and congruence to write proofs
involving segments.
Write a two-column proof.
Given: QT RT, TS TP
Prove: QS RP
1
2
5.6 x (Substitution)
6.x 6 (Sym. Prop.)
2-7 Proving Segment Relationships
Example
4.2(3) 2 x (Mult. Prop)
43 – 46. See margin.
See pages
101–106.
1
2.5 2 2 x 2
2
(Subt. Prop.)
1
3.3 x (Substitution)
R
Q
S
Proof:
Statements (Reasons)
1.MN PQ, PQ RS (Given)
2.MN RS (Transitive Prop.)
B
C
Proof:
Statements (Reasons)
1.AC AB,AC 4x 1,
AB 6x 13 (Given)
2.4x 1 6x 13 (Subst.)
3.4x 1 1 6x 13 1
(Subt. Prop.)
4.4x 6x 14 (Subst.)
5.4x 6x 6x 14 6x
(Subt. Prop.)
6.2x 14 (Subst.)
2x
2
14
2
7. (Div. Prop.)
8.x 7 (Subst.)
Chapter 2 Study Guide and Review 119
• Extra Practice, see pages 756–758.
• Mixed Problem Solving, see page 783.
Study Guide and Review
PROOF
1. Sample answer: Formal is the
two-column proof, informal can
be paragraph proofs.
2. Sample answer: You can use a
counterexample.
reasons to justify statements
7. 3 2 and 3x 12 when x 4.
8. 3 2 or 3x 12 when x 4.
9. 3 2, or 3x 12 when x 4
and an equilateral triangle is also
equiangular.
10. H: you eat an apple a day; C: the
doctor will stay away; If you eat
an apple a day, then the doctor
will stay away. Converse: If the
doctor stays away, then you eat an
apple a day. Inverse: If you do not
eat an apple a day, then the
doctor will not stay away.
Contrapositive: If the doctor does
not stay away, then you do not eat
an apple a day.
11. H: a stone is rolling; C: it gathers
no moss; If a stone is rolling, then
it gathers no moss. Converse: If a
stone gathers no moss, then it is
rolling. Inverse: If a stone is not
rolling, then it gathers moss.
Contrapositive: If a stone gathers
moss, then it is not rolling.
16. Given: y 4x 9; x 2
Prove: y 17
Proof:
Statements (Reasons)
1.y 4x 9; x 2 (Given)
2.y 4(2) 9 (Substitution)
3.y 8 9 (Substitution)
4.y 17 (Substitution)
17. Given: AM CN, MB ND
A
M
B
53. Given:
Prove:
54. Given: AB CD
Prove: AC BD
E
A B
C D
C
D
A
53 – 54. See p. 123G.
2-8 Proving Angle Relationships
See pages
107–114.
Example
Concept Summary
• The properties of equality and congruence can be applied to angle
relationships.
Find the measure of each numbered angle.
m1 55, since 1 is a vertical angle to the 55° angle.
2 and the 55° angle form a linear pair.
2
1
55°
55 + m2 180 Def. of supplementary m2 125 Subtract 55 from each side.
Exercises Find the measure of each numbered angle.
55. m6 145
56. m7 23
57. m8 90
58. PROOF
See Example 2 on page 108.
7
157°
6
35°
8
Copy and complete the proof.
See Example 3 on page 109.
Given: 1 and 2 form a linear pair.
m2 2(m1)
Prove: m1 60
Proof:
Statements
Reasons
a. 1 and 2 form a linear pair.
a.
?
b.
?
b. 1 and 2 are supplementary.
c.
?
m1 m2 180
d. m2 2(m1)
e.
?
f.
?
m1 2(m1) 180
3(m1) 180
3(m1)
180
g. 3
3
h.
?
m1 60
120 Chapter 2 Reasoning and Proof
Prove: AB CD
Chapter 2 Reasoning and Proof
BC EC, CA CD
BA DE
B
Paragraph Proof:
We are given that AM CN, MB ND. By the
D
N
C
Addition Property, AM MB CN MB. By
Substitution, AM MB CN ND. Using
the Segment Addition Postulate, AB AM MB,
and CD CN ND. Then, by Substitution AB CD.
120
Write a two-column proof. See Examples 1 and 2 on pages 102 and 103.
18. H: you are a hard-working person;
C: you deserve a great vacation;
If you are a hard-working person,
then you deserve a great vacation.
Given
Supplement Theorem
c. Definition of supplementary angles
d.
?
Given
e. Substitution
f. Substitution
g.
?
Division Property
h. Substitution
Practice Test
Vocabulary and Concepts
Assessment Options
1. Explain the difference between formal and informal proofs. 1–3. See margin.
2. Explain how you can prove that a conjecture is false.
3. Describe the parts of a two-column proof.
Vocabulary Test A vocabulary
test/review for Chapter 2 can be
found on p. 118 of the Chapter 2
Resource Masters.
Skills and Applications
Chapter Tests There are six
Chapter 2 Tests and an OpenEnded Assessment task available
in the Chapter 2 Resource Masters.
Determine whether each conjecture is true or false. Explain your answer and
give a counterexample for any false conjecture.
4. Given: A B
5. Given: y is a real number
6. Given: 3a2 48
Conjecture: B A
Conjecture: y 0
Conjecture: a 4
true; Symmetric Prop.
false; y 2
false; a 4
Use the following statements to write a compound statement for each conjunction or
disjunction. Then find its truth value. 7–9. See margin for statements.
p: 3 2
q: 3x 12 when x 4.
r: An equilateral triangle is also equiangular.
7. p and q false
Form
9. p (q r) true
8. p or q true
Identify the hypothesis and conclusion of each statement and write each statement in
if-then form. Then write the converse, inverse, and contrapositive of each conditional. 10–11. See margin.
10. An apple a day keeps the doctor away.
11. A rolling stone gathers no moss.
12. Determine whether statement (3) follows from statements (1) and (2) by the
Law of Detachment or the Law of Syllogism. If it does, state which law was
used. If it does not, write invalid.
(1) Perpendicular lines intersect.
(2) Lines m and n are perpendicular.
(3) Lines m and n intersect. valid; Law of Detachment
Find the measure of each numbered angle.
13. 1 22
14. 2 85
15. 3See
85margin.
16–17.
16.Write a two-column proof.
If y 4x 9 and x 2, then y 17.
D
N
Pages
105–106
107–108
109–110
111–112
113–114
115–116
ExamView® Pro
B
C
18. ADVERTISING Identify the hypothesis and conclusion of the following statement, then write it
in if-then form. Hard working people deserve a great vacation. See margin.
19. STANDARDIZED TEST PRACTICE If two planes intersect, their intersection can be A
I a line.
II three noncollinear points.
III two intersecting lines.
A I only
B II only
C
D I and II only
III only
Chapter 2 Practice Test
www.geometryonline.com/chapter_test
basic
average
average
average
average
Open-Ended Assessment
can be found on p. 117 of the
Chapter 2 Resource Masters. A
sample scoring rubric for these
17. Write a paragraph proof.
Given: AM CN, MB ND
Prove: AB CD
M
MC
MC
MC
FR
FR
FR
MC = multiple-choice questions
FR = free-response questions
95˚
2 3
1 73˚
A
1
2A
2B
2C
2D
3
Chapter 2 Tests
Type
Level
121
Portfolio Suggestion
Introduction In a chapter of diverse material, highlight concepts that are
important but may not be used often or regularly, so that they will remain
fresh in students’ minds.
Ask Students Search the chapter for items you found the most difficult.
Record these items in your portfolio and write about how you were able to
master the concepts. If you are still having difficulty with the concepts, write
about the steps you could take to better your understanding of them.
Use the networkable
ExamView® Pro to:
• Create multiple versions of
tests.
• Create modified tests for
Inclusion students.
• Edit existing questions and
• Use built-in state curriculum
correlations to create tests
aligned with state standards.
• Apply art to your tests from a
program bank of artwork.
Chapter 2 Practice Test 121
Standardized
Test Practice
These two pages contain practice
questions in the various formats
that can be found on the most
frequently given standardized
tests.
6. Which of the following is an example of
inductive reasoning? (Lesson 2-1) C
Part 1 Multiple Choice
provided by your teacher or on a sheet
of paper.
A
B
C
Practice
2Standardized
Standardized Test
Test Practice
Student Record
Sheet (Use with Sheet,
pages 122–123 ofp.
the Student
Student
Recording
A1 Edition.)
D
Part 1 Multiple Choice
B
C
D
4
A
B
C
D
7
A
B
C
D
2
A
B
C
D
5
A
B
C
D
8
A
B
C
D
3
A
B
C
D
6
A
B
C
D
B
Carlos reads in his textbook that the
measure of all right angles is 90. He
conjectures that the measure of each
right angle in a square equals 90.
1
7
1
7 2, 7, , 7
7
1
7, , 7, 7 2
7
1
7 2, , 7, 7
7
C
Carlos measures the angles of several
triangles and finds that their measures
all add up to 180. He conjectures that
the sum of the measures of the angles
in any triangle is always 180.
D
Carlos knows that the sum of the
measures of the angles in a square is
always 360. He conjectures that if he
draws a square, the sum of the measures
of the angles will be 360.
7, 7, , 7 2
Select the best answer from the choices given and fill in the corresponding oval.
A
Carlos learns that the measures of
all acute angles are less than 90. He
conjectures that if he sees an acute
angle, its measure will be less than 90.
1
1. Arrange the numbers 7, , 7, 7 2 in order
7
from least to greatest. (Prerequisite Skill) D
A practice answer sheet for these
two pages can be found on p. A1
of the Chapter 2 Resource Masters.
1
A
2. Points A and B lie on the line y 2x 3.
Which of the following are coordinates of a
point noncollinear with A and B? (Lesson 1-1)C
Part 2 Short Response/Grid In
For Questions 9 and 11, also enter your answer by writing each number or symbol
in a box. Then fill in the corresponding oval for that number or symbol.
9
(grid in)
10
11
(grid in)
12
9
A
(7, 11)
B
(4, 5)
C
(–2, –10)
D
(–5, –13)
11
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
3. Dana is measuring distance on a map. Which
of the following tools should Dana use to
make the most accurate measurement?
(Lesson 1-2)
A
7. Which of the following is the contrapositive of
the statement If Rick buys hamburgers for lunch,
then Denzel buys French fries and a large soda?
(Lesson 2-2)
A
If Denzel does not buy French fries and
a large soda, then Rick does not buy
hamburgers for lunch.
B
If Rick does not buy hamburgers for
lunch, then Denzel does not buy French
fries and a large soda.
C
If Denzel buys French fries and a large
for lunch.
D
If Rick buys hamburgers for lunch, then
Denzel does not buy French fries and a
large soda.
Part 3 Extended Response
Record your answers for Questions 13–15 on the back of this paper.
A
centimeter ruler
B
protractor
C
yardstick
D
calculator
4. Point E is the midpoint of D
F
. If DE 8x 3
and EF 3x 7, what is x? (Lesson 1-3) B
A
See pp. 123–124 in the Chapter 2
standardized test practice.
1
B
2
C
4
D
E
A
8. Which property could justify the first step in
A
B
C
14x 6
8
solving 3 18? (Lesson 2-5) A
A
complementary angles
A
Division Property of Equality
B
congruent angles
B
Substitution Property of Equality
C
supplementary angles
C
D
vertical angles
D
Transitive Property of Equality
122 Chapter 2 Reasoning and Proof
ExamView® Pro
Special banks of standardized test
questions similar to those on the SAT,
ACT, TIMSS 8, NAEP 8, and state
proficiency tests can be found on this
CD-ROM.
122 Chapter 2 Reasoning and Proof
13
F
5. What is the relationship
between ACF and DCF?
(Lesson 1-6)
D
A
Preparing for Standardized Tests
For test-taking strategies and more
practice, see pages 795– 810.
Evaluating Extended
Response Questions
Part 2 Short Response/Grid In
provided by your teacher or on a sheet
of paper.
Test-Taking Tip
9. Two cheerleaders stand at opposite corners
of a football field. What is the shortest
distance between them, to the nearest yard?
(Lesson 1-3)
Question 6
every answer choice and eliminate those you decide are
definitely wrong. This way, you may deduce the correct
131 yd
120 yd
Part 3 Extended Response
10 20 30 40 50 40 30 20 10
1
53 3 yd
10 20 30 40 50 40 30 20 10
10. Consider the conditional If I call in sick, then
I will not get paid for the day. Based on the
original conditional, what is the name of the
conditional If I do not call in sick, then I will
get paid for the day? (Lesson 2-2) inverse
11. Examine the following statements.
p: Martina drank a cup of soy milk.
q: A cup is 8 ounces.
r: Eight ounces of soy milk contains
300 milligrams of calcium.
Using the Law of Syllogism, how many
milligrams of calcium did Martina get
from drinking a cup of soy milk?
(Lesson 2-4)
13. In any right triangle, the sum of the squares
of the lengths of the legs equals the square
of the length of the hypotenuse. From a
single point in her yard, Marti measures and
marks distances of 18 feet and 24 feet for
two sides of her garden. Explain how Marti
can ensure that the two sides of her garden
form a right angle. (Lesson 1-3) See margin.
Extended Response questions are
rubric that guides you in
assessing a student’s knowledge
of a particular concept.
Goal: Find measures and prove
an angle measure.
Sample Scoring Rubric: The
following rubric is a sample
scoring device. You may wish to
add more detail to this sample to
needs.
Score
Criteria
4
A correct solution that is
supported by well-developed,
accurate explanations
A generally correct solution,
but may contain minor flaws
in reasoning or computation
A partially correct interpretation
and/or solution to the problem
A correct solution with no
supporting evidence or
explanation
An incorrect solution indicating
no mathematical understanding
of the concept or task, or no
solution is given
14. A farmer needs to make a 100-square-foot
rectangular enclosure for her chickens. She
wants to save money by purchasing the
least amount of fencing possible to enclose
the area. (Lesson 1-4) a–c. See margin.
3
a. What whole-number dimensions, to the
nearest yard, will require the least
amount of fencing?
1
b. Explain your procedure for finding the
dimensions that will require the least
amount of fencing.
0
2
300
12. In the following proof, what property
justifies statement c? (Lesson 2-7)
Given: AC
MN
Postulate
Prove: AB BC MN
B
C
A
N
M
Proof:
Statements
C
MN
a. A
b. AC MN
c. AC AB BC
d. AC BC MN
c. Explain how the amount of fencing
required to enclose the area changes
as the dimensions change.
Reasons
a. Given
b. Definition of
congruent
segments
c.
?
d. Substitution
www.geometryonline.com/standardized_test
15. Given: 1 and 3 are vertical angles.
m1 3x 5, m3 2x 8
Prove:
m1 14 (Lesson 2-8) See p. 123G.
1
2
4
3
Chapter 2 Standardized Test Practice 123
13. Sample answer: Marti can measure
a third distance c, the distance
between the ends of the two
sides, and make sure it satisfies
the equation a 2 b 2 c 2.
14a. 10 yd by 10 yd
14b. Sample answer: Make a list of all
possible whole-number lengths
and widths that will form a
100-square-foot area. Then find
the perimeter of each rectangle.
Choose the length and width
combination that has the
smallest perimeter.
14c. As the length and width get closer
to having the same measure as
one another, the amount of
fencing required decreases.
Chapter 2 Standardized Test Practice 123
Pages 63–66, Lesson 2-1
1. Sample answer: After the news is over, it’s time for
dinner.
2. Sometimes; the conjecture is true when E is between
D and F; otherwise it is false.
3. Sample answer: When it is cloudy, it rains.
Counterexample: It is often cloudy and it does not rain.
19. 30
20. 20
14.
p q ( p q) r
p
q
r
T
T
T
T
T
T
T
F
T
T
T
F
T
T
T
T
F
F
T
T
F
T
T
T
T
F
T
F
T
T
F
F
T
F
T
F
F
F
F
F
8 and an equilateral triangle has three
18. 64
congruent sides.
21.
22.
y
8
4
m
B(2, 1)
–8
C (5, 10)
–4
O
4
8x
–4
23.
3
–8
4
A(–2, –11)
24.
25.
A
y
P
D
R(6, 5)
(–1, 7)
C
B
26. H
I
x
O
Q (6, –2)
K
J
27. P
Q
S
R
28. A
C
B
Pages 71–74, Lesson 2-2
11. p q p q
13.
123A
12.
q
r
qr
19. 64
8 or an equilateral triangle has three
congruent sides.
20. 64
8 and 0 0.
21. 0 0 and an obtuse angle measures greater than 90°
and less than 180°.
22. An equilateral triangle has three congruent sides or
0 0.
23. An equilateral triangle has three congruent sides and
an obtuse angle measures greater than 90° and less
than 180°.
24. 64
8 and an obtuse angle measures greater
than 90° and less than 180°.
25. An equilateral triangle has three congruent sides and
0 0.
26. 0 0 or 64
8
27. An obtuse angle measures greater than 90° and less
than 180° or an equilateral triangle has three
congruent sides.
28. 64
8 and an equilateral triangle has three
congruent sides, or an obtuse angle measures greater
than 90° and less than 180°.
29. An obtuse angle measures greater than 90° and less
than 180°, or an equilateral triangle has three
congruent sides and 0 0.
33. q r q and r
34. p q p or q
T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
F
F
F
F
p
r
p or r
p
q
p and q
T
T
T
T
T
T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
T
T
F
F
F
F
F
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
F
F
F
F
p p r
p
r
T
T
F
F
T
F
F
F
F
T
T
T
F
F
T
F
35.
36.
37.
39.
40.
r q r
38.
q p q p q
F
T
T
T
T
T
F
F
F
F
T
F
F
F
p
q
r
T
T
T
F
F
F
F
T
T
F
F
T
T
T
T
F
T
F
F
F
F
T
F
F
F
T
F
F
F
T
T
T
F
F
T
F
T
F
T
T
T
T
F
F
T
T
F
F
T
F
F
F
T
T
F
T
p
q
r
T
T
T
T
T
T
F
T
F
F
T
F
T
F
F
F
F
F
r
T
T
F
T
F
F
T
F
F
F
F
F
F
T
F
T
T
F
F
T
T
T
p r q r p (q r )
q r q r
p (q r)
F
F
F
F
F
F
T
T
T
T
T
F
T
T
F
T
T
T
T
T
F
F
F
F
F
F
T
T
F
T
T
F
T
F
T
T
T
F
Pages 78–80, Lesson 2-3
15. If you are in Colorado, then aspen trees cover high
areas of the mountains. If you are in Florida, then
cypress trees rise from the swamps. If you are in
Vermont, then maple trees are prevalent.
16. H: 2x 6 10, C: x 2
17. H: you are a teenager; C: you are at least 13 years old
18. H: you have a driver’s license; C: you are at least
16 years old
19. H: three points lie on a line; C: the points are collinear
20. H: a man hasn’t discovered something he will die for;
C: he isn’t fit to live
21. H: an angle measures between 0 and 90; C: the angle
is acute
22. If you buy a 1-year fitness plan, then you get a free visit.
23. If you are a math teacher, then you love to solve
problems.
24. If I think, then I am.
25. If two angles are adjacent, then they have a common
side.
26. If two angles are vertical, then they are congruent.
27. If two triangles are equiangular, then they are
equilateral.
40. Converse: If you live in Texas, then you live in Dallas.
False; you could live in Austin. Inverse: If you don’t live
in Dallas, then you don’t live in Texas. False; you
could live in Austin. Contrapositive: If you don’t live in
Texas, then you don’t live in Dallas; true.
41. Converse: If you are in good shape, then you exercise
regularly; true. Inverse: If you do not exercise
regularly, then you are not in good shape; true.
Contrapositive: If you are not in good shape, then you
do not exercise regularly. False; an ill person may
exercise a lot, but still not be in good shape.
42. Converse: If the sum of two angles is 90, then they are
complementary; true. Inverse: If two angles are not
complementary, then their sum is not 90; true.
Contrapositive: If the sum of two angles is not 90, then
they are not complementary; true.
43. Converse: If a figure is a quadrilateral, then it is a
rectangle; false, rhombus. Inverse: If a figure is not a
rectangle, then it is not a quadrilateral; false, rhombus.
Contrapositive: If a figure is not a quadrilateral, then it
is not a rectangle; true.
44. Converse: If an angle has a measure of 90, then it is a
right angle; true. Inverse: If an angle is not a right
angle, then its measure is not 90; true. Contrapositive:
If an angle does not have a measure of 90, then it is
not a right angle; true.
45. Converse: If an angle has measure less than 90, then
it is acute; true. Inverse: If an angle is not acute, then
its measure is not less than 90; true. Contrapositive: If
an angle’s measure is not less than 90, then it is not
acute; true.
daylight than darkness, then it is summer; true. In
Alaska, if there are more hours of darkness than
daylight, then it is winter; true.
Page 80, Practice Quiz 1
1. W
X
3.
Y
4.
p
q
r
qr
p (q r )
T
T
T
T
T
T
T
F
F
T
T
F
T
F
T
T
F
F
F
T
F
T
T
T
T
F
T
F
F
F
F
F
T
F
F
F
F
F
F
F
p
q p p q
T
T
F
F
T
F
F
F
F
T
T
T
F
F
T
F
p
q
5. Converse: If two angles have a
C
A
common vertex, then the angles
are adjacent. False; ABD is not
D
B
Inverse: If two angles are not
A
C
adjacent, then they do not have
B
a common vertex. False, ABC
E
D
and DBE have a common vertex
Contrapositive: If two angles do not have a common
vertex, then they are not adjacent; true.
Pages 84–87, Lesson 2-4
33. Sample answer: Doctors and nurses use charts to
assist in determining medications and their doses for
patients. Answers should include the following.
• Doctors need to note a patient’s symptoms to
determine which medication to prescribe, then
determine how much to prescribe based on weight,
age, severity of the illness, and so on.
• Doctors use what is known to be true about diseases
and when symptoms appear, then deduce that the
patient has a particular illness.
39. q r q r
40. p r p p r
41.
42.
T
T
T
T
T
F
T
T
F
F
T
F
F
F
F
T
F
F
T
T
T
F
F
F
F
F
T
T
Animal
p
q
r
qr
p (q r )
Arthropod
T
T
T
T
T
Butterfly
T
T
F
T
T
T
F
T
T
T
T
F
F
F
F
F
T
T
T
F
F
T
F
T
F
F
F
T
T
F
F
F
F
F
F
p
q
r
T
T
T
F
F
T
T
T
F
F
F
T
T
F
T
T
T
T
T
F
F
T
F
T
F
T
T
F
F
F
F
T
F
F
F
F
F
F
T
T
T
T
F
F
F
T
F
F
q q r
p (q r )
Pages 91–93, Lesson 2-5
6. Sometimes; if the planes have a common intersection,
then their intersection is one line.
123C
10. Since P is the midpoint of Q
R
and ST
, PQ PR 1
1
QR and PS PT ST by the definition of
2
2
midpoint. We are given QR
S
T
so QR ST by the
definition of congruent segments. By the Multiplication
1
1
Property, QR ST. So, by substitution, PQ PT.
2
2
16. Sometimes; the three points cannot be on the same line.
17. Always; if two points lie in a plane, then the entire line
containing those points lies in that plane.
18. Never; the intersection of a line and a plane can be a
point, but the intersection of two planes is a line.
19. Sometimes; the three points cannot be on the same line.
20. Always; one plane contains at least three points, so it
must contain two.
21. Sometimes; and m could be skew, so they would not
lie in the same plane.
22. Postulate 2.1; through any two points, there is exactly
one line.
23. Postulate 2.5; if two points lie in a plane, then the
entire line containing those points lies in that plane.
24. Postulate 2.2; through any three points not on the
same line, there is exactly one plane.
25. Postulate 2.5; if two points lie in a plane, then the
entire line containing those points lies in the plane.
26. Postulate 2.1; through any two points, there is exactly
one line.
27. Postulate 2.2; through any three points not on the
same line, there is exactly one plane.
38.
Pages 97–100, Lesson 2-6
10. Given: 25 7(y 3) 5y
Prove: 2 y
Proof:
Statements (Reasons)
1. 25 7(y 3) 5y (Given)
2. 25 7y 21 5y (Dist. Prop.)
3. 25 2y 21 (Substitution)
4. 4 2y (Subt. Prop.)
5. 2 y (Div. Prop.)
11. Given: Rectangle ABCD,
10
A
B
3
3
Prove: AC BD
D
C
10
Proof:
Statements (Reasons)
1. Rectangle ABCD, AD 3, AB 10 (Given)
2. Draw segments AC and DB. (Two points determine
a line.)
3. ABC and BCD are right triangles. (Def. of rt. )
4. AC 32 1
02, DB 32 1
02 (Pythag. Th.)
5. AC BD (Substitution)
12. Given: c 2 a2 b 2
Prove: a c 2 b2
Proof:
Statements (Reasons)
1. c 2 a2 b2 (Given)
2. c 2 b2 a2 (Subt. Prop.)
3. a2 c 2 b2 (Reflexive Prop.)
4. a2 c 2 b2 (Square Root Prop.)
5. a c 2 b2 (Square Root Prop.)
1
7
26. Given: 4 a a
2
2
Prove: a 1
Proof:
Statements (Reasons)
1
1. 4 a 2
1
2. 2 4 a
2
3
27. Given: 2y 2 8
13
4
Prove: y Proof:
Statements (Reasons)
3
2
1. 2y 8 (Given)
3
2
2. 2 2y 2(8) (Mult. Prop.)
3. 4y 3 16 (Dist. Prop.)
4. 4y 13 (Subt. Prop.)
13
5. y (Div. Prop.)
4
1
28. Given: m 9
2
Prove: m 18
Proof:
Statements (Reasons)
1
2
1. m 9 (Given)
1
2
2
3
3. 15 2x 3 (Dist. Prop.)
4. 15 2x 15 3 15 (Subt. Prop.)
5. 2x 12 (Substitution)
2x
2
8 a 7 2a (Dist. Prop.)
1 a 2a (Subt. Prop.)
1 a (Div. Prop.)
a 1 (Symmetric Prop.)
2. 3 5 z 3(1) (Mult. Prop.)
12
2
6. (Div. Prop.)
7
a (Given)
2
7
2 a (Mult. Prop.)
2
2
3
1. 5 z 1 (Given)
2. 2 m 2(9) (Mult. Prop.)
7. x 6 (Substitution)
30. Given: XZ ZY, XZ 4x 1,
and ZY 6x 13
Prove: x 7
X 4x 1 Z
6x 13
Proof:
Y
Statements (Reasons)
1. XZ ZY, XZ 4x 1, and ZY 6x 13 (Given)
2. 4x 1 6x 13 (Substitution)
3. 4x 1 4x 6x 13 4x (Subt. Prop.)
4. 1 2x 13 (Substitution)
5. 1 13 2x 13 13 (Add. Prop.)
6. 14 2x (Substitution)
14
2
2x
2
7. (Div. Prop.)
8. 7 x (Substitution)
9. x 7 (Symmetric Prop.)
31. Given: mACB mABC
Prove: mXCA mYBA
A
Proof:
X
C
B
Statements (Reasons)
1. mACB mABC (Given)
2. mXCA mACB 180,
mYBA mABC 180 (Def. of supp. )
3. mXCA mACB mYBA mABC
(Substitution)
4. mXCA mACB mYBA mACB
(Substitution)
5. mXCA mYBA (Subt. Prop.)
Y
3. m 18 (Substitution)
3.
4.
5.
6.
7.
2
29. Given: 5 z 1
3
Prove: z 6
Proof:
Statements (Reasons)
Page 100, Practice Quiz 2
5. Given: 2(n 3) 5 3(n 1)
Prove: n 2
Proof:
Statements (Reasons)
1. 2(n 3) 5 3(n 1) (Given)
2. 2n 6 5 3n 3 (Dist. Prop.)
3. 2n 1 3n 3 (Substitution)
4. 2n 1 2n 3n 3 2n (Subt. Prop.)
5. 1 n 3 (Substitution)
6. 1 3 n 3 3 (Add. Prop.)
7. 2 n (Substitution)
8. n 2 (Symmetric Prop.)
20. Given: AB
A
C
and PC
Q
B
Prove: AP
AQ
Pages 103–106, Lesson 2-7
8. Given: A
P
CP
A
P
P
B
DP
C
Prove: AB
CD
Proof:
Statements (Reasons)
1. A
P
and BP
D
P
(Given)
CP
2. AP CP and BP DP (Def. of segs.)
3. AP PB AB (Seg. Add. Post.)
4. CP DP AB (Substitution)
5. CP PD CD (Seg. Add. Post.)
6. AB CD (Transitive Prop.)
7. AB
(Def. of segs.)
CD
9. Given: HI TU
and
H
H
J TV
I
Prove: IJ U
V
22.
T
Proof:
J
Statements (Reasons)
1. H
I TU
and HJ T
V
(Given)
2. HI TU and HJ TV (Def. of segs.)
3. HI IJ HJ (Seg. Add. Post.)
4. TU IJ TV (Substitution)
5. TU UV TV (Seg. Add. Post.)
6. TU IJ TU UV (Substitution)
7. TU TU (Reflexive Prop.)
8. IJ UV (Subt. Prop.)
9. IJ U
V
(Def. of segs.)
19. Given: XY
WZ
and WZ
AB
Prove: XY
AB
Proof:
Statements (Reasons)
U
D
B
23.
V
24.
B
X
Proof:
Y
Statements (Reasons)
1. X
Y
WZ
and WZ
AB
(Given)
2. XY WZ and WZ AB (Def. of segs.)
3. XY AB (Transitive Prop.)
4. XY
AB
(Def. of segs.)
Q
P
W
25.
A
123E
B
C
A
1. A
B
AC
and PC
Q
B
(Given)
2. AB AC, PC QB (Def. of segs.)
3. AB AQ QB, AC AP PC (Seg. Add. Post.)
4. AQ QB AP PC (Substitution)
5. AQ QB AP QB (Substitution)
6. QB QB (Reflexive Prop.)
7. AP AQ (Subt. Prop.)
8. AP
AQ
(Def. of segs.)
Given: LM
P
N
and
P
L
X
M
XN
M
X
N
Prove: LX
P
X
Proof:
Statements (Reasons)
1. L
M
PN
and X
M
XN
(Given)
2. LM PN and XM XN (Def. of segs.)
3. LM LX XM, PN PX XN (Seg. Add. Post.)
4. LX XM PX XN (Substitution)
5. LX XN PX XN (Substitution)
6. XN XN (Reflexive Prop.)
7. LX PX (Subt. Prop.)
8. L
X
P
X
(Def. of segs.)
Given: AB BC
A
B
C
Prove: AC 2BC
Proof:
Statements (Reasons)
1. AB BC (Given)
2. AC AB BC (Seg. Add. Post.)
3. AC BC BC (Substitution)
4. AC 2BC (Substitution)
Given: A
B
A
B
Prove: AB
AB
Proof:
Statements (Reasons)
1. A
B
(Given)
2. AB AB (Reflexive Prop.)
3. AB
AB
(Def. of segs.)
Given: AB
DE
, C is the
midpoint of BD
.
Prove: AC
CE
Z
Proof:
A B C D E
Statements (Reasons)
1. A
B
DE
, C is the midpoint of BD
. (Given)
2. BC CD (Def. of midpoint)
3. AB DE (Def. of segs.)
4. AB BC CD DE (Add. Prop.)
5. AB BC AC, CD DE CE (Seg. Add. Post.)
6. AC CE (Substitution)
7. AC
CE
(Def. of segs.)
26. Given: A
B
and BC
EF
DE
Prove: AC
DF
A
C
D
B
E
F
Proof:
Statements (Reasons)
1. A
B
EF
and B
C
D
E
(Given)
2. AB EF and BC DE (Def. of segs.)
3. AB BC DE EF (Add. Prop.)
4. AC AB BC, DF DE EF (Seg. Add. Post.)
5. AC DF (Substitution)
6. A
C
DF
(Def. of segs.)
QO
and
L
M
M
N
RS
S
T
Q
P
P
O
1
2
3 4
1 2
1
Proof:
Statements (Reasons)
1. ⊥ m (Given)
2. 1 and 2 are rt. . (⊥ lines intersect to form 4 rt. .)
3. 1 2 (All rt. are .)
36. Given: 1 2, 1 and 2
are supplementary.
1 2
Prove: 1 and 2 are rt. .
Proof:
Statements (Reasons)
1. 1 2, 1 and 2 are supplementary. (Given )
2. m1 m2 180 (Def. of supplementary )
3. m1 m2 (Def. of angle)
4. m1 m1 180 (Substitution)
6. m1 90 (Div. Prop.)
7. m2 90 (Substitution (steps 3, 6))
8. 1 and 2 are rt. . (Def. of rt. )
37. Given: ABD CBD, ABD and
D
CBD form a linear pair.
Prove: ABD and CBD are rt. .
A
m
Proof:
Statements (Reasons)
1. ⊥ m (Given)
2. 1 is a right angle. (Def. of ⊥)
3. m1 90 (Def. of rt. )
4. 1 4 (Vert. are )
5. m1 m4 (Def. of )
6. m4 90 (Substitution)
7. 1 and 2 form a linear pair; 3 and 4 form a
linear pair. (Def. of linear pair)
8. m1 m2 180, m4 m3 180 (Linear
pairs are supplementary.)
9. 90 m2 180, 90 m3 180 (Substitution)
10. m2 90, m3 90 (Subt. Prop.)
11. 2, 3, 4 are rt. . (Def. of rt. (steps 6, 10))
m
2
3 4
B
C
Proof:
Statements (Reasons)
1. ABD CBD, ABD and CBD form a linear
pair. (Given)
2. ABD and CBD are supplementary. (Linear pairs
are supplementary.)
3. ABD and CBD are rt. . (If are and suppl.,
they are rt. .)
Pages 111–114, Lesson 2-8
25. Given: A
Prove: A A
Proof:
A
Statements (Reasons)
1. A is an angle. (Given)
2. mA mA (Reflexive Prop)
3. A A (Def. of angles)
26. Given: 1 2,
2 3
1
2
3
Prove: 1 3
Proof:
Statements (Reasons)
1. 1 2, 2 3 (Given)
2. m1 m 2, m 2 m 3 (Def. of angles)
3. m1 m3 (Trans. Prop.)
4. 1 3 (Def. of angles)
33. Given: ⊥ m
Prove: 2, 3, 4 are rt. 34. Given: 1 and 2 are rt. .
Prove: 1 2
Proof:
Statements (Reasons)
1. 1 and 2 are rt. . (Given)
2. m1 90, m2 90 (Def. of rt. )
3. m1 m2 (Substitution)
4. 1 2 (Def. of angles)
35. Given: ⊥ m
Prove: 1 2
38. Given: ABD YXZ
Prove: CBD WXZ
W
Y
Z
B
C
Proof:
Statements (Reasons)
1. ABD YXZ; ABD and CBD form a linear
pair; YXZ and WXZ form a linear pair. (Given;
from the figure)
2. mABD mCBD 180, mYXZ mWXZ 180 (Linear pairs are supplementary.)
3. mABD mCBD mYXZ mWXZ (Subst.)
4. mABD mYXZ (Def. of )
5. mYXZ mCBD mYXZ mWXZ (Subst.)
6. mYXZ mYXZ (Reflexive Prop.)
7. mCBD mWXZ (Subt. Prop.)
8. CBD WXZ (Def. of )
39. Given: mRSW mTSU
Prove: mRST mWSU
T
R
W
U
S
Proof:
Statements (Reasons)
1. mRSW mTSU (Given)
2. mRSW mRST mTSW, mTSU mTSW mWSU (Angle Addition Postulate)
3. mRST mTSW mTSW mWSU
(Substitution)
4. mTSW mTSW (Reflexive Prop.)
5. mRST mWSU (Subt. Prop.)
48. Given: AC BD
A B
C D
Prove: AB CD
Proof:
Statements (Reasons)
1. AC BD (Given)
2. AB BC AC, BC CD BD (Segment Addition
Postulate)
3. BC BC (Reflexive Prop.)
4. AB BC BC CD (Substitution (2 and 3))
5. AB CD (Subt. Prop.)
123G
E
C
D
A
X
Page 115-120, Chapter 2 Study Guide and Review
53. Given: BC EC, CA CD
B
Prove: BA DE
D
Proof:
A
Statements (Reasons)
1. BC EC, CA CD (Given)
2. BC CA EC CA (Add. Prop.)
3. BC CA EC CD (Substitution)
4. BC CA BA, EC CD DE (Seg. Add. Post.)
5. BA DE (Substitution)
54. Given: AB CD
A B
C D
Prove: AC BD
Proof:
Statements (Reasons)
1. AB CD (Given)
2. BC BC (Reflexive Prop.)
3. AB BC CD BC (Add. Prop.)
4. AB BC AC, CD BC BD (Seg. Add. Post.)
5. AC BD (Substitution)
Page 123, Chapter 2 Standardized Test Practice
15. Given: 1 and 3 are vertical angles.
m1 3x 5, m3 2x 8
Prove: m1 14
1
2
4
3
Proof:
Statements (Reasons)
a. 1 and 3 are vertical angles; m1 3x 5,
m3 2x 8 (Given)
b. 1 3 (Vert. are .)
c. m1 m3 (Def. of )
d. 3x 5 2x 8 (Substitution)
e. x 5 8 (Subt. Prop.)
f. x 3 (Subt. Prop.)
g. m1 3(3) 5 (Substitution)
h. m1 14 (Substitution)
Notes