Chapter 15
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1 2 3 4 5 6 7 8 9 10
Geometry: Concepts and Applications
Chapter 15 Resource Masters
024
11 10 09 08 07 06 05 04
Contents
Lesson 15-5
Study Guide and Intervention . . . . . . . . . . . . . . . 649
Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
Reading to Learn Mathematics . . . . . . . . . . . . . . 652
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
Vocabulary Builder . . . . . . . . . . . . . . . . . vii-viii
Lesson 15-1
Study Guide and Intervention . . . . . . . . . . . . . . . 629
Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
Reading to Learn Mathematics . . . . . . . . . . . . . . 632
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
Lesson 15-6
Study Guide and Intervention . . . . . . . . . . . . . . . 654
Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656
Reading to Learn Mathematics . . . . . . . . . . . . . . 657
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
Lesson 15-2
Study Guide and Intervention . . . . . . . . . . . . . . . 634
Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
Reading to Learn Mathematics . . . . . . . . . . . . . . 637
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
Chapter 15 Assessment
Chapter 15 Test, Form 1A. . . . . . . . . . . . . . . 659-660
Chapter 15 Test, Form 1B . . . . . . . . . . . . . . . 661-662
Chapter 15 Test, Form 2A. . . . . . . . . . . . . . . 663-664
Chapter 15 Test, Form 2B . . . . . . . . . . . . . . . 665-666
Chapter 15 Extended Response
Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
Chapter 15 Mid-Chapter Test . . . . . . . . . . . . . . . . 668
Chapter 15 Quizzes A & B. . . . . . . . . . . . . . . . . . 669
Chapter 15 Cumulative Review . . . . . . . . . . . . . . 670
Chapter 15 Standardized
Test Practice . . . . . . . . . . . . . . . . . . . . . . . 671-672
Lesson 15-3
Study Guide and Intervention . . . . . . . . . . . . . . . 639
Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
Reading to Learn Mathematics . . . . . . . . . . . . . . 642
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
Lesson 15-4
Study Guide and Intervention . . . . . . . . . . . . . . . 644
Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Reading to Learn Mathematics . . . . . . . . . . . . . . 647
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
© Glencoe/McGraw-Hill
Standardized Test Practice
Student Recording Sheet . . . . . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . . . . . A2-A23
iii
Geometry: Concepts and Applications
A Teacher’s Guide to Using the
Chapter 15 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the
resources you use most often. The Chapter 15 Resource Masters include the core
materials needed for Chapter 15. These materials include worksheets, extensions,
and assessment options. The answers for these pages appear at the back of this
booklet.
All of the materials found in this booklet are included for viewing and printing in
the Geometry: Concepts and Applications TeacherWorks CD-ROM.
Vocabulary Builder Pages vii-viii include a
student study tool that presents the key
vocabulary terms from the chapter. Students are
to record definitions and/or examples for each
term. You may suggest that students highlight or
star the terms with which they are not familiar.
Practice There is one master for each lesson.
These problems more closely follow the
structure of the Practice section of the Student
Edition exercises. These exercises are of average
difficulty.
When to Use Give these pages to students
practice options or may be used as homework
for second day teaching of the lesson.
When to Use These provide additional
before beginning Lesson 15-1. Encourage them
to add these pages to their Geometry: Concepts
and Applications Interactive Study Notebook.
Remind them to add definitions and examples as
they complete each lesson.
Reading to Learn Mathematics One
master is included for each lesson. The first
section of each master presents key terms from
the lesson. The second section contains
questions that ask students to interpret the
context of and relationships among terms in the
lesson. Finally, students are asked to summarize
what they have learned using various
representation techniques.
Study Guide There is one Study Guide
master for each lesson.
When to Use Use these masters as reteaching
activities for students who need additional
reinforcement. These pages can also be used in
conjunction with the Student Edition as an
instructional tool for those students who have
been absent.
When to Use This master can be used as a
study tool when presenting the lesson or as an
informal reading assessment after presenting the
lesson. It is also a helpful tool for ELL (English
Language Learners) students.
Skills Practice There is one master for each
lesson. These provide computational practice at
a basic level.
When to Use These worksheets can be used
with students who have weaker mathematics
backgrounds or need additional reinforcement.
© Glencoe/McGraw-Hill
iv
Geometry: Concepts and Applications
Enrichment There is one master for each
lesson. These activities may extend the concepts
in the lesson, offer a historical or multicultural
look at the concepts, or widen students’
perspectives on the mathematics they are
learning. These are not written exclusively for
honors students, but are accessible for use with
all levels of students.
Intermediate Assessment
• A Mid-Chapter Test provides an option to
assess the first half of the chapter. It is
composed of free-response questions.
• Two free-response quizzes are included to
offer assessment at appropriate intervals in
the chapter.
When to Use These may be used as extra
Continuing Assessment
credit, short-term projects, or as activities for
days when class periods are shortened.
• The Cumulative Review provides students
an opportunity to reinforce and retain skills
as they proceed through their study of
geometry. It can also be used as a test. The
master includes free-response questions.
Assessment Options
The assessment section of the Chapter 15
Resources Masters offers a wide range of
assessment tools for intermediate and final
assessment. The following lists describe each
assessment master and its intended use.
• The Standardized Test Practice offers
continuing review of geometry concepts in
multiple choice format.
Answers
Chapter Assessments
Chapter Tests
• Page A1 is an answer sheet for the
Standardized Test Practice questions that
appear in the Student Edition on page 673.
This improves students’ familiarity with the
answer formats they may encounter in test
taking.
• Forms 1A and 1B contain multiple-choice
questions and are intended for use with
average-level and basic-level students,
respectively. These tests are similar in
format to offer comparable testing
situations.
• The answers for the lesson-by-lesson
masters are provided as reduced pages with
answers appearing in red.
• Forms 2A and 2B are composed of freeresponse questions aimed at the averagelevel and basic-level student, respectively.
These tests are similar in format to offer
comparable testing situations.
• Full-size answer keys are provided for the
assessment options in this booklet.
All of the above tests include a challenging
Bonus question.
• The Extended Response Assessment
includes performance assessment tasks that
are suitable for all students. A scoring rubric
is included for evaluation guidelines.
Sample answers are provided for
assessment.
© Glencoe/McGraw-Hill
v
Geometry: Concepts and Applications
Chapter 15 Leveled Worksheets
Glencoe’s leveled worksheets are helpful for meeting the needs of every
student in a variety of ways. These worksheets, many of which are found
in the FAST FILE Chapter Resource Masters, are shown in the chart
below.
• The Prerequisite Skills Workbook provides extra practice on the basic
skills students need for success in geometry.
• Study Guide and Intervention masters provide worked-out examples
as well as practice problems.
• Reading to Learn Mathematics masters help students improve reading
skills by examining lesson concepts more closely.
• Noteables™: Interactive Study Notebook with Foldables™ helps
students improve note-taking and study skills.
• Skills Practice masters allow students who are progressing at a slower
pace to practice concepts using easier problems. Practice masters
provide average-level problems for students who are moving at a regular pace.
• Each chapter’s Vocabulary Builder master provides students the
opportunity to write out key concepts and definitions in their own words.
The Proof Builder master provides students the opportunity to write the
chapter’s postulates and theorems in their own words.
• Enrichment masters offer students the opportunity to extend their
learning.
Ten Different Options to Meet the Needs of
Every Student in a Variety of Ways
primarily skills
primarily concepts
primarily applications
BASIC
AVERAGE
1
Prerequisite Skills Workbook
2
Study Guide and Intervention
3
Reading to Learn Mathematics
4
NoteablesTM: Interactive Study Notebook with FoldablesTM
5
Skills Practice
6
Vocabulary Builder
7
Proof Builder
8
Parent and Student Study Guide (online)
© Glencoe/McGraw-Hill
9
Practice
10
Enrichment
vi
ADVANCED
Geometry: Concepts and Applications
15
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Reading to Learn Mathematics
Vocabulary Builder
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 15.
As you study the chapter, complete each term’s definition or description.
Remember to add the page number where you found the term.
Vocabulary Term
Found
on Page
Definition/Description/Example
compound statement
conjunction
contrapositive
coordinate proof
deductive reasoning
dee•DUK•tiv
disjunction
indirect proof
indirect reasoning
inverse
(continued on the next page)
© Glencoe/McGraw-Hill
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Geometry: Concepts and Applications
15
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Reading to Learn Mathematics
Vocabulary Builder (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
Law of Detachment
Law of Syllogism
SIL•oh•jiz•um
logically equivalent
negation
paragraph proof
proof
proof by contradiction
statement
truth table
truth value
two-column proof
© Glencoe/McGraw-Hill
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Geometry: Concepts and Applications
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Study Guide
Logic and Truth Tables
A statement is any sentence that is either true or false, but not
both. The table below lists different kinds of statements.
Term
negation
conjunction
disjunction
conditional
converse
Symbol Definition
~p
not p
pq
p and q
pq
p or q
p→q
if p, then q
q→p
if q, then p
Every statement has a truth value. It is convenient
to organize the truth values in a truth table like the
one shown at the right.
Conjunction
p
q
pq
T
T
F
F
T
F
T
F
T
F
F
F
Complete a truth table for each compound statement.
1. p q
2. ~p q
p
q
pq
p
~p
q
~p q
T
T
F
F
T
F
T
F
T
T
T
F
T
T
F
F
F
F
T
T
T
F
T
F
T
F
T
T
~( p q)
3. p ~q
4. ~(p q)
p
q
~q
p ~q
p
q
pq
T
T
F
F
T
F
T
F
F
T
F
T
F
T
F
F
T
T
F
F
T
F
T
F
T
F
F
F
© Glencoe/McGraw-Hill
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F
T
T
T
Geometry: Concepts and Applications
15–1
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Skills Practice
Logic and Truth Tables
For Exercises 1–16, use conditionals a, b, c and d.
a: A triangle has three sides.
b: January is a day of the week.
c: 5 5 20
d: Parallel lines do not intersect.
Write the statements for each negation.
1. a
A triangle does not have three sides.
2. b
January is not a day of the week.
3. c
5 5 20
4. d
Parallel lines intersect.
Write a statement for each conjunction or disjunction. Then find the truth value.
5. a b A triangle has three sides or January is a day of the week; true.
6. a b A triangle has three sides and January is a day of the week; false.
7. a c A triangle has three sides or 5 5 20; true.
8. a c A triangle has three sides and 5 5 20; false.
9. a d A triangle has three sides or parallel lines do not intersect; true.
10. a d A triangle has three sides and parallel lines do not intersect; true.
11. b c January is a day of the week or 5 5 20; false.
12. b c January is a day of the week and 5 5 20; false.
13. b d January is a day of the week or parallel lines do not intersect; true.
14. b d January is a day of the week and parallel lines do not intersect; false.
15. c d 5 5 20 or parallel lines do not intersect; true.
16. c d 5 5 20 and parallel lines do not intersect; false.
© Glencoe/McGraw-Hill
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Geometry: Concepts and Applications
15–1
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DATE
PERIOD
Practice
Logic and Truth Tables
Use conditionals p, q, r, and s for Exercises 1–9.
p: Labor Day is in April.
q: A quadrilateral has 4 sides.
r: There are 30 days in September.
s: (5 3) 3 5
Write the statements for each negation.
1. p
Labor Day is not
in April.
2. q
A quadrilateral does
not have 4 sides.
3. r
There are not
30 days in
September.
Write a statement for each conjunction or disjunction.
Then find the truth value.
4. p q
5. p q
6. p r
7. q s
8. p s
9. q r
Labor Day is in April or a
quadrilateral has 4 sides; true.
Labor Day is in April and a
quadrilateral has 4 sides; false.
Labor Day is not in April or there
are 30 days in September; true.
A quadrilateral does not have
4 sides and (5 3) 3 5;
false.
Labor Day is in April and
(5 3) 3 5; false.
A quadrilateral does not have
4 sides or there are not 30 days
in September; false.
Construct a truth table for each compound statement.
10. p q
11. p q
p
q
~p
~q
~(p ~q)
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
F
T
T
T
© Glencoe/McGraw-Hill
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p ~p
q
~p q
T
T
F
F
T
F
T
F
F
F
T
F
F
F
T
T
Geometry: Concepts and Applications
15–1
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Reading to Learn Mathematics
Logic and Truth Tables
Key Terms
statement a statement that is either true or false, but not both
truth value the true or false nature of a statement
negation the negative of a statement
truth table a convenient way to organize truth values
compound statement two or more logic statements joined by
and or or
inverse a statement formed by negating both p and q in the
conditional p → q
Reading the Lesson
1. Supply one or two words to complete each sentence.
a. Any two statements can be joined to form a
compound
statement.
b. A statement that is formed by joining two statements with the word or is called a
disjunction.
truth value.
d. A statement that is formed by joining two statements with the word and is called a
conjunction.
e. The statement represented by not p is the negation of p.
f. If you negate both p and q in a statement p → q, the new statement is called the
inverse.
2. Let p represent “you live in the United States,” q represent “July is a month in the
c. The true or false nature of a statement is called its
summer,” and r represent “red is a color.” For each exercise, explain what the symbols
mean and then write the statement indicated by the symbols.
a. p q The symbols mean the conjunction of p and q. You live in the
United States and July is a month in the summer.
b. p q The symbols mean the disjunction of p and q. You live in the
United States or July is a month in the summer.
c. q r The symbols mean the conjunction of the negation of q and r.
July is not a month in the summer and red is a color.
Helping You Remember
3. Prefixes can often help you to remember the meaning of words or to distinguish between
similar words. Use the dictionary to find the meanings of the prefixes con and dis. 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.
© Glencoe/McGraw-Hill
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Geometry: Concepts and Applications
15–1
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Enrichment
Counterexamples
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.
Example: Is the statement 1 1 true when you replace x with 1,
x
2, and 3? Is the statement true for all reals? If
possible, find a counterexample.
1
1
1
1
1
1, 1, and 1. But when x , then 2. This
1
2
3
2
x
counterexample shows that the statement is not
always true.
Answer each question.
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
January? no
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
3. Is the equation k
2 k true when you
replace k with 1, 2, and 3? Is the
equation true for all integers? If
possible, find a counterexample.
4. Is the statement 2x x x true when
1
you replace x with 2, 4, and 0.7? Is the
statement true for all real numbers?
If possible, find a counterexample.
It is true for 1, 2, and 3. It is
not true for negative
integers. Sample: 2
5. Suppose you draw four points A, B, C,
and D and then draw A
B
, B
C
, C
D
,
and D
A
. Does this procedure give a
quadrilateral always or only
sometimes? Explain your answers
with figures. only sometimes
Example:
© Glencoe/McGraw-Hill
It is true for all real
numbers.
6. Suppose you draw a circle, mark
three points on it, and connect them.
Will the angles of the triangle be
acute? Explain your answers with
figures no, only sometimes
Example:
Counterexample:
Counterexample:
633
Geometry: Concepts and Applications
15–2
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Study Guide
Deductive Reasoning
Two important laws used frequently in deductive reasoning
are the Law of Detachment and the Law of Syllogism. In
both cases you reach conclusions based on if-then statements.
Law of Detachment
Law of Syllogism
If p → q is a true conditional and
p is true, then q is true.
If p → q and q → r are true
conditionals, then p → r is also true.
Example: Determine if 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.
(1) If you break an item in a store, you must pay for it.
(2) Jill broke a vase in Potter’s Gift Shop.
(3) Jill must pay for the vase.
Yes, statement (3) follows from statements (1) and
(2) by the Law of Detachment.
Determine if a valid conclusion can be reached from the two
true statements using the Law of Detachment or the Law of
Syllogism. If a valid conclusion is possible, state it and the
law that is used. If a valid conclusion does not follow, write no
valid conclusion.
1. (1) If a number is a whole number, then it is an integer.
(2) If a number is an integer, then it is a rational number. If a number
is a whole number, then it is a rational number; Syllogism.
2. (1) If a dog eats Dogfood Delights, the dog is happy.
(2) Fido is a happy dog. no conclusion
3. (1) If people live in Manhattan, then they live in New York.
(2) If people live in New York, then they live in the United
States. If people live in Manhattan, then they live in
the United States; Syllogism.
4. (1) Angles that are complementary have measures with a sum
of 90.
(2) A and B are complementary.
m A m B 90; Detachment
5. (1) All fish can swim.
(2) Fonzo can swim. no conclusion
6. Look for a Pattern Find the next number in the list 83,
77, 71, 65, 59 and make a conjecture about the pattern.
53; Each number is 6 less than the preceding one.
© Glencoe/McGraw-Hill
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Geometry: Concepts and Applications
15–2
NAME
DATE
PERIOD
Skills Practice
Deductive Reasoning
Use the Law of Detachment to determine a conclusion that follows from statements
(1) and (2). If a valid conclusion does not follow, write no valid conclusion.
1. (1) If a figure is a triangle, then it is a polygon.
(2) The figure is a triangle.
The figure is a polygon.
2. (1) If I sell my skis, then I will not be able to go skiing.
(2) I did not sell my skis.
no valid conclusion
3. (1) If two angles are complementary, then the sum of their measures is 90.
(2) Angle A and B are complementary.
The sum of the measures of angles A and B is 90.
4. (1) If the measures of the lengths of two sides of a triangle are equal, then the triangle
is isosceles.
(2) Triangle ABC has two sides with lengths of equal measure.
Triangle ABC is isosceles.
5. (1) If it rains, we will not go on a picnic.
(2) We do not go on a picnic.
no valid conclusion
Use the Law of Syllogism to determine a conclusion that follows from statements
(1) and (2). If a valid conclusion does not follow, write no valid conclusion.
6. (1) If my dog does not bark all night, I will give him a treat.
(2) If I give my dog a treat, then he will wag his tail.
If my dog does not bark all night, then he will wag his tail.
7. (1) If a polygon has three sides, then the figure is a triangle.
(2) If a figure is a triangle, then the sum of the measures of the interior angles is 180.
If a polygon has three sides, then the sum of the measures of the
interior angles is 180.
8. (1) If the concert is postponed, then I will be out of town.
(2) If the concert is postponed, then it will be held in the gym.
no valid conclusion
9. (1) All whole numbers are rational numbers.
(2) All whole numbers are real numbers.
no valid conclusion
10. (1) If the temperature reaches 70°, then the swimming pool will open.
(2) If the swimming pool opens, then we will not go to the beach.
If the temperature reaches 70°, then we will not go to the beach.
© Glencoe/McGraw-Hill
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Geometry: Concepts and Applications
15–2
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PERIOD
Practice
Deductive Reasoning
Determine if a valid conclusion can be reached from the two true
statements using the Law of Detachment or the Law of Syllogism.
If a valid conclusion is possible, state it and the law that is used. If
a valid conclusion does not follow, write no valid conclusion.
1. If Jim is a Texan, then he is an American.
Jim is a Texan.
Jim is an American; Detachment.
2. If Spot is a dog, then he has four legs.
Spot has four legs.
no valid conclusion
3. If Rachel lives in Tampa, than Rachel lives in Florida.
If Rachel lives in Florida, then Rachel lives in the United States.
If Rachel
lives in Tampa, then Rachel lives in the United States; Syllogism.
4. If October 12 is a Monday, then October 13 is a Tuesday.
October 12 is a Monday.
October 13 is a Tuesday; Detachment.
5. If Henry studies his algebra, then he passes the test.
If Henry passes the test, then he will get a good grade. If Henry studies his
algebra, then he will get a good grade; Syllogism.
Determine if 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 no valid conclusion.
6. (1) If the measure of an angle is greater than 90, then it is obtuse.
(2) m T is greater than 90.
(3) T is obtuse. yes; Detachment
7. (1) If Pedro is taking history, then he will study about World War II.
(2) Pedro will study about World War II.
(3) Pedro is taking history. no valid conclusion
8. (1) If Julie works after school, then she works in a department store.
(2) Julie works after school.
(3) Julie works in a department store. yes; Detachment
9. (1) If William is reading, then he is reading a magazine.
(2) If William is reading a magazine, then he is reading a magazine about
computers.
(3) If William is reading, then he is reading a magazine about computers.
yes; Syllogism
10. Look for a Pattern Tanya likes to burn candles. She has
found that, once a candle has burned, she can melt 3 candle
stubs, add a new wick, and have one more candle to burn. How
many total candles can she burn from a box of 15 candles? 22
© Glencoe/McGraw-Hill
636
Geometry: Concepts and Applications
15–2
NAME
DATE
PERIOD
Reading to Learn Mathematics
Deductive Reasoning
Key Terms
deductive reasoning (dee•DUK•tiv) the process of using facts, rules,
definitions, or properties in logical order to reach a conclusion
Law of Detachment a logic rule that states “if p → q is a true
conditional and p is true, then q is true”
Law of Syllogism (SIL•oh•jiz•um) a logic rule that states “if p → q
and q → r are true conditionals, then p → r is also true”
Reading the Lesson
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 u e
a. s u
2. conjunction of s and u g
b. If s → t is true and s is true, then t is true.
3. negation of t h
c. s → u
4. disjunction of s and u a
d. s t
5. Law of Detachment b
e. u
6. inverse of u → t j
f. If s → t and t → u are true, then s → u is true.
7. inverse of s → u c
g. s u
8. conjunction of s and t d
h. t
9. Law of Syllogism f
i. t
10. negation of t i
j. u → t
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 A and do not intersect.
(3) Lines and m are parallel. yes; Law of Detachment
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. Explain
this rule in a way that will help him to understand. 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.
© Glencoe/McGraw-Hill
637
Geometry: Concepts and Applications
15–2
NAME
DATE
PERIOD
Enrichment
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.
Conclusion: Justin’s luggage will
survive airline travel. valid
2. (1) If you buy Tuff Cote luggage, it
will survive airline travel.
(2) Justin’s luggage survived airline
travel.
Conclusion: Justin has Tuff Cote
luggage. faulty
3. (1) If you use Clear Line long
distance service, you will have clear
reception.
(2) Anna has clear long distance
reception.
Conclusion: Anna uses Clear Line
long distance service. faulty
4. (1) If you read the book Beautiful
Braids, you will be able to make
beautiful braids easily.
(2) Nancy read the book Beautiful
Braids.
Conclusion: Nancy can make
beautiful braids easily. valid
5. (1) If you buy a word processor, you
will be able to write letters faster.
(2) Tania bought a word processor.
Conclusion: Tania will be able to
write letters faster. valid
6. (1) Great swimmers wear AquaLine
swimwear.
(2) Gina wears AquaLine swimwear.
Conclusion: Gina is a great swimmer.
faulty
7. Write an example of faulty logic that
you have seen in an advertisement.
Answers will vary.
© Glencoe/McGraw-Hill
638
Geometry: Concepts and Applications
15–3
NAME
DATE
PERIOD
Study Guide
Paragraph Proofs
A proof is a logical argument in which each statement you make is
backed up by a reason that is accepted as true. In a paragraph
proof, you write your statements and reasons in paragraph form.
Example: Write a paragraph proof for the conjecture.
Given:
Prove:
WXYZ
W and X are supplementary.
X and Y are supplementary.
Y and Z are supplementary.
Z and W are supplementary.
Z
Y
and W
Z
X
Y.
By the definition of a parallelogram, WX
X
and Z
Y
, WZ
and X
Y
are transversals; for
For parallels W
and X
Y
, WX
and Z
Y
are transversals. Thus, the
parallels WZ
consecutive interior angles on the same side of a transversal
are supplementary. Therefore, W and X, X and Y, Y
and Z, Z and W are supplementary.
Write a paragraph proof for each conjecture.
1. Given:
Prove:
PSU PTR
TR
SU
2. Given:
S
P
TP
DEF and RST are rt.
triangles. E and S are
ST
and
right angles. EF
ED
SR
Prove: DEF RST
1. We know that PSU PTR and SU
TR
. By the Reflexive Property of
Congruent Angles,P P. Then SUP TRP by AAS and
TP
by CPCTC.
SP
F
ST
, ED
SR
, and E and S are right angles. Since
2. We know that E
all right angles are congruent, E S. Therefore, by SAS, DEF RST.
© Glencoe/McGraw-Hill
639
Geometry: Concepts and Applications
15–3
NAME
DATE
PERIOD
Skills Practice
Paragraph Proofs
Write a paragraph proof for each conjecture.
A
1. If ABD is an isosceles triangle with base BD
and C is the midpoint of B
D
, then ACD ACB.
D
B
C
If ABD is an isosceles triangle with base B
D
, then AD
AB
. If C is the
, then CD
CB
. AC
AC
by the Reflexive Property, so
midpoint of BD
ACD ACB by SSS.
2. If lines a and b are parallel and W
X
X
Y
,
then WXS YXZ.
S
W
a
X
b
Z
Y
If lines a and b are parallel, then SWX XYZ since they are alternate
interior angles. WXS YXZ since they are vertical angles. Then it is
XY
, so WXS YXZ by ASA.
given that WX
3. If ACDE is an isosceles trapezoid with bases
and E
D
, then AED CDE.
AC
A
C
B
E
D
If ACDE is an isosceles trapezoid with bases A
C
and E
D
, then the legs
CD
. Also, an isosceles trapezoid has congruent
are congruent, so AE
ED
by the Reflexive Property,
base angles, so AED CDE. Now, ED
so AED CDE by SAS.
4. If RSTX is a rhombus, then RXT RST.
S
R
X
T
If RSTX is a rhombus, then RS RX and XT ST. RT RT by the
Reflexive Property, so RXT RST by SSS.
© Glencoe/McGraw-Hill
640
Geometry: Concepts and Applications
15–3
NAME
DATE
PERIOD
Practice
Paragraph Proofs
Write a paragraph proof for each conjecture.
1. If p q and p and q are cut by a transversal t,
then 1 and 3 are supplementary.
Since p q, we know that 1 2
since they are corresponding angles.
We also know that 2 3 180
since they form a linear pair. Therefore,
by substitution, 1 3 180. So, 1
and 3 are supplementary.
2. If E bisects B
D
and A
C
, then BA CD .
Since E bisects B
D
and A
C
, we know
that BE ED and CE EA. We also
know that BEA CED since they
are vertical angles. Therefore,
BEA DEC by SAS. So, BAE DCE because corresponding parts
of congruent triangles are congruent.
So, line BA line CD since we have
alternate interior angles that are
congruent.
3. If 3 4, then ABC is isosceles.
We know that 3 1 180 and
2 4 180 since they form linear
pairs. Since 3 4, we can write
3 1 180 and 2 3 180. So,
1 180 3 and 2 180 3.
Therefore, 1 2 by substitution.
This implies that AC BC. So, ABC
is isosceles by definition of isosceles.
© Glencoe/McGraw-Hill
641
Geometry: Concepts and Applications
15–3
NAME
DATE
PERIOD
Reading to Learn Mathematics
Paragraph Proofs
Key Terms
proof a logical argument used to validate a conjecture in which each
statement you make is backed up by a reason that is accepted
as true
paragraph proof a logical argument used to validate a conjecture in
paragraph form
Reading the Lesson
1. Complete each sentence with one or two words to make a true statement.
a. In a proof, the given information comes from the hypothesis of the conditional.
b. A proof is a logical argument in which each statement
you make is backed up by a
reason.
c. A paragraph proof is written in paragraph form.
d. One
problem-solving strategy that you might use for writing a proof is work backward.
e. In mathematics, proofs are used to validate a conjecture.
2. Use the diagram and the information. Write a plan for
proving the conjecture. You do not need to write the proof.
Given: a b; XY XZ
Prove: 1 3
1
a
X
3
Y 2 Z
b
Sample answer: First, use parallel lines and corresponding angles to
show that 1 2. Then use the fact that XYZ is isosceles to show
that 2 3. Then use the Transitive Property of Congruence to
conclude that 1 3.
3. Write a paragraph proof for the conjecture. First, write a
plan for the proof.
Given: C; A
T
is tangent to C at T.
Prove: CAT is a right triangle.
T
A
C
Sample answer: Plan: Since AT
is a tangent to the circle, it is
perpendicular to a radius at point T. Perpendicular segments form
90° angles. If a triangle has a 90° angle, then it is a right triangle.
T
is a tangent to C, then it is
Proof: By Theorem 14-4, if A
. By the definition of perpendicular
perpendicular to the radius TC
T
C
, then CTA is a right angle. By the definition of right
lines, if AT
triangle, if CTA is a right angle, then CAT is a right triangle.
Helping You Remember
4. Some students like to use sayings like “My Dear Aunt Sally” to help them remember
a mathematical idea. My Dear Aunt Sally stands for multiplication, division, addition,
and subtraction for order of operations. Think of a saying to help you remember that
definitions, postulates, and theorems can be used to justify statements when you write
a proof. Sample answer: Down the Parallel Tracks for definition,
postulate, theorem
© Glencoe/McGraw-Hill
642
Geometry: Concepts and Applications
NAME
15–3
DATE
PERIOD
Enrichment
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.
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
Orange
Banana
Nancy
Olivia
Mario
X
X
X
X
X
X
Apple
X
X
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
Kenji
X
X
X
Nancy-apple,
Olivia-banana,
Mario-orange,
Kenji-peach
3. Mr. Guthrie, Mrs. Hakoi, Mr. Mirza,
and Mrs. Riva have jobs of doctor,
accountant, teacher, and office
manager. Mr. Mirza lives near the
doctor and the teacher. Mrs. Riva is
not the doctor or the office manager.
Mrs. Hakoi is not the accountant or
the office manager. Mr. Guthrie went
to lunch with the doctor. Mrs. Riva’s
son is a high school student and is
only seven years younger than his
algebra teacher. Which person has
each occupation?
4. Yvette, Lana, Boris, and Scott each
have a dog. The breeds are collie,
beagle, poodle, and terrier. Yvette and
Boris walked to the library with the
student who has a collie. Boris does
not have a poodle or terrier. Scott
does not have a collie. Yvette is in
math class with the student who has
a terrier. Which student has each
breed of dog?
Yvette, poodle;
Lana, collie;
Boris, beagle;
Scott, terrier
Mr. Guthrie-teacher,
Mrs. Hakoi-doctor,
Mr. Mirza-office manager,
Mrs. Riva-accountant
© Glencoe/McGraw-Hill
643
Geometry: Concepts and Applications
15–4
NAME
DATE
PERIOD
Study Guide
Preparing for Two-Column Proofs
Many rules from algebra are used in geometry.
Properties of Equality for Real Numbers
Reflexive Property
Symmetric Property
Transitive Property
Addition Property
Subtraction Property
Multiplication Property
Division Property
Substitution Property
Distributive Property
aa
If a b, then b a.
If a b and b c, then a c.
If a b, then a c b c.
If a b, then a c b c.
If a b, then a c b c.
a
b
If a b and c 0, then .
c
c
If a b, then a may be replaced
by b in any equation or expression.
a(b c) ab ac
Example: Prove that if 4x 8 8, then x 0.
Given: 4x 8 8
Prove: x 0
Proof:
Statements
Reasons
a. 4x 8 8
b. 4x 0
c. x 0
a. Given
b. Addition Property ()
c. Division Property ()
Name the property that justifies each statement.
1. Prove that if 3x 9, then x 15.
5
Given: 3x 9
5
Prove: x 15
Proof:
Statements
Reasons
a. 3x 9
a. Given
5
b. 3x 45
b. Multiplication Property ()
c. x 15
c. Division Property ()
2. Prove that if 3x 2 x 8, then x 3.
Given: 3x 2 x 8
Prove: x 3
Proof:
Statements
Reasons
a.
b.
c.
d.
3x 2 x 8
2x 2 8
2x 6
x 3
© Glencoe/McGraw-Hill
a.
b.
c.
d.
Given
Subtraction Property ()
Addition Property ()
Division Property ()
644
Geometry: Concepts and Applications
15–4
NAME
DATE
PERIOD
Skills Practice
Preparing for Two-Column Proofs
Complete each proof.
1. If m1 m2 and m3 m4, then
mABC mROD.
Given: m1 m2, m3 m4
Prove: mABC mROD
Proof:
Statements
a. m1 m2, m3 m4
b. mABC m1 m3
mROD m2 m4
c. m1 m3 m2 m4
d. mABC mROD
2. If
7x
6
D
R
2 4
O
A
1 3
B
Reasons
C
a. Given
b. Angle Addition Postulate
c. Addition Property of Equality
d. Substitution Property of Equality
14, then x 12.
Given:
7x
6
14
Prove: x 12
Proof:
Statements
Reasons
7x
a. 14
a. Given
b. 7x 84
c. x 12
b. Multiplication Property of Equality
c. Division Property of Equality
6
3. If ABC is a right triangle with C a right
angle and mA mB, then mA 45.
Given: ABC is a right triangle with C a
right angle and mA mB.
Prove: mA 45
Proof:
Statements
Reasons
a. ABC is a right triangle with C a
right angle and mA mB
b. mA mB mC 180
c. mC 90
d. mA mB 90
e. mA mA 90
f. 2mA 90
g. mA 45
© Glencoe/McGraw-Hill
A
C
B
a. Given
b.
c.
d.
e.
f.
g.
Angle Sum Theorem
Definition of Right Angle
Subtraction Property of Equality
Substitution Property of Equality
Substitution Property of Equality
Division Property of Equality
645
Geometry: Concepts and Applications
NAME
15–4
DATE
PERIOD
Practice
Preparing for Two-Column Proofs
Name the property or equality that justifies each statement.
1. If mA mB, then mB mA.
2. If x 3 17, then x 14.
3. xy xy
4. If 7x 42, then x 6.
5. If XY YZ XM, then XM YZ XY.
6. 2(x 4) 2x 8
7. If mA mB 90, and mA 30,
then 30 mB 90.
8. If x y 3 and y 3 10, then
x 10.
Symmetric
Subtraction
Reflective
Division
Addition
Distributive
Substitution
Transitive
Complete each proof by naming the property that justifies
each statement.
9. Prove that if 2(x 3) 8, then x 7.
Given: 2(x 3) 8
Prove: x 7
Proof:
Statements
Reasons
a. 2(x 3) 8
a. Given
b. 2x 6 8
b. Distributive Property
c. 2x 14
c. Addition Property ()
d. x 7
d. Division Property ()
10. Prove that if 3x 4 1x 6, then x 4.
2
Given: 3x 4 Prove: x 4
Proof:
1
x
2
6
Statements
Reasons
a. 3x 4 1x 6
a. Given
b. 5x 4 6
b. Subtraction Property ()
2
c.
2
5
x
2
c. Addition Property ()
10
d. Multiplication Property ()
d. x 4
© Glencoe/McGraw-Hill
646
Geometry: Concepts and Applications
15–4
NAME
DATE
PERIOD
Reading to Learn Mathematics
Preparing for Two-Column Proofs
Key Terms
two-column proof a deductive argument that contains statements and reasons
organized in two columns
Reading the Lesson
1. State whether each statement is true or false. If the statement is false, explain why.
a. Algebraic properties can be used as reasons in proofs.
true
b. When you solve an equation, you are using inductive reasoning.
False; you are
using deductive reasoning.
c. In a two-column proof, you must give a reason for each statement.
true
d. The last statement in a two-column proof is the given information.
False; the last
statement is what you want to prove.
2. Fill in the missing statements and reasons in the two-column proof.
Given: a b, c d
Prove: m2 m7 m8
c
1
2
3
4
d
6
5
7
8
a
b
Proof:
Statements
Reasons
a. a b, c d
a. Given
b. 2 4
b. Postulate 4-1 Corresponding Angles
c. 4 5
c. Theorem 4-1 Alternate Interior Angles
d. 2 5
d. Transitive Property of Congruence
e. m2 m5
e. Definition of Congruent Angles
f. m5 m7 m8
f. Exterior Angle Theorem
g. m2 m7 m8
g. Substitution Property of Equality
Helping You Remember
3. A good way to remember some terms is to compare them. Write several sentences
comparing the similarities and differences between paragraph proofs and two-column
proofs. Sample answer: Similarities: Both types of proofs contain a
figure, the given information, a statement about what to prove, and a
justification for each statement. Difference: A paragraph proof is
written in paragraph form, while a two-column proof is written in two
columns where one column has the statements and the second
column has the reasons.
© Glencoe/McGraw-Hill
647
Geometry: Concepts and Applications
NAME
15–4
DATE
PERIOD
Enrichment
More Counterexamples
Some statements in mathematics can be proven false by
counterexamples. Consider the following statement.
For any numbers a and b, a b b a.
You can prove that this statement is false in general if you can
find one example for which the statement is false.
Let a 7 and b 3. Substitute these values in the equation
above.
7337
4 4
In general, for any numbers a and b, the statement a b b a
is false. You can make the equivalent verbal statement:
subtraction is not a commutative operation.
In each of the following exercises a, b, and c are any numbers.
Prove that the statement is false by counterexample. Sample answers are given.
1. a (b c) (a b) c
2. a (b c) (a b) c
6 (4 2) (6 4) 2
6222
40
6 (4 2) (6 4) 2
1.5
2
3 0.75
6
2
3. a b b a
4. a (b c) (a b) (a c)
6446
3
2
6 (4 2) (6 4) (6 2)
6 6 1.5 3
1 4.5
2
3
5. a (bc) (a b)(a c)
6. a2 a2 a4
6 (4 2) (6 4)(6 2)
6 8 (10)(8)
14 80
62 62 64
36 36 1296
72 1296
7. Write the verbal equivalents for Exercises 1, 2, and 3.
1. Subtraction is not an associative operation.
2. Division is not an associative operation.
3. Division is not a commutative operation.
8. For the Distributive Property a(b c) ab ac it is said
that multiplication distributes over addition. Exercises 4 and
5 prove that some operations do not distribute. Write a
statement for each exercise that indicates this.
4. Division does not distribute over addition.
5. Addition does not distribute over multiplication.
© Glencoe/McGraw-Hill
648
Geometry: Concepts and Applications
15–5
NAME
DATE
PERIOD
Study Guide
Two-Column Proofs
The reasons necessary to complete the following proof are
scrambled up below. To complete the proof, number the reasons
to match the corresponding statements.
Given:
Prove:
 
CD ⊥ BE
 
AB ⊥ BE
 
AD CE
 
BD DE
 
AD CE
Proof:
Statements
Reasons
⊥ BE
1. CD
1. Definition of Right Triangle
2. AB
⊥ BE
2. Given
3. 3 and 4 are right angles.
3. Given 2
4. ABD and CDE are right
triangles.
4. Definition of Perpendicular Lines
CE
5. AD
5. Given
6. BD
DE
6. CPCTC
7. ABD CDE
7. In a plane, if two lines are cut
by a transversal so that a pair of
corresponding angles is congruent,
then the lines are parallel.
(Postulate 4-2) 9
8. 1 2
8. Given 6
CE
9. AD
9. HL 7
© Glencoe/McGraw-Hill
649
4
1
3
5
8
Geometry: Concepts and Applications
15–5
NAME
DATE
PERIOD
Skills Practice
Two-Column Proofs
Write a two-column proof.
1. Given: ITC is an isosceles triangle with base
IC
, TS bisects ITC
C
S
Prove: IS
Proof:
Statements Sample Answer:
Reasons
I
S
C
T
a. ITC is an isosceles triangle
a. Given
b. TS bisects ITC
b. Given
c. IT
C
T
c. Definition of isosceles triangle
d. ITS CTS
d. Definition of angle bisector
e. S
T
S
T
e. Reflexive Property
f. ITS CTS
f. SAS
g. IS
C
S
g. CPCTC
with base IC
.
A
B
D
C
2. Given: ABCD is a square.
Prove: AC
D
B
Proof:
Statements Sample Answer:
Reasons
a. ABCD is a square.
a. Given
b. A
D
BC
b. Definition of a square
c. ADC and BCD are right
c. Definition of a square
d. ADC BCD
d. Definition of congruent angles
e. D
C
DC
e. Reflexive Property
f. ∆ADC ∆BCD
f. SAS
g. A
C
BD
g. CPCTC
angles.
© Glencoe/McGraw-Hill
650
Geometry: Concepts and Applications
15–5
NAME
DATE
PERIOD
Practice
Two-Column Proofs
Write a two-column proof.

1. Given: B is the midpoint of AC.
Prove: AB CD BD
Proof:
Statements
a.
b.
c.
d.
Reasons

B is the midpoint of AC.
AB BC
BC CD BD
AB CD BD
a.
b.
c.
d.
Given
Definition of midpoint
Segment Addition Postulate
Substitution Property ()
2. Given: AEC DEB
Prove: AEB DEC
Proof:
Statements
Reasons
a. AEC DEB
a. Given
b. mAEC mDEB
b. Definition of congruent angles
c. mAEC mAEB mBEC,
mDEB mDEC mBEC
c. Angle Addition Postulate
d. mAEB mBEC mDEC mBEC
d. Transitive Property ()
e. mAEB mDEC
e. Subtraction Property ()
f. AEB DEC
f. Definition of congruent angles
© Glencoe/McGraw-Hill
651
Geometry: Concepts and Applications
15–5
NAME
DATE
PERIOD
Reading to Learn Mathematics
Two-Column Proofs
Reading the Lesson
1. Determine whether each statement is true or false. If the statement is false, explain why.
a. The given information for a proof can be found in the conclusion of the conjecture.
False; the given information is found in the hypothesis.
b. You cannot use definitions of geometric terms as a reason for a statement in a proof.
False; definitions are one of the three things that can be used for a
reason.
c. If the figure you are given to work with for a proof has overlapping triangles, you can
redraw the triangles as separate triangles. true
False; the given
information is always used in a two-column proof.
d. The given information is never used in a two-column proof.
2. Write the statements and reasons for a two-column proof for each set of information. First,
write a short plan for your proof.
Given: UR
, R
S
UT
TS
Prove: RUS TUS
R
Plan: Sample answer: Show that the two
triangles are congruent and then use
CPCTC.
U
S
T
Proof:
Statements
a.
b.
c.
d.
e.
U
R
U
T
R
S
T
S
U
SU
S
RUS TUS
RUS TUS
Reasons
a.
b.
c.
d.
e.
Given
Given
Reflexive Prop. of SSS Postulate
CPCTC
Helping You Remember
3. Sometimes it is helpful to summarize information that you need to remember. Summarize
the steps you would take to write a two-column proof. Sample answer: First, write
the Given and Prove and draw a diagram for the situation. Look at
the given information and mark the diagram with that information.
Look at what you are to prove and make a plan for using the given
information to reach that conclusion. You can use the work backward
strategy as well. Then write each statement and its reason in a logical
order to arrive at the conclusion.
© Glencoe/McGraw-Hill
652
Geometry: Concepts and Applications
15–5
NAME
DATE
PERIOD
Enrichment
Pythagorean Theorem
Use the Pythagorean Theorem to find the area of the shaded
region in the figure at the right.
Think of the figure as four
triangles and a square.
b
c
c
a
a
c
c
b
a
2
a
b
2
a
b
b
b
a
area of large square







b
area of the center square







2
a
b
a
b
area of the 4 triangles
a
a
1
4 2 a b
b






a
b
Think of the figure as a
large square.
c2
a2 a b b2 a b
c2
a2 b2 a b a b
c2
c2 2 a b
c2
a2 b2 2 a b
a2 b2
a2 b2
The relationship c2 a2 b2 is true for all right triangles.
Use the Pythagorean Theorem to find the area of A, B, and C in each of the
following. Then, answer true or false for the statement A B C.
1.
2.
C
3.
C
1.5
3
C
A
9
15
A
2.5
3
3
5
A 3
5
4
4
12
Squares
B
Equilateral
Triangles
B
2 3
Semicircles
B
A: 81,
B: 144,
C: 225; true
© Glencoe/McGraw-Hill
A: 2.253
,
B: 43
,
C: 6.253
; true
653
A: 1.125,
B: 2.000,
C: 3.125; true
Geometry: Concepts and Applications
15–6
NAME
DATE
PERIOD
Study Guide
Coordinate Proofs
You can place figures in the coordinate plane and use algebra to
prove theorems. The following guidelines for positioning figures
can help keep the algebra simple.
•
•
•
•
Use the origin as a vertex or center.
Place at least one side of a polygon on an axis.
Keep the figure within the first quadrant if possible.
Use coordinates that make computations simple.
The Distance Formula, Midpoint Formula, and Slope Formula are
useful tools for coordinate proofs.
Example: Use a coordinate proof to prove that the diagonals
of a rectangle are congruent.
Use (0, 0) as one vertex. Place another vertex on
the x-axis at (a, 0) and another on the y-axis at
(0, b). The fourth vertex must then be (a, b).
Use the Distance Formula to find OB and AC.
2
2
)
02
b
(
)
0
a
b2
OB (a
2
2
AC (0
)
a2
b
(
)
0
a
b2
Since OB AC, the diagonals are congruent.
Name the missing coordinates in terms of the given variables.
1. ABCD is a rectangle.
2. HIJK is a parallelogram.
D(a, c)
A(a, 0)
H (0, 0)
K(0, c)
J(a, b c)
3. Use a coordinate proof to show that the opposite sides of any
parallelogram are congruent. Label the vertices A(0, 0), B(a, 0), C(b, c), and
D(a b, c). Then use the Distance Formula to
find AB, CD, AC, and BD.
2
)
02
0
(
)
0
a
2 a
AB (a
2
CD ((
a
)
b
)
b2
c
(
)
c
a
2 a
2
a
)
b
)
a2
c
(
)
0
b
2
c2
BD ((
2
AC (b
)
02
c
(
)
0
b
2
c2
So AB CD and AC BD. Therefore, the opposite sides of a
parallelogram are congruent.
© Glencoe/McGraw-Hill
654
Geometry: Concepts and Applications
15–6
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DATE
PERIOD
Skills Practice
Coordinate Proofs
Position and label each figure on a coordinate plane. 1–5. Sample answers given.
1. a square with sides m units long
2. a right triangle with legs p and r units long
3. a rhombus with sides c units long
4. an isosceles triangle with base d units long and heights s units long
5. a rectangle with length x units and width y units
© Glencoe/McGraw-Hill
655
Geometry: Concepts and Applications
15–6
NAME
DATE
PERIOD
Practice
Coordinate Proofs
Name the missing coordinates in terms of the given variables.
1. XYZ is a right isosceles triangle.
2. MART is a rhombus.
M(0, 0), R(a b, a
2
b2)
X(0, 0) Y(a, 0)
3. RECT is a rectangle.
4. DEFG is a parallelogram.
R(0, 0) C(a, b)
D(0, 0), F(a c, b)
5. Use a coordinate proof to prove that the diagonals of a
rhombus are perpendicular. Draw the diagram at the right.
Sample proof:
 a
b2 0
b2
2
a
2
slope of AC ab0
ab
 a
b2 0
b2
2
a
2
slope of BD ba
ba
2
2
2
2
2
2
b a
b
a
a b 1
ab
ba
b2 a2
© Glencoe/McGraw-Hill
656
Geometry: Concepts and Applications
15–6
NAME
DATE
PERIOD
Reading to Learn Mathematics
Coordinate Proofs
Key Terms
coordinate proof a proof that uses figures on a coordinate plane
Reading the Lesson
1. Complete each sentence with one or two words to make a true statement.
a. If you are writing a coordinate proof and need to show that two segments are congruent,
Distance Formula
a formula you may want to use is the ______________________.
b. When drawing a diagram for a coordinate proof, try to place a vertex of the figure at the
origin
________.
c. If you are writing a coordinate proof and want to show that two segments are parallel, a
Slope Formula
formula you may want to use is the ___________________.
d. When drawing a diagram for a coordinate proof, try to place at least on side of the
axis
polygon on a(n) ______.
e. If you are writing a coordinate proof and want to show that a segment has been bisected,
Midpoint Formula
a formula you may want to use is the _______________________.
first
f. When drawing a diagram for a coordinate proof, try to keep the figure in the ______
quadrant.
2. Find the missing coordinates in each figure. Explain how you find the coordinates.
a. isosceles triangle
b. isosceles trapezoid
y
y
R(?, b)
S(?, c)
T(b, ?)
T(a, ?)
S(?, ?)
R(?, ?) O
x
R is (0, b) because the point
is on the y–axis; S is (0, 0)
because the point is the
b
origin; T is a, because it
2
is half way between R and S
in vertical distance.
U(a, ?) x
R is (–a, 0) since it is a units in the
negative direction horizontally and
lies on the x–axis; S is (–b, c)
since it is b units in the negative
direction horizontally; T is (b, c)
since it is c units in a positive
vertical direction; U is (a, 0) since
it is on the x–axis.
Helping You Remember
3. What is an easy way to remember how best to draw a diagram that will help you devise a
coordinate proof? Sample answer: A key point in the coordinate plane is
the origin. The everyday meaning of origin is place where something
begins. So look to see if there is a good way to begin by placing a
vertex of the figure at the origin.
© Glencoe/McGraw-Hill
657
Geometry: Concepts and Applications
15–6
NAME
DATE
PERIOD
Enrichment
Coordinate Proofs with Circles
You can prove many theorems about circles by using coordinate
geometry. Whenever possible locate the circle so that its center
is at the origin.
1. Prove that an angle inscribed in a
semicircle is a right angle. Use the figure at
right. (Hint: Write an equation for the
circle. Use your equation to help show that
P) (slope of PB
(slope of A
) 1).
b0
slope of AP
b
slope of PB
y
P (a, b)
B (r, O)
a ( r)
ar
b0
b
ar
ar
A(r, O)
x
O
(slope of AP
PB
) (slope of )
2
b
2
b b 2
ar
ar
a r
a 2 b 2 r 2, since (a, b) is on the
graph of x 2 y 2 r 2. Therefore
b
r a
2 2 1.
b 2 r 2 a 2, and 2
2
2
a r
2
2
a r
2. Suppose PQ
B, Q is between A and B,
A
and PQ
is the geometric mean between A
Q
and Q
B
.
Prove
that
P
is
on
the
circle
that
has A
B
as a diameter. Use the figure at the
right.
(PQ)2 (AQ) (QB)
(b)2 (a r) (r a)
b 2 (r a) (r a)
b2 r2 a2
Therefore a 2 b 2 r 2, which means
that (a, b) is on the circle with the
equation x 2 y 2 r 2. This is the
circle that has A
B
as a diameter.
© Glencoe/McGraw-Hill
658
y
P (a, b)
A (r, O)
B (r, O)
O
Q (a, O)
x
Geometry: Concepts and Applications
NAME
15
DATE
PERIOD
Chapter 15 Test, Form 1A
Write the letter for the correct answer in the blank at the right of each
problem.
For Questions 1–3, refer to the following statements.
p: Quadrilateral ABCD has four right angles.
q: Opposite sides of quadrilateral ABCD are parallel.
r: Quadrilateral ABCD is a rhombus.
1. Which statement is the negation of p?
A. Quadrilateral ABCD has three right angles.
B. Quadrilateral ABCD is not a square.
C. Quadrilateral ABCD could be a trapezoid.
D. Quadrilateral ABCD does not have four right angles.
1.
2. How would you write the statement below using symbols?
Opposite sides of quadrilateral ABCD are parallel and
quadrilateral ABCD is not a rhombus.
A. q ∧ r
B. q ∨ r
C. q ∧ r
D. q ∨ r
2.
3. If quadrilateral ABCD is a square, which compound statement is true?
A. p ∧ q
B. p ∨ r
C. q ∨ p
D. r ∧ q
3.
For Questions 4–5, use the Law of Detachment or Law of Syllogism to
determine a conclusion that follows from statements (1) and (2). If a valid
conclusion does not follow, choose no valid conclusion.
4. (1) All prime numbers except 2 are odd.
(2) 37 is an odd number.
A. 37 is a prime number.
B. 35 is not a prime number.
C. The next prime after 37 is 39. D. no valid conclusion
4.
5. (1) In two triangles, if two pairs of corresponding angles are congruent,
then the triangles are similar.
(2) If two triangles are similar, then the measures of their corresponding
sides are proportional.
A. If the corresponding sides of two triangles are proportional, then two
pairs of corresponding angles are congruent.
B. In two triangles, if two pairs of corresponding angles are congruent,
then the measures of their corresponding sides are proportional.
C. In two triangles, if all three pairs of corresponding angles are
congruent, then the two triangles are similar.
D. no valid conclusion
5.
6. If ABC at the right is an equilateral triangle,
what are the coordinates of point C?
A. (2a, a3
)
a a3
C. , 2
2
B. (a, a3
)
a3
D. , a3
2
y
C
O A(0, 0)
B(2a, 0)
x
6.
7. Rectangle ABCD has been positioned on a coordinate plane so that its
vertices have the coordinates A(0, 0), B(a, 0), C(a, b), and D(0, b). Which
expression is the length of diagonal AC?
A. a b
©
Glencoe/McGraw-Hill
B. ab
2 b2
C. a
659
D. (a
b
)2
7.
Geometry: Concepts and Applications
NAME
15
DATE
PERIOD
Chapter 15 Test, Form 1A (continued)
For Questions 8–11, complete the proof by selecting the
B
missing information for each corresponding location.
E
D
Given: ABC is a right triangle, and quadrilateral
ADEF is a rectangle.
A
C
F
Prove: BDE EFC
Since quadrilateral ADEF is a rectangle, it is a (Question 8) by the definition
of a rectangle. Then DE
AF
and (Question 9) by the definition of a
parallelogram. This means DE
AC
FE
and A
B
. Then B FEC and
(Question 10) C since if two parallel lines are cut by a transversal, then
corresponding angles are congruent. Therefore, DBE FEC by
(Question 11) Similarity.
B. rhombus
EF
B. A
D
C. polygon
D. parallelogram
8.
C. A
E
⊥
DF
D. D
E
AF
9.
10. A. EBD
B. FEC
C. BED
D. B
10.
11. A. AA
B. SAS
C. SSS
D. ASA
11.
8. A. square
EF
9. A. AD
For Questions 12–16, complete the proof below by choosing
the statement or reason for each location.
Theorem: If two segments from the same exterior point are
tangent to a circle, then they are congruent.
Given: X
A
is tangent to C at A; X
B
is tangent to C at B.
Prove: X
A
XB
C
A
B
X
Statements
Reasons
A
is tangent to C at A.
a. X
X
B
is tangent to C at B.
a. Given
A
and C
B
; CAX and
b. Draw C
(Question 12) are right angles.
b. (Question 13)
X
; C
X
CX
c. Draw C
c. (Question 14)
d. A
C
BC
d. All radii of a circle are congruent.
e. ACX (Question 15)
e. (Question 16)
A
XB
f. X
f. CPCTC
12. A. ACB
13. A.
B.
C.
D.
B. BXC
C. CBX
D. AXB
Pythagorean Theorem
A tangent is perpendicular to a radius at the point of tangency.
Definition of tangent
Definition of perpendicular
12.
13.
14. A. Transitive
B. Substitution
C. Symmetric
D. Reflexive
14.
15. A. CBX
B. CXB
C. BCX
D. BXC
15.
16. A. HA
B. LA
C. LL
D. HL
16.
Bonus The coordinates of the vertices of ABC are A(0, 0), B(2a, 0), and
B
.
C(0, b). Write an equation of the line containing the median to A
b
a
A. y x b B. x 2a
©
Glencoe/McGraw-Hill
C. y 2b
660
b
a
D. y x
Bonus
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Test, Form 1B
Write the letter for the correct answer in the blank at the right of each
problem.
For Questions 1–3, refer to the following statements.
p: ABC is an isosceles right triangle.
q: In ABC, B is a right angle.
r: In ABC, AB BC.
1. Which statement is the negation of p?
A. ABC is an equilateral triangle.
B. ABC is a right triangle that is not isosceles.
C. ABC is a scalene triangle.
D. ABC is not an isosceles right triangle.
1.
2. How would you write the statement below using symbols?
In ABC, B is not a right angle or AB BC.
A. q ∨ r
B. q ∧ r
C. q ∨ r
D. q ∧ r
2.
3. If ABC is equilateral, which compound statement is true?
A. p ∨ q
B. p ∧ r
C. q ∨ r
D. q ∧ r
3.
For Questions 4–5, use the Law of Detachment or Law of Syllogism to
determine a conclusion that follows from statements (1) and (2). If a valid
conclusion does not follow, choose no valid conclusion.
4. (1)
(2)
A.
C.
All right angles are congruent.
A and B are right angles.
mA 90
B. mA mB
A B
D. no valid conclusion
4.
5. (1)
(2)
A.
B.
C.
If a parallelogram is a rhombus, then its diagonals bisect each other.
All squares are rhombuses.
All squares are parallelograms.
All squares have diagonals that bisect each other.
If the diagonals of a parallelogram bisect each other, then the
parallelogram is a square.
D. no valid conclusion
6. ABC at the right is an isosceles triangle. Which
ordered pair could be the coordinates of point C?
A. (b, a)
B. (0, b)
C. (a, b)
D. (b, 0)
y
5.
C
B(2b, 0) x
O A(0, 0)
6.
7. Rectangle WXYZ has been positioned on a coordinate plane so that its
vertices have the coordinates W(0, 0), X(2a, 0), Y(2a, 2b), and Z(0, 2b).
What are the coordinates of the midpoint of diagonal WY ?
A. (a, a)
©
Glencoe/McGraw-Hill
B. (b, b)
C. (a, b)
661
ab
2
ab
2
D. , 7.
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Test, Form 1B (continued)
For Questions 8–11, complete the proof by selecting the
missing information for each corresponding location.
DE
Given: AB
Prove: ABC EDC
A
D
C
B
E
DE
, then BAC (Question 8) because when two parallel lines
Since AB
are cut by a transversal the (Question 9) angles are congruent. Also,
BCA (Question 10) because vertical angles are congruent. Therefore,
ABC EDC by (Question 11) Similarity.
8. A. CED
B. CDE
9. A. corresponding
C. alternate interior
C. ABC
D. BCE
B. alternate exterior
D. consecutive interior
8.
9.
10. A. BCA
B. DCE
C. EDC
D. CED
10.
11. A. SSS
B. SAS
C. ASA
D. AA
11.
For Questions 12–16, complete the proof below by
choosing the statement or reason for each location.
X
Given: AB
and C
D
are diameters of X.
Prove: A
C
B
D
Reasons
B
and C
D
are diameters of X.
a. A
a. Given
A
XB
; X
C
XD
b. X
b. (Question 12)
c. (Question 13) BXD
c. (Question 14)
d. ACX (Question 15)
d. (Question 16)
BD
e. AC
e. CPCTC
Chords equidistant from the center of a circle are congruent.
All radii of a circle are congruent.
Distance Formula
Definition of midpoint
13. A. AXC
14. A.
B.
C.
D.
B
D
Statements
12. A.
B.
C.
D.
C
A
B. BXC
C. CBX
D. AXB
The angles opposite congruent sides of a triangle are congruent.
Alternate interior angles are congruent.
Vertical angles are congruent.
The central angles of a circle are congruent.
12.
13.
14.
15. A. XBD
B. CXA
C. BDX
D. XBD
15.
16. A. AA
B. SSS
C. ASA
D. SAS
16.
Bonus The coordinates of the vertices of ABC are A(0, 0), B(2a, 0), and
C(0, 2b). Write an equation of the line containing the median to B
C
.
A. x y 2b
©
Glencoe/McGraw-Hill
B. x 2a
C. y a
662
b
a
D. y x
Bonus
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Test, Form 2A
Let p represent “A and B are acute angles,” q represent
“A and B are vertical angles,” and r represent “A and B
are complementary angles.”
1.
1. Write the statement for the negation of r.
2. Use symbols to represent the disjunction below.
A and B are not acute angles or they are vertical angles.
2.
3. Suppose mA 35 and mB 55. What is the truth value
of the conjunction p ∧ r?
3.
4. If mA 35 and mB 55, what is the truth value of the
conditional q → r?
4.
Use the Law of Detachment or the Law of Syllogism to
determine a conclusion that follows from statements (1) and (2).
If a valid conclusion does not follow, write no valid conclusion.
5. (1) If a rectangle is a square, then all four of its sides are
congruent.
(2) Quadrilateral ABCD has four congruent sides.
5.
6. (1) If two angles are complementary, then the sum of their
measures is 90.
(2) 1 and 2 are complementary.
6.
7. (1) If two solids are similar, then the ratio of their volumes is
equal to the cube of the ratio of their heights.
(2) All cubes are similar.
7.
For Questions 8–13, complete the proof below by supplying the
missing information for each corresponding location.
Given: BAC BCA; BD
BE
Prove: BDC BEA
8.
B
9.
You know that BAC BCA. So
BC
(Question 8) because in a triangle
(Question 9). Also, B B because
(Question 10). It was also given that B
D
BE
.
So, BDC (Question 11) by (Question 12).
Therefore, BDC BEA by (Question 13).
D
E
10.
A
C
11.
12.
13.
©
Glencoe/McGraw-Hill
663
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Test, Form 2A (continued)
Name the property of equality that justifies each statement.
14. If mABC mBCA 90 and mBCA 32, then
mABC 32 90.
14.
15. If PQ BC AB BC, then PQ AB.
15.
16. If a b, then b a.
16.
For Questions 17–22, complete the two-column proof below by
supplying the statement or reason for each location.
Given: Quadrilateral ABCD is a
rectangle with YZ
⊥
DX
.
Prove: ADX ZYD
X
A
17.
B
Z
D
C
Y
18.
Statements
Reasons
a. ABCD is a rectangle.
a. Given
b. DAB is a right angle.
b. (Question 17)
Z
⊥
DX
c. Y
c. Given
d. DZY is a right angle.
d. Definition of perpendicular
e. mZDY mZYD 90
e. (Question 18)
f. mADX mZDY mADY
f. (Question 19)
g. mADY 90
g. Definition of rectangle
h. mADX mZDY 90
h. (Question 20)
i. mZDY mZYD mADX mZDY
i. Substitution Property
j. mZYD mADX
j. (Question 21)
k. ADX ZYD
k. (Question 22)
19.
20.
21.
22.
23. Quadrilateral ABCD is an isosceles trapezoid. If the
coordinates of three of its vertices are A(a, 0), B(b, c), and
D(a, 0), find the coordinates of C.
23.
24. Parallelogram PQRS is placed on a coordinate plane to be
used for a coordinate proof. What could be the coordinates of
its vertices?
24.
25. Refer to Question 24. Show that opposite sides of the
parallelogram are congruent.
25.
Bonus You are asked to prove the theorem given
below, using the figure shown at the right. What
specifically do you need to prove in the figure?
The altitude drawn to the hypotenuse of a right
triangle divides it into two similar right triangles.
©
Glencoe/McGraw-Hill
664
B
A
D
C
Bonus
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Test, Form 2B
Let p represent “The area of rectangle ABCD is 50 square
inches,” q represent “The perimeter of rectangle ABCD is
30 inches,” and r represent “The length of rectangle ABCD is
twice its width.”
1. Write the statement for the negation of q.
1.
2. Use symbols to represent the conjunction below.
The area of rectangle ABCD is 50 square inches and its length
is not twice its width.
2.
3. If rectangle ABCD is 9 inches long and 6 inches wide, what is
the truth value of the disjunction q ∨ p?
3.
4. Suppose rectangle ABCD is 12.5 inches long and 4 inches
wide. What is the truth value of the conditional p → r?
4.
Use the Law of Detachment or the Law of Syllogism to
determine a conclusion that follows from statements (1) and (2).
If a valid conclusion does not follow, write no valid conclusion.
5. (1) If a triangle is a right triangle, then it has two acute
angles.
(2) ABC is a right triangle.
5.
6. (1) If three points are noncollinear, then they determine a
plane.
(2) Points A, B, and C are coplanar.
6.
7. (1) All circles have diameters whose measure is twice the
measure of their radii.
(2) AB
is a radius of B.
7.
For Questions 8–13, complete the proof below by supplying the
missing information for each corresponding location.
Theorem: The angles opposite the congruent sides of an isosceles
triangle are congruent.
Given: isosceles ABC with AB
BC
B
Prove: A C
B
BC
. Let M be the
You know that A
midpoint of A
C
and draw B
M
. So
A
M
(Question 8) by the (Question 9).
Also, B
M
BM
because (Question 10). So,
ABM (Question 11) by (Question 12).
Therefore, A C by (Question 13).
8.
9.
10.
A
M
C
11.
12.
13.
©
Glencoe/McGraw-Hill
665
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Test, Form 2B (continued)
Name the property of equality that justifies each statement.
, PQ PQ.
14. For any PQ
14.
15. If m1 m2 m1 m3, then m2 m3.
15.
16. If AB CD and CD EF, then AB EF.
16.
For Questions 17–22, complete the two-column proof below by
supplying the reason for each location.
Given: Quadrilateral ABCD is a
YB
.
rectangle with DX
A
Prove: AXD CYB
X
D
17.
B
Y
C
18.
Statements
Reasons
a. ABCD is a rectangle.
a. Given
b. DAX and BCY are
right angles.
b. (Question 17)
c. DAX BCY
c. (Question 18)
d. ABCD is a parallelogram.
d. Definition of rectangle
D
BC
e. A
CD
B
f. A
e. (Question 19)
g. BYC XBY
YB
X
h. D
g. (Question 20)
h. Given
i. XBY AXD
i. (Question 21)
j. BYC AXD
j. Transitive Property
k. AXD CYB
k. (Question 22)
19.
20.
f. Definition of parallelogram
21.
22.
23. ABCD is a square. If the coordinates of three of its vertices
are A(a, 2a), B(a, 2a), and C(a, 0), find the coordinates of D.
23.
24. Isosceles triangle XYZ is placed on a coordinate plane to be
used for a coordinate proof. What could be the coordinates of
its vertices?
24.
25. Refer to Question 24. Show that the legs of the isosceles
triangle are congruent.
25.
Bonus You are asked to prove the theorem
B
D
given below, using the figure shown at the
right. What specifically do you need to
A
C
prove in the figure?
The median drawn to the hypotenuse of a right triangle divides
it into two isosceles triangles.
©
Glencoe/McGraw-Hill
666
Bonus
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Extended Response Assessment
Instructions: Demonstrate your knowledge by giving a clear, concise
solution for each problem. Be sure to include all relevant drawings
and to justify your answers. You may show your solution in more
than one way or investigate beyond the requirements of the problem.
1. Write a paragraph proof of the following theorem.
In an isosceles right triangle, the angle bisector of the right angle is
the perpendicular bisector of the hypotenuse.
2. Supply a reason for each statement to complete the proof below.
Theorem: The measure of an inscribed angle is equal to
one-half the measure of its intercepted arc.
Given: P with inscribed angle ABC
1
Prove: mABC (mAC )
B
C
P
2
A
Statements
Reasons
A
, PB
, and PC
;
a. Draw P
mAPC mBPC mAPB 360
a.
b. AP BP; BP CP
b.
c. mPBC mPCB; mPAB mPBA
c.
d. mBPC mPBC mPCB 180;
mAPB mPBA mPAB 180
d.
e. mBPC 2(mPBC) 180;
mAPB 2(mPBA) 180
e.
f. mBPC 180 2(mPBC);
mAPB 180 2(mPBA)
f.
g. mAPC [180 2(mPBC)] [180 2(mPBA)] 360
g.
h. mAPC 2(mPBC) 2(mPBA) 0
h.
i. mAPC 2(mPBC) 2(mPBA)
i.
j. mAPC 2(mPBC mPBA)
j.
k. mABC mPBC mPBA
k.
l. mAPC 2(mABC)
m. mAPC mAC
n. mAC 2(mABC)
1
o. mABC (mAC )
l.
m.
n.
o.
2
3. Write a two-column proof of the following theorem.
The segments joining consecutive midpoints of the sides
of a rhombus form a parallelogram.
Glencoe/McGraw-Hill
667
B
Z
D
©
W
A
X
Y
C
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Mid-Chapter Test
(Lessons 15–1 through 15–3)
Let p represent “x is divisible by 5,” q represent “100 x 120,”
and r represent “x is divisible by 10.”
1. Write the statement for the negation of q.
1.
2. For what whole number value(s) of x is the conjunction p ∧ r
true?
2.
3. If x 115, what is the truth value of q ∨ p?
3.
4. If x 113, which conditional statements below are true?
p→q
q→r
r→p
4.
Use the Law of Detachment or the Law of Syllogism to
determine a conclusion that follows from statements (1) and (2).
If a valid conclusion does not follow, write no valid conclusion.
5. (1) If both pairs of opposite sides in a quadrilateral are
congruent, then it is a parallelogram.
(2) If a quadrilateral is a parallelogram, then both pairs of its
opposite sides are parallel.
5.
6. (1) All regular hexagons are similar.
(2) Hexagons ABCDEF and UVWXYZ are regular.
6.
7. (1) If two angles are vertical, then they are congruent.
(2) Two right angles are congruent.
7.
Determine whether each situation is an example of inductive or
deductive reasoning.
8. The numbers 3, 13, and 23 are prime, so Raina conjectures
that all numbers with 3 as their ones digit are prime.
8.
9. Mike knows that the product of two odd numbers is an odd
number, so he concludes that the product 193 207 is odd.
9.
For Questions 10–14, complete the proof by supplying the
missing information for each corresponding location.
Given: AE
DE
; EBC ECB
Prove: AB
DC
A
D
10.
E
You know that EBC ECB, so
B
C
11.
(Question 10) because in a triangle the
EB
12.
sides opposite congruent angles are
E
. And
DE
congruent. You also know that A
AEB DEC because (Question 11). So
13.
AEB (Question 12) by (Question 13).
DC
by (Question 14).
Therefore, AB
©
Glencoe/McGraw-Hill
14.
668
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Quiz A
(Lessons 15–1 and 15–2)
Let p represent “A hexagon has six sides,” q represent “Skew
lines are coplanar,” and r represent “72 12 5.”
1. What is the truth value of the conjunction r ∧ q?
1.
2. What is the truth value of the disjunction p ∨ r?
2.
3. Determine the truth value of the conditional p → q.
3.
Use the Law of Detachment or the Law of Syllogism to
determine a conclusion that follows from statements (1) and (2).
If a valid conclusion does not follow, write no valid conclusion.
4. (1) If a quadrilateral is a square, then it is a rectangle.
(2) All rectangles have four right angles.
4.
5. (1) If a triangle has one right angle, then it has two acute
angles.
(2) ABC has two acute angles.
5.
15
NAME
DATE
PERIOD
Chapter 15 Quiz B
(Lessons 15–3 through 15–6)
Complete the proof of the theorem by supplying the missing
information for each corresponding location.
Theorem: The diagonal of a rhombus bisects
a pair of opposite angles.
Given: rhombus ABCD with diagonal BD
Prove: BD
bisects ABC and ADC.
A
D
B
C
. Since
BC
CD
DA
By the definition of rhombus, AB
congruence of segments is reflexive, B
D
BD
. So,
ABD CBD by (Question 1). Then, ABD CBD and
ADB (Question 2) by CPCTC. Thus, by the (Question 3),
BD
bisects ABC and ADC.
1.
2.
3.
4. Draw and label a figure for the conjecture below.
The medians of an equilateral triangle are congruent.
4.
5. Name the property of equality that justifies the following
statement.
If mA mB 180 and mA 110, then 110 mB 180.
6. Parallelogram ABCD is placed on a coordinate plane as part of
a coordinate proof. If three of the vertices are labeled A(0, 0),
B(a, b), and D(c, 0), what are the coordinates of vertex C?
©
Glencoe/McGraw-Hill
669
5.
6.
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Cumulative Review
1. Write this statement in if-then form. (Lesson 1–4)
The sum of two odd integers x and y is even.
bisects CXE.
2. In the figure, XD
If mDXE 55, find mAXE.
(Lessons 3–3 and 3–6)
B
A
X
C
1.
E
D
2.
3. Find the slope of the line passing through points at (2, 7) and
(4, 3). (Lesson 4–5)
3.
4. In the figure, ACB DCE and
BC
CE
. What other congruence
relation is needed to prove
ABC DEC by SAS? (Lesson 5–5)
4.
A
E
C
B
D
5. If AX
is the perpendicular bisector of BC
in ABC, then
ABC must be a(n) ? triangle with A its ? angle and
BC
its ? . (Lesson 6–4)
6. Refer to the figure at the right. Find
the value of x. (Lesson 7–2)
5.
(3x 5)
x
115
6.
7. In parallelogram ABCD, mA 2x 30 and mD x. Find
mB. (Lesson 8–2)
8. Determine if the pair of triangles shown
at the right are similar. If so, state the
reason and find the value of x. (Lesson 9–3)
7.
x
3
5
8.
10
9. What is the measure of one interior angle of a regular
nonagon? (Lesson 10–2)
For Questions 10–11, refer to the figure.
10. In the figure, BF
is a ? of C.
(Lesson 11–1)
11. If mAB 72 and mEF 58, find mAGB.
(Lesson 15–3)
9.
B
A
C
D
G
F
10.
E
11.
12. Find the lateral area, to the nearest tenth, of a cone with
height 9 feet and radius 12 feet. (Lesson 12–4)
12.
For Questions 13–14, refer to the theorem and figure below.
(Lesson 15–2)
Theorem: The diagonals of a rectangle
are congruent.
13. What is the hypothesis of the theorem?
Glencoe/McGraw-Hill
D
F
E
13.
14.
14. What is to be proved?
©
C
670
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Standardized Test Practice
(Chapters 1–15)
Write the letter for the correct answer in the blank at the right of each
problem.
1. What is the intersection of BF and plane EGH in the
figure at the right?
A. B
B. E
C. F
D. G
E
A
H
B
D
F
G
C
1.
2. Identify the hypothesis of the converse of the statement below.
All right triangles have two acute angles.
A. A triangle has two acute angles.
B. A triangle has one right angle.
C. An isosceles triangle has two acute angles.
D. A right triangle can be isosceles.
2.
3. Points R, Q, and T are collinear with point R between points Q and T. If
QR 6.8 and RT 9.2, find QT.
A. 2.4
B. 3.6
C. 11.5
D. 16
3.
4. If CAT and DAT form a linear pair and CAT is acute, then DAT is
what kind of angle?
A. acute
B. obtuse
C. right
D. straight
4.
5. Find the value of x in the figure at the right so
that m .
A. 2.5
B. 17
C. 61
D. 119
7x m
(3x 10)
n
5.
6. Find the equation of the line passing through the point at (6, 4) and
perpendicular to the line y 2x 1.
A. y 2x 8
1
2
B. y x 2
1
2
1
2
C. y x 7 D. y x 1
7. Which congruence test can be used to prove
ABC CDA?
A. SSS
B. ASA
C. AAS
D. SAS
8. In ABC, AB
AC
and mB 38. Find mA.
A. 38
B. 76
C. 104
A
B
6.
D
C
7.
8.
D. 142
9. In XYZ, mX 58 and mY 49. List the sides of the triangle in
order from least to greatest measure.
A. XZ
, YZ
, XY
B. X
Y
, Y
Z
, X
Z
C. YZ
, X
Y
, X
Z
D. X
Z
, XY
, YZ
9.
10. Quadrilateral WXYZ is a parallelogram whose diagonals intersect at
point A. If YA 2t, WA 3t 4, and XZ 5t, find XA.
A. 4
B. 9
C. 10
D. 20
10.
©
Glencoe/McGraw-Hill
671
Geometry: Concepts and Applications
15
NAME
DATE
PERIOD
Chapter 15 Standardized Test Practice
(Chapters 1–15) (continued)
BC
11. If XY
in the figure, find AC.
A. 15
B. 25
C. 35
D. 45
A
12
X
9
B
20
Y
11.
C
12. If RST XYZ, RS 18, XY 30, and the perimeter of RST is 57,
find the perimeter of XYZ.
A. 83
B. 95
C. 102
D. 110
12.
13. What is the area of a trapezoid with altitude 12 centimeters and bases
10 centimeters and 18 centimeters long?
A. 168 cm2
B. 336 cm2
C. 968 cm2
D. 2160 cm2
13.
14. Which of the following statements is false?
A.
B.
C.
D.
A
is a chord of X.
AD
mAB 180 mBD
mEBD mEA 180
EB
is a radius of X.
E
B
X
C
D
14.
15. To the nearest tenth, what is the volume of a cone with diameter 3 feet
and height 5 feet?
A. 11.8 ft3
B. 28.3 ft3
C. 35.3 ft3
D. 47.1 ft3
15.
16. The measure of the longer leg of a 30°-60°-90° triangle is 12 inches.
What is the measure of the hypotenuse?
A. 63
in.
B. 83
in.
C. 123
in.
17. Find the value of a to the nearest tenth.
A. 10.7
B. 11.9
C. 14.4
D. 17.8
18. Find the value of x.
A. 6
B. 9
C. 10
D. 12
D. 24 in.
16.
B
a
C
42
16
A
17.
C
x 12
3 4
18.
19. Use the Law of Detachment to determine a conclusion that follows from
statements (1) and (2).
(1) Acute angles have measures less than 90.
(2) 1 and 2 are acute angles.
A. 1 2
B. m1 m2
C. m1 m2 90
D. 1 and 2 both have measure less than 90.
19.
20. Which of the following could be the coordinates of the vertices of
parallelogram ABCD?
A. A(0, 0), B(2a, 0), C(a, b), D(2a, b)
B. A(0, 0), B(a, 0), C(a b, c), D(b, c)
C. A(0, 0), B(a, 0), C(a b, c), D(b, c)
D. A(0, 0), B(a, 0), C(a, b), D(a, b)
20.
©
Glencoe/McGraw-Hill
672
Geometry: Concepts and Applications
Preparing for Standardized Tests
Answer Sheet
1.
A
B
C
D
E
2.
A
B
C
D
E
3.
A
B
C
D
E
4.
A
B
C
D
E
5.
A
B
C
D
E
6.
A
B
C
D
E
7.
A
B
C
D
E
8.
A
B
C
D
E
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
10. Show your work.
© Glencoe/McGraw-Hill
A1
Geometry: Concepts and Applications
Study Guide
NAME
DATE
© Glencoe/McGraw-Hill
p→q
q→p
if p, then q
if q, then p
Symbol Definition
~p
not p
pq
p and q
pq
p or q
A2
T
F
T
F
T
T
F
F
© Glencoe/McGraw-Hill
q
T
F
T
F
p
T
T
F
F
3. p ~q
q
p
1. p q
F
T
F
T
~q
T
T
T
F
pq
F
T
F
F
p ~q
629
T
T
F
F
p
4. ~(p q)
T
T
F
F
p
2. ~p q
Complete a truth table for each compound statement.
Every statement has a truth value. It is convenient
to organize the truth values in a truth table like the
one shown at the right.
conditional
converse
Term
negation
conjunction
disjunction
T
F
F
F
pq
T
F
T
F
F
T
T
T
~( p q)
T
F
T
T
~p q
T
F
F
F
Geometry: Concepts and Applications
T
F
T
F
q
F
F
T
T
q
T
F
T
F
T
T
F
F
~p
q
pq
PERIOD
Conjunction
p
A statement is any sentence that is either true or false, but not
both. The table below lists different kinds of statements.
Logic and Truth Tables
15–1
Skills Practice
NAME
DATE
PERIOD
d: Parallel lines do not intersect.
© Glencoe/McGraw-Hill
630
Geometry: Concepts and Applications
16. c d 5 5 20 and parallel lines do not intersect; false.
15. c d 5 5 20 or parallel lines do not intersect; true.
14. b d January is a day of the week and parallel lines do not intersect; false
13. b d January is a day of the week or parallel lines do not intersect; true.
12. b c January is a day of the week and 5 5 20; false.
11. b c January is a day of the week or 5 5 20; false.
10. a d A triangle has three sides and parallel lines do not intersect; true.
9. a d A triangle has three sides or parallel lines do not intersect; true.
8. a c A triangle has three sides and 5 5 20; false.
7. a c A triangle has three sides or 5 5 20; true.
6. a b A triangle has three sides and January is a day of the week; false.
Write a statement for each conjunction or disjunction. Then find the truth value.
5. a b A triangle has three sides or January is a day of the week; true.
Parallel lines intersect.
4. d
5 5 20
3. c
January is not a day of the week.
2. b
A triangle does not have three sides.
Write the statements for each negation.
1. a
c: 5 5 20
For Exercises 1–16, use conditionals a, b, c and d.
a: A triangle has three sides.
b: January is a day of the week.
Logic and Truth Tables
15–1
Answers
(Lesson 15-1)
Geometry: Concepts and Applications
Practice
NAME
DATE
PERIOD
© Glencoe/McGraw-Hill
A quadrilateral does
not have 4 sides.
2. q
A3
T
F
T
F
T
T
F
F
F
F
T
T
~p
© Glencoe/McGraw-Hill
q
p
10. p q
F
T
F
T
~q
F
T
T
T
~(p ~q)
T
T
F
F
F
F
T
T
p ~p
F
F
T
F
~p q
Geometry: Concepts and Applications
T
F
T
F
q
A quadrilateral does not have
4 sides or there are not 30 days
in September; false.
9. q r
A quadrilateral does not have
4 sides and (5 3) 3 5;
false.
7. q s
11. p q
631
There are not
30 days in
September.
3. r
Labor Day is in April and a
quadrilateral has 4 sides; false.
5. p q
Construct a truth table for each compound statement.
Labor Day is in April and
(5 3) 3 5; false.
8. p s
Labor Day is not in April or there
are 30 days in September; true.
6. p r
Labor Day is in April or a
quadrilateral has 4 sides; true.
4. p q
Write a statement for each conjunction or disjunction.
Then find the truth value.
Labor Day is not
in April.
1. p
Write the statements for each negation.
Use conditionals p, q, r, and s for Exercises 1–9.
p: Labor Day is in April.
q: A quadrilateral has 4 sides.
r: There are 30 days in September.
s: (5 3) 3 5
Logic and Truth Tables
15–1
NAME
statement.
negation of p.
© Glencoe/McGraw-Hill
632
Geometry: Concepts and Applications
conjunction is an and (or both together) statement and a disjunction is an
or statement.
3. Prefixes can often help you to remember the meaning of words or to distinguish between
similar words. Use the dictionary to find the meanings of the prefixes con and dis. 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
Helping You Remember
July is not a month in the summer and red is a color.
c. q r The symbols mean the conjunction of the negation of q and r.
United States or July is a month in the summer.
b. p q The symbols mean the disjunction of p and q. You live in the
United States and July is a month in the summer.
a. p q The symbols mean the conjunction of p and q. You live in the
2. Let p represent “you live in the United States,” q represent “July is a month in the
summer,” and r represent “red is a color.” For each exercise, explain what the symbols
mean and then write the statement indicated by the symbols.
f. If you negate both p and q in a statement p → q, the new statement is called the
inverse.
e. The statement represented by not p is the
truth value.
d. A statement that is formed by joining two statements with the word and is called a
conjunction.
c. The true or false nature of a statement is called its
compound
PERIOD
b. A statement that is formed by joining two statements with the word or is called a
disjunction.
a. Any two statements can be joined to form a
1. Supply one or two words to complete each sentence.
Reading the Lesson
statement a statement that is either true or false, but not both
truth value the true or false nature of a statement
negation the negative of a statement
truth table a convenient way to organize truth values
compound statement two or more logic statements joined by
and or or
inverse a statement formed by negating both p and q in the
conditional p → q
Key Terms
DATE
Reading to Learn Mathematics
Logic and Truth Tables
15–1
Answers
(Lesson 15-1)
Geometry: Concepts and Applications
15–2
Study Guide
NAME
DATE
© Glencoe/McGraw-Hill
A4
C
B
© Glencoe/McGraw-Hill
D
A
Example:
D
A
B
C
Counterexample:
5. Suppose you draw four points A, B, C,
and D and then draw A
B
, B
C
, C
D
,
and D
A
. Does this procedure give a
quadrilateral always or only
sometimes? Explain your answers
with figures. only sometimes
It is true for 1, 2, and 3. It is
not true for negative
integers. Sample: 2
3. Is the equation k
2 k true when you
replace k with 1, 2, and 3? Is the
equation true for all integers? If
possible, find a counterexample.
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
January? no
Answer each question.
633
Example:
Geometry: Concepts and Applications
Counterexample:
6. Suppose you draw a circle, mark
three points on it, and connect them.
Will the angles of the triangle be
acute? Explain your answers with
figures no, only sometimes
It is true for all real
numbers.
4. Is the statement 2x x x true when
1
you replace x with 2, 4, and 0.7? Is the
statement true for all real numbers?
If possible, find a counterexample.
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
Example: Is the statement 1 1 true when you replace x with 1,
x
2, and 3? Is the statement true for all reals? If
possible, find a counterexample.
1
1
1
1
1
1, 1, and 1. But when x , then 2. This
1
2
3
2
x
counterexample shows that the statement is not
always true.
If p → q is a true conditional and
p is true, then q is true.
© Glencoe/McGraw-Hill
634
PERIOD
Geometry: Concepts and Applications
53; Each number is 6 less than the preceding one.
6. Look for a Pattern Find the next number in the list 83,
77, 71, 65, 59 and make a conjecture about the pattern.
5. (1) All fish can swim.
(2) Fonzo can swim. no conclusion
m A m B 90; Detachment
(2) A and B are complementary.
4. (1) Angles that are complementary have measures with a sum
of 90.
the United States; Syllogism.
3. (1) If people live in Manhattan, then they live in New York.
(2) If people live in New York, then they live in the United
States. If people live in Manhattan, then they live in
2. (1) If a dog eats Dogfood Delights, the dog is happy.
(2) Fido is a happy dog. no conclusion
is a whole number, then it is a rational number; Syllogism.
1. (1) If a number is a whole number, then it is an integer.
(2) If a number is an integer, then it is a rational number. If a number
Determine if a valid conclusion can be reached from the two
true statements using the Law of Detachment or the Law of
Syllogism. If a valid conclusion is possible, state it and the
law that is used. If a valid conclusion does not follow, write no
valid conclusion.
Yes, statement (3) follows from statements (1) and
(2) by the Law of Detachment.
(1) If you break an item in a store, you must pay for it.
(2) Jill broke a vase in Potter’s Gift Shop.
(3) Jill must pay for the vase.
Example: Determine if 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.
Law of Syllogism
If p → q and q → r are true
conditionals, then p → r is also true.
Law of Detachment
Two important laws used frequently in deductive reasoning
are the Law of Detachment and the Law of Syllogism. In
both cases you reach conclusions based on if-then statements.
PERIOD
Deductive Reasoning
DATE
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.
Enrichment
NAME
Counterexamples
15–1
Answers
(Lessons 15-1 and 15-2)
Geometry: Concepts and Applications
15–2
Practice
NAME
DATE
© Glencoe/McGraw-Hill
A5
© Glencoe/McGraw-Hill
635
Geometry: Concepts and Applications
If the temperature reaches 70°, then we will not go to the beach.
(2) If the swimming pool opens, then we will not go to the beach.
10. (1) If the temperature reaches 70°, then the swimming pool will open.
no valid conclusion
(2) All whole numbers are real numbers.
9. (1) All whole numbers are rational numbers.
no valid conclusion
(2) If the concert is postponed, then it will be held in the gym.
8. (1) If the concert is postponed, then I will be out of town.
If a polygon has three sides, then the sum of the measures of the
interior angles is 180.
(2) If a figure is a triangle, then the sum of the measures of the interior angles is 180.
7. (1) If a polygon has three sides, then the figure is a triangle.
If my dog does not bark all night, then he will wag his tail.
(2) If I give my dog a treat, then he will wag his tail.
6. (1) If my dog does not bark all night, I will give him a treat.
Use the Law of Syllogism to determine a conclusion that follows from statements
(1) and (2). If a valid conclusion does not follow, write no valid conclusion.
no valid conclusion
(2) We do not go on a picnic.
5. (1) If it rains, we will not go on a picnic.
Triangle ABC is isosceles.
is isosceles.
(2) Triangle ABC has two sides with lengths of equal measure.
4. (1) If the measures of the lengths of two sides of a triangle are equal, then the triangle
The sum of the measures of angles A and B is 90.
(2) Angle A and B are complementary.
3. (1) If two angles are complementary, then the sum of their measures is 90.
no valid conclusion
(2) I did not sell my skis.
2. (1) If I sell my skis, then I will not be able to go skiing.
The figure is a polygon.
(2) The figure is a triangle.
1. (1) If a figure is a triangle, then it is a polygon.
PERIOD
© Glencoe/McGraw-Hill
636
Geometry: Concepts and Applications
10. Look for a Pattern Tanya likes to burn candles. She has
found that, once a candle has burned, she can melt 3 candle
stubs, add a new wick, and have one more candle to burn. How
many total candles can she burn from a box of 15 candles? 22
yes; Syllogism
9. (1) If William is reading, then he is reading a magazine.
(2) If William is reading a magazine, then he is reading a magazine about
computers.
(3) If William is reading, then he is reading a magazine about computers.
8. (1) If Julie works after school, then she works in a department store.
(2) Julie works after school.
(3) Julie works in a department store. yes; Detachment
7. (1) If Pedro is taking history, then he will study about World War II.
(2) Pedro will study about World War II.
(3) Pedro is taking history. no valid conclusion
6. (1) If the measure of an angle is greater than 90, then it is obtuse.
(2) m T is greater than 90.
(3) T is obtuse. yes; Detachment
Determine if 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 no valid conclusion.
If Henry studies his
algebra, then he will get a good grade; Syllogism.
5. If Henry studies his algebra, then he passes the test.
If Henry passes the test, then he will get a good grade.
October 13 is a Tuesday; Detachment.
4. If October 12 is a Monday, then October 13 is a Tuesday.
October 12 is a Monday.
If Rachel
lives in Tampa, then Rachel lives in the United States; Syllogism.
3. If Rachel lives in Tampa, than Rachel lives in Florida.
If Rachel lives in Florida, then Rachel lives in the United States.
no valid conclusion
2. If Spot is a dog, then he has four legs.
Spot has four legs.
Jim is an American; Detachment.
1. If Jim is a Texan, then he is an American.
Jim is a Texan.
Determine if a valid conclusion can be reached from the two true
statements using the Law of Detachment or the Law of Syllogism.
If a valid conclusion is possible, state it and the law that is used. If
a valid conclusion does not follow, write no valid conclusion.
PERIOD
Use the Law of Detachment to determine a conclusion that follows from statements
(1) and (2). If a valid conclusion does not follow, write no valid conclusion.
DATE
Deductive Reasoning
Skills Practice
NAME
Deductive Reasoning
15–2
Answers
(Lesson 15-2)
Geometry: Concepts and Applications
© Glencoe/McGraw-Hill
PERIOD
A6
© Glencoe/McGraw-Hill
637
Geometry: Concepts and Applications
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.
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. Explain
this rule in a way that will help him to understand. Sample answer: The word
Helping You Remember
b. (1) If two lines that lie in the same plane do not intersect, they are parallel.
(2) Lines ᐉ and m lie in plane A and do not intersect.
(3) Lines ᐉ and m are parallel. yes; Law of Detachment
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
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 u e
a. s u
2. conjunction of s and u g
b. If s → t is true and s is true, then t is true.
3. negation of t h
c. s → u
4. disjunction of s and u a
d. s t
5. Law of Detachment b
e. u
6. inverse of u → t j
f. If s → t and t → u are true, then s → u is true.
7. inverse of s → u c
g. s u
8. conjunction of s and t d
h. t
9. Law of Syllogism f
i. t
10. negation of t i
j. u → t
Reading the Lesson
deductive reasoning (dee•DUK•tiv) the process of using facts, rules,
definitions, or properties in logical order to reach a conclusion
Law of Detachment a logic rule that states “if p → q is a true
conditional and p is true, then q is true”
Law of Syllogism (SIL•oh•jiz•um) a logic rule that states “if p → q
and q → r are true conditionals, then p → r is also true”
Key Terms
DATE
Reading to Learn Mathematics
NAME
Deductive Reasoning
15–2
Enrichment
NAME
(1) Boots is a cat.
(2) Boots is purring.
(3) A cat purrs if it is happy.
DATE
© Glencoe/McGraw-Hill
Answers will vary.
7. Write an example of faulty logic that
you have seen in an advertisement.
5. (1) If you buy a word processor, you
will be able to write letters faster.
(2) Tania bought a word processor.
Conclusion: Tania will be able to
write letters faster. valid
3. (1) If you use Clear Line long
distance service, you will have clear
reception.
(2) Anna has clear long distance
reception.
Conclusion: Anna uses Clear Line
long distance service. faulty
1. (1) If you buy Tuff Cote luggage, it
will survive airline travel.
(2) Justin buys Tuff Cote luggage.
Conclusion: Justin’s luggage will
survive airline travel. valid
Decide if each argument is valid or faulty.
PERIOD
638
faulty
Geometry: Concepts and Applications
6. (1) Great swimmers wear AquaLine
swimwear.
(2) Gina wears AquaLine swimwear.
Conclusion: Gina is a great swimmer.
4. (1) If you read the book Beautiful
Braids, you will be able to make
beautiful braids easily.
(2) Nancy read the book Beautiful
Braids.
Conclusion: Nancy can make
beautiful braids easily. valid
2. (1) If you buy Tuff Cote luggage, it
will survive airline travel.
(2) Justin’s luggage survived airline
travel.
Conclusion: Justin has Tuff Cote
luggage. faulty
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.
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.
Consider the statements at the right.
What conclusions can you make?
Valid and Faulty Arguments
15–2
Answers
(Lesson 15-2)
Geometry: Concepts and Applications
15–3
Skills Practice
NAME
© Glencoe/McGraw-Hill
WXYZ
W and X are supplementary.
X and Y are supplementary.
Y and Z are supplementary.
Z and W are supplementary.
A7
SP
TP
PSU PTR
TR
SU
DEF and RST are rt.
triangles. E and S are
ST
and
right angles. EF
ED
SR
Prove: DEF RST
2. Given:
© Glencoe/McGraw-Hill
639
Geometry: Concepts and Applications
1. We know that PSU PTR and SU
TR
. By the Reflexive Property of
Congruent Angles,P P. Then SUP TRP by AAS and
TP
by CPCTC.
SP
F
ST
, ED
S
R
, and E and S are right angles. Since
2. We know that E
all right angles are congruent, E S. Therefore, by SAS, DEF RST.
Prove:
1. Given:
Write a paragraph proof for each conjecture.
Z
Y
and W
Z
X
Y.
By the definition of a parallelogram, WX
X
and Z
Y
, WZ
and X
Y
are transversals; for
For parallels W
and X
Y
, WX
and Z
Y
are transversals. Thus, the
parallels WZ
consecutive interior angles on the same side of a transversal
are supplementary. Therefore, W and X, X and Y, Y
and Z, Z and W are supplementary.
Given:
Prove:
Example: Write a paragraph proof for the conjecture.
D
DATE
C
A
B
PERIOD
Y
S
X
Z
W
b
a
E
A
B
C
D
X
R
T
S
© Glencoe/McGraw-Hill
640
Geometry: Concepts and Applications
If RSTX is a rhombus, then RS RX and XT ST. RT RT by the
Reflexive Property, so RXT RST by SSS.
4. If RSTX is a rhombus, then RXT RST.
If ACDE is an isosceles trapezoid with bases A
C
and E
D
, then the legs
CD
. Also, an isosceles trapezoid has congruent
are congruent, so AE
ED
by the Reflexive Property,
base angles, so AED CDE. Now, ED
so AED CDE by SAS.
3. If ACDE is an isosceles trapezoid with bases
and E
D
, then AED CDE.
AC
If lines a and b are parallel, then SWX XYZ since they are alternate
interior angles. WXS YXZ since they are vertical angles. Then it is
X
Y
, so WXS YXZ by ASA.
given that WX
2. If lines a and b are parallel and WX
X
Y
,
then WXS YXZ.
If ABD is an isosceles triangle with base B
D
, then AD
A
B
. If C is the
D
, then CD
C
B
. AC
AC
by the Reflexive Property, so
midpoint of B
ACD ACB by SSS.
1. If ABD is an isosceles triangle with base BD
and C is the midpoint of B
D
, then ACD ACB.
Write a paragraph proof for each conjecture.
PERIOD
Paragraph Proofs
DATE
A proof is a logical argument in which each statement you make is
backed up by a reason that is accepted as true. In a paragraph
proof, you write your statements and reasons in paragraph form.
Study Guide
NAME
Paragraph Proofs
15–3
Answers
(Lesson 15-3)
Geometry: Concepts and Applications
Practice
DATE
Reading to Learn Mathematics
NAME
© Glencoe/McGraw-Hill
A8
© Glencoe/McGraw-Hill
3. If 3 4, then ABC is isosceles.
2. If E bisects B
D
and A
C
, then BA CD .
641
Geometry: Concepts and Applications
We know that 3 1 180 and
2 4 180 since they form linear
pairs. Since 3 4, we can write
3 1 180 and 2 3 180. So,
1 180 3 and 2 180 3.
Therefore, 1 2 by substitution.
This implies that AC BC. So, ABC
is isosceles by definition of isosceles.
Since E bisects B
D
and A
C
, we know
that BE ED and CE EA. We also
know that BEA CED since they
are vertical angles. Therefore,
BEA DEC by SAS. So, BAE DCE because corresponding parts
of congruent triangles are congruent.
So, line BA line CD since we have
alternate interior angles that are
congruent.
Since p q, we know that 1 2
since they are corresponding angles.
We also know that 2 3 180
since they form a linear pair. Therefore,
by substitution, 1 3 180. So, 1
and 3 are supplementary.
PERIOD
1
Y 2 Z
X
3
b
a
C
T
A
© Glencoe/McGraw-Hill
postulate, theorem
642
Geometry: Concepts and Applications
4. Some students like to use sayings like “My Dear Aunt Sally” to help them remember
a mathematical idea. My Dear Aunt Sally stands for multiplication, division, addition,
and subtraction for order of operations. Think of a saying to help you remember that
definitions, postulates, and theorems can be used to justify statements when you write
a proof. Sample answer: Down the Parallel Tracks for definition,
Sample answer: Plan: Since AT
is a tangent to the circle, it is
perpendicular to a radius at point T. Perpendicular segments form
90° angles. If a triangle has a 90° angle, then it is a right triangle.
T
is a tangent to C, then it is
Proof: By Theorem 14-4, if A
. By the definition of perpendicular
perpendicular to the radius TC
T
C
, then CTA is a right angle. By the definition of right
lines, if AT
triangle, if CTA is a right angle, then CAT is a right triangle.
Helping You Remember
3. Write a paragraph proof for the conjecture. First, write a
plan for the proof.
Given: C; A
T
is tangent to C at T.
Prove: CAT is a right triangle.
Sample answer: First, use parallel lines and corresponding angles to
show that 1 2. Then use the fact that XYZ is isosceles to show
that 2 3. Then use the Transitive Property of Congruence to
conclude that 1 3.
2. Use the diagram and the information. Write a plan for
proving the conjecture. You do not need to write the proof.
Given: a b; XY XZ
Prove: 1 3
1. Complete each sentence with one or two words to make a true statement.
a. In a proof, the given information comes from the hypothesis of the conditional.
b. A proof is a logical argument in which each statement
you make is backed up by a
reason.
c. A paragraph proof is written in paragraph form.
d. One
problem-solving strategy that you might use for writing a proof is work backward.
e. In mathematics, proofs are used to validate a conjecture.
Reading the Lesson
proof a logical argument used to validate a conjecture in which each
statement you make is backed up by a reason that is accepted
as true
paragraph proof a logical argument used to validate a conjecture in
paragraph form
Paragraph Proofs
15–3
Key Terms
PERIOD
1. If p q and p and q are cut by a transversal t,
then 1 and 3 are supplementary.
DATE
Write a paragraph proof for each conjecture.
Paragraph Proofs
15–3
NAME
Answers
(Lesson 15-3)
Geometry: Concepts and Applications
15–4
Study Guide
NAME
© Glencoe/McGraw-Hill
A9
X
X
X
X
X
X
X
X
Mario
X
X
X
Kenji
© Glencoe/McGraw-Hill
Mr. Guthrie-teacher,
Mrs. Hakoi-doctor,
Mr. Mirza-office manager,
Mrs. Riva-accountant
3. Mr. Guthrie, Mrs. Hakoi, Mr. Mirza,
and Mrs. Riva have jobs of doctor,
accountant, teacher, and office
manager. Mr. Mirza lives near the
doctor and the teacher. Mrs. Riva is
not the doctor or the office manager.
Mrs. Hakoi is not the accountant or
the office manager. Mr. Guthrie went
to lunch with the doctor. Mrs. Riva’s
son is a high school student and is
only seven years younger than his
algebra teacher. Which person has
each occupation?
X
Olivia
Nancy
Nancy-apple,
Olivia-banana,
Mario-orange,
Kenji-peach
Apple
Banana
Orange
Peach
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?
Solve each problem.
643
Geometry: Concepts and Applications
Yvette, poodle;
Lana, collie;
Boris, beagle;
Scott, terrier
4. Yvette, Lana, Boris, and Scott each
have a dog. The breeds are collie,
beagle, poodle, and terrier. Yvette and
Boris walked to the library with the
student who has a collie. Boris does
not have a poodle or terrier. Scott
does not have a collie. Yvette is in
math class with the student who has
a terrier. Which student has each
breed of dog?
Victor-flute,
Leon-viola,
Kasha-clarinet,
Sheri-trumpet
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?
aa
If a b, then b a.
If a b and b c, then a c.
If a b, then a c b c.
If a b, then a c b c.
If a b, then a c b c.
a
b
If a b and c 0, then .
c
c
If a b, then a may be replaced
by b in any equation or expression.
a(b c) ab ac
9
3x 2 x 8
2x 2 8
2x 6
x 3
© Glencoe/McGraw-Hill
a.
b.
c.
d.
PERIOD
a. Given
b. Multiplication Property ()
c. Division Property ()
Reasons
Geometry: Concepts and Applications
Given
Subtraction Property ()
Addition Property ()
Division Property ()
644
a.
b.
c.
d.
2. Prove that if 3x 2 x 8, then x 3.
Given: 3x 2 x 8
Prove: x 3
Proof:
Statements
Reasons
a.
b. 3x 45
c. x 15
3
x
5
1. Prove that if 3x 9, then x 15.
5
Given: 3x 9
5
Prove: x 15
Proof:
Statements
DATE
a. Given
b. Addition Property ()
c. Division Property ()
Name the property that justifies each statement.
a. 4x 8 8
b. 4x 0
c. x 0
Example: Prove that if 4x 8 8, then x 0.
Given: 4x 8 8
Prove: x 0
Proof:
Statements
Reasons
Distributive Property
Reflexive Property
Symmetric Property
Transitive Property
Addition Property
Subtraction Property
Multiplication Property
Division Property
Substitution Property
Properties of Equality for Real Numbers
Many rules from algebra are used in geometry.
PERIOD
Preparing for Two-Column Proofs
DATE
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.
Enrichment
NAME
Logic Problems
15–3
Answers
(Lessons 15-3 and 15-4)
Geometry: Concepts and Applications
1 3
2 4
D
15–4
Practice
NAME
DATE
© Glencoe/McGraw-Hill
A10
14
B
C
O
d. Subtraction Property of Equality
e. Substitution Property of Equality
f. Substitution Property of Equality
g. Division Property of Equality
d. mA mB 90
e. mA mA 90
f. 2mA 90
g. mA 45
Geometry: Concepts and Applications
c. Definition of Right Angle
645
b. Angle Sum Theorem
B
c. mC 90
a. Given
C
A
10
6
© Glencoe/McGraw-Hill
d. x 4
c.
b. 5x 4 6
2
5
x
2
a. 3x 4 1
x
2
Statements
Given: 3x 4 1x 6
2
Prove: x 4
Proof:
646
Geometry: Concepts and Applications
d. Multiplication Property ()
c. Addition Property ()
b. Subtraction Property ()
a. Given
Reasons
d. Division Property ()
d. x 7
10. Prove that if 3x 4 1x 6, then x 4.
c. Addition Property ()
c. 2x 14
b. Distributive Property
b. 2x 6 8
a. Given
b. Multiplication Property of Equality
c. Division Property of Equality
a. Given
a. 2(x 3) 8
Reasons
2
9. Prove that if 2(x 3) 8, then x 7.
Given: 2(x 3) 8
Prove: x 7
Proof:
Reasons
a. ABC is a right triangle with C a
right angle and mA mB
b. mA mB mC 180
© Glencoe/McGraw-Hill
Transitive
8. If x y 3 and y 3 10, then
x 10.
Distributive
6. 2(x 4) 2x 8
Division
4. If 7x 42, then x 6.
Subtraction
PERIOD
2. If x 3 17, then x 14.
Complete each proof by naming the property that justifies
each statement.
Substitution
7. If mA mB 90, and mA 30,
then 30 mB 90.
Addition
5. If XY YZ XM, then XM YZ XY.
Reflective
3. xy xy
Symmetric
1. If mA mB, then mB mA.
Statements
c. Addition Property of Equality
d. Substitution Property of Equality
a. Given
b. Angle Addition Postulate
Reasons
3. If ABC is a right triangle with C a right
angle and mA mB, then mA 45.
Given: ABC is a right triangle with C a
right angle and mA mB.
Prove: mA 45
Proof:
Statements
Reasons
b. 7x 84
c. x 12
a.
7x
6
Prove: x 12
Proof:
Statements
6
7x
Given: 14
6
7x
2. If 14, then x 12.
d. mABC mROD
a. m1 m2, m3 m4
b. mABC m1 m3
mROD m2 m4
c. m1 m3 m2 m4
Given: m1 m2, m3 m4
Prove: mABC mROD
Proof:
Statements
1. If m1 m2 and m3 m4, then
mABC mROD.
Name the property or equality that justifies each statement.
R
PERIOD
Complete each proof.
A
DATE
Preparing for Two-Column Proofs
Skills Practice
NAME
Preparing for Two-Column Proofs
15–4
Answers
(Lesson 14-4)
Geometry: Concepts and Applications
PERIOD
© Glencoe/McGraw-Hill
true
A11
c. Theorem 4-1 Alternate Interior Angles
d. Transitive Property of Congruence
e. Definition of Congruent Angles
f. Exterior Angle Theorem
g. Substitution Property of Equality
c. 4 5
d. 2 5
e. m2 m5
f. m5 m7 m8
g. m2 m7 m8
© Glencoe/McGraw-Hill
647
Geometry: Concepts and Applications
figure, the given information, a statement about what to prove, and a
justification for each statement. Difference: A paragraph proof is
written in paragraph form, while a two-column proof is written in two
columns where one column has the statements and the second
column has the reasons.
3. A good way to remember some terms is to compare them. Write several sentences
comparing the similarities and differences between paragraph proofs and two-column
proofs. Sample answer: Similarities: Both types of proofs contain a
Helping You Remember
b. Postulate 4-1 Corresponding Angles
b. 2 4
b
a
a. Given
8
7
Reasons
5
6
a. a b, c d
3
4
1
2
d
Proof:
Statements
2. Fill in the missing statements and reasons in the two-column proof.
Given: a b, c d
Prove: m2 m7 m8
c
False; the last
d. The last statement in a two-column proof is the given information.
statement is what you want to prove.
true
False; you are
c. In a two-column proof, you must give a reason for each statement.
using deductive reasoning.
b. When you solve an equation, you are using inductive reasoning.
a. Algebraic properties can be used as reasons in proofs.
1. State whether each statement is true or false. If the statement is false, explain why.
Reading the Lesson
two-column proof a deductive argument that contains statements and reasons
organized in two columns
Key Terms
DATE
Reading to Learn Mathematics
NAME
Preparing for Two-Column Proofs
15–4
Enrichment
DATE
PERIOD
2
62 62 64
36 36 1296
72 1296
6. a2 a2 a4
6 (4 2) (6 4) (6 2)
6 6 1.5 3
1 4.5
4. a (b c) (a b) (a c)
6
2
1.5
2
3 0.75
6 (4 2) (6 4) 2
2. a (b c) (a b) c
© Glencoe/McGraw-Hill
648
Geometry: Concepts and Applications
4. Division does not distribute over addition.
5. Addition does not distribute over multiplication.
8. For the Distributive Property a(b c) ab ac it is said
that multiplication distributes over addition. Exercises 4 and
5 prove that some operations do not distribute. Write a
statement for each exercise that indicates this.
1. Subtraction is not an associative operation.
2. Division is not an associative operation.
3. Division is not a commutative operation.
7. Write the verbal equivalents for Exercises 1, 2, and 3.
6 (4 2) (6 4)(6 2)
6 8 (10)(8)
14 80
3
5. a (bc) (a b)(a c)
3
2
6446
3. a b b a
6 (4 2) (6 4) 2
6222
40
1. a (b c) (a b) c
In each of the following exercises a, b, and c are any numbers.
Prove that the statement is false by counterexample. Sample answers are given.
In general, for any numbers a and b, the statement a b b a
is false. You can make the equivalent verbal statement:
subtraction is not a commutative operation.
7337
4 4
Let a 7 and b 3. Substitute these values in the equation
above.
You can prove that this statement is false in general if you can
find one example for which the statement is false.
For any numbers a and b, a b b a.
Some statements in mathematics can be proven false by
counterexamples. Consider the following statement.
More Counterexamples
15–4
NAME
Answers
(Lesson 14-4)
Geometry: Concepts and Applications
Study Guide
NAME
DATE
© Glencoe/McGraw-Hill
Statements
 
AD CE
A12
8. Given
7
9. HL
649
8. 1 2
CE
9. AD
© Glencoe/McGraw-Hill
6
3
Geometry: Concepts and Applications
7. In a plane, if two lines are cut
by a transversal so that a pair of
corresponding angles is congruent,
then the lines are parallel.
(Postulate 4-2) 9
7. ABD CDE
8
5
6. CPCTC
5. Given
6. BD
DE
CE
5. AD
4. Definition of Perpendicular Lines
4. ABD and CDE are right
triangles.
2
3. Given
3. 3 and 4 are right angles.
4
2. Given
1
1. Definition of Right Triangle
Reasons
PERIOD
2. AB
⊥ BE
⊥ BE
1. CD
Proof:
Prove:
Given:
 
CD ⊥ BE
 
AB ⊥ BE
 
AD CE
 
BD DE
The reasons necessary to complete the following proof are
scrambled up below. To complete the proof, number the reasons
to match the corresponding statements.
Two-Column Proofs
15–5
Skills Practice
NAME
Geometry: Concepts and Applications
g. CPCTC
g. A
C
BD
650
f. SAS
f. ∆ADC ∆BCD
© Glencoe/McGraw-Hill
e. Reflexive Property
e. D
C
DC
d. ADC BCD
C
d. Definition of congruent angles
c. Definition of a square
c. ADC and BCD are right
angles.
b. Definition of a square
b. A
D
BC
D
a. Given
Reasons
a. ABCD is a square.
Statements Sample Answer:
2. Given: ABCD is a square.
Prove: AC
B
D
Proof:
B
f. SAS
f. ITS CTS
A
e. Reflexive Property
e. S
T
S
T
g. CPCTC
d. Definition of angle bisector
d. ITS CTS
g. IS
C
S
c. Definition of isosceles triangle
C
c. IT
C
T
T
S
PERIOD
b. Given
a. Given
I
DATE
b. TS bisects ITC
.
with base IC
a. ITC is an isosceles triangle
1. Given: ITC is an isosceles triangle with base
IC
, TS bisects ITC
C
S
Prove: IS
Proof:
Statements Sample Answer:
Reasons
Write a two-column proof.
Two-Column Proofs
15–5
Answers
(Lesson 15-5)
Geometry: Concepts and Applications
Practice
NAME
© Glencoe/McGraw-Hill

B is the midpoint of AC.
AB BC
BC CD BD
AB CD BD
A13
Geometry: Concepts and Applications
f. Definition of congruent angles
f. AEB DEC
© Glencoe/McGraw-Hill
e. Subtraction Property ()
c. Angle Addition Postulate
c. mAEC mAEB mBEC,
mDEB mDEC mBEC
e. mAEB mDEC
b. Definition of congruent angles
b. mAEC mDEB
d. Transitive Property ()
a. Given
a. AEC DEB
d. mAEB mBEC mDEC mBEC
Reasons
651
PERIOD
Given
Definition of midpoint
Segment Addition Postulate
Substitution Property ()
Reasons
a.
b.
c.
d.
DATE
Statements
2. Given: AEC DEB
Prove: AEB DEC
Proof:
a.
b.
c.
d.
Statements

1. Given: B is the midpoint of AC.
Prove: AB CD BD
Proof:
Write a two-column proof.
Two-Column Proofs
15–5
DATE
Reading to Learn Mathematics
NAME
PERIOD
U
R
U
T
R
S
T
S
U
SU
S
RUS TUS
RUS TUS
a.
e.
d.
c.
b.
Given
Given
Reflexive Prop. of SSS Postulate
CPCTC
Reasons
U
S
T
R
© Glencoe/McGraw-Hill
652
Geometry: Concepts and Applications
the Given and Prove and draw a diagram for the situation. Look at
the given information and mark the diagram with that information.
Look at what you are to prove and make a plan for using the given
information to reach that conclusion. You can use the work backward
strategy as well. Then write each statement and its reason in a logical
order to arrive at the conclusion.
3. Sometimes it is helpful to summarize information that you need to remember. Summarize
the steps you would take to write a two-column proof. Sample answer: First, write
Helping You Remember
e.
d.
c.
b.
a.
Statements
Proof:
Plan: Sample answer: Show that the two
triangles are congruent and then use
CPCTC.
Given: UR
UT
, R
S
TS
Prove: RUS TUS
2. Write the statements and reasons for a two-column proof for each set of information. First,
write a short plan for your proof.
d. The given information is never used in a two-column proof.
False; the given
information is always used in a two-column proof.
c. If the figure you are given to work with for a proof has overlapping triangles, you can
redraw the triangles as separate triangles. true
False; definitions are one of the three things that can be used for a
reason.
b. You cannot use definitions of geometric terms as a reason for a statement in a proof.
False; the given information is found in the hypothesis.
a. The given information for a proof can be found in the conclusion of the conjecture.
1. Determine whether each statement is true or false. If the statement is false, explain why.
Reading the Lesson
Two-Column Proofs
15–5
Answers
(Lesson 15-5)
Geometry: Concepts and Applications
b
c
c
c
c
a
b
area of the 4 triangles
a
b
a
A14
a
b
2
a
b
2
a
b
2
a
b
1
2
b
b
a
a
a
area of the center square
c2
c2 2 a b
c2
c2
c2
DATE
b
a
area of large square
PERIOD
a2 b2 2 a b
a2 b2
a2 b2
9
B
12
15
C
© Glencoe/McGraw-Hill
A: 81,
B: 144,
C: 225; true
Squares
A
1.
3
4
B
2 3
5
2.5
3
653
A: 2.253
,
B: 43
,
C: 6.253
; true
Equilateral
Triangles
A
1.5
3
2.
C
Semicircles
B
4
5
Geometry: Concepts and Applications
A: 1.125␲,
B: 2.000␲,
C: 3.125␲; true
A 3
3.
C
a2 b2 a b a b
Use the Pythagorean Theorem to find the area of A, B, and C in each of the
following. Then, answer true or false for the statement A B C.
a2 a b b2 a b
b
b
Think of the figure as a
large square.
The relationship c2 a2 b2 is true for all right triangles.
4







a
Think of the figure as four
triangles and a square.
Use the Pythagorean Theorem to find the area of the shaded
region in the figure at the right.
Pythagorean Theorem
Enrichment
NAME







© Glencoe/McGraw-Hill






15–5
Study Guide
NAME
DATE
Use the origin as a vertex or center.
Place at least one side of a polygon on an axis.
Keep the figure within the first quadrant if possible.
Use coordinates that make computations simple.
H (0, 0)
K(0, c)
J(a, b c)
© Glencoe/McGraw-Hill
654
Geometry: Concepts and Applications
So AB CD and AC BD. Therefore, the opposite sides of a
parallelogram are congruent.
2
)
02
0
(
)
0
a
2 a
AB (a
2
2
CD ((
a
)
b
)
b
c
(
)
c a
2 a
2
a
)
b
)
a2
c
(
)
0
b
2
c2
BD ((
2
AC (b
)
02
c
(
)
0
b
2
c2
D(a b, c). Then use the Distance Formula to
find AB, CD, AC, and BD.
D(a, c)
A(a, 0)
2. HIJK is a parallelogram.
PERIOD
3. Use a coordinate proof to show that the opposite sides of any
parallelogram are congruent. Label the vertices A(0, 0), B(a, 0), C(b, c), and
1. ABCD is a rectangle.
Name the missing coordinates in terms of the given variables.
Since OB AC, the diagonals are congruent.
2
2
)
02
b
(
)
0
a
b2
OB (a
2 a
2 AC (0
)
a2
b
(
)
0
b2
Use the Distance Formula to find OB and AC.
Use (0, 0) as one vertex. Place another vertex on
the x-axis at (a, 0) and another on the y-axis at
(0, b). The fourth vertex must then be (a, b).
Example: Use a coordinate proof to prove that the diagonals
of a rectangle are congruent.
The Distance Formula, Midpoint Formula, and Slope Formula are
useful tools for coordinate proofs.
•
•
•
•
You can place figures in the coordinate plane and use algebra to
prove theorems. The following guidelines for positioning figures
can help keep the algebra simple.
Coordinate Proofs
15–6
Answers
(Lessons 15-5 and 15-6)
Geometry: Concepts and Applications
15–6
Practice
NAME
DATE
© Glencoe/McGraw-Hill
B (m, 0) x
A (0, 0)
O
x
Y (p, 0)
A15
R (c a, d)
O
x
B (12 d, 0)
© Glencoe/McGraw-Hill
x
B ( x2, 0)
A ( x2, 0)
O
C ( x2 , y)
D ( x2, y)
y
655
5. a rectangle with length x units and width y units
A ( 12d, 0)
C (0, s)
y
Geometry: Concepts and Applications
4. an isosceles triangle with base d units long and heights s units long
O P (0, 0) Q (c, 0) x
S (a, d)
y
3. a rhombus with sides c units long
O X (0, 0)
Z (0, r )
y
2. a right triangle with legs p and r units long
C (m, m)
D (0, m)
y
D(0, 0), F(a c, b)
4. DEFG is a parallelogram.
656
Sample proof:
 a
b2 0
b2
2
a
2
slope of AC ab0
ab
2
2
 a
b
b2
0
a2
slope of BD ba
ba
b2 a
b2 a2 b2 1
a2
2
ab
ba
b2 a2
© Glencoe/McGraw-Hill
PERIOD
M(0, 0), R(a b, a
2
b2)
2. MART is a rhombus.
Geometry: Concepts and Applications
5. Use a coordinate proof to prove that the diagonals of a
rhombus are perpendicular. Draw the diagram at the right.
R(0, 0) C(a, b)
3. RECT is a rectangle.
X(0, 0) Y(a, 0)
1. XYZ is a right isosceles triangle.
Name the missing coordinates in terms of the given variables.
PERIOD
Coordinate Proofs
DATE
Position and label each figure on a coordinate plane. 1–5. Sample answers given.
1. a square with sides m units long
Skills Practice
NAME
Coordinate Proofs
15–6
Answers
(Lesson 15-6)
Geometry: Concepts and Applications
© Glencoe/McGraw-Hill
PERIOD
A16
x
T(a, ?)
U(a, ?) x
T(b, ?)
R is (–a, 0) since it is a units in the
negative direction horizontally and
lies on the x–axis; S is (–b, c)
since it is b units in the negative
direction horizontally; T is (b, c)
since it is c units in a positive
vertical direction; U is (a, 0) since
it is on the x–axis.
R(?, ?) O
S(?, c)
y
b. isosceles trapezoid
© Glencoe/McGraw-Hill
657
Geometry: Concepts and Applications
the origin. The everyday meaning of origin is place where something
begins. So look to see if there is a good way to begin by placing a
vertex of the figure at the origin.
3. What is an easy way to remember how best to draw a diagram that will help you devise a
coordinate proof? Sample answer: A key point in the coordinate plane is
Helping You Remember
R is (0, b) because the point
is on the y–axis; S is (0, 0)
because the point is the
b
origin; T is a, because it
2
is half way between R and S
in vertical distance.
S(?, ?)
R(?, b)
y
a. isosceles triangle
2. Find the missing coordinates in each figure. Explain how you find the coordinates.
1. Complete each sentence with one or two words to make a true statement.
a. If you are writing a coordinate proof and need to show that two segments are congruent,
Distance Formula
a formula you may want to use is the ______________________.
b. When drawing a diagram for a coordinate proof, try to place a vertex of the figure at the
origin
________.
c. If you are writing a coordinate proof and want to show that two segments are parallel, a
Slope Formula
formula you may want to use is the ___________________.
d. When drawing a diagram for a coordinate proof, try to place at least on side of the
axis
polygon on a(n) ______.
e. If you are writing a coordinate proof and want to show that a segment has been bisected,
Midpoint Formula
a formula you may want to use is the _______________________.
first
f. When drawing a diagram for a coordinate proof, try to keep the figure in the ______
quadrant.
Reading the Lesson
coordinate proof a proof that uses figures on a coordinate plane
Key Terms
DATE
Reading to Learn Mathematics
NAME
Coordinate Proofs
15–6
Enrichment
NAME
DATE
ar
a r
2
2
© Glencoe/McGraw-Hill
658
(PQ)2 (AQ) (QB)
(b)2 (a r) (r a)
b 2 (r a) (r a)
b2 r2 a2
Therefore a 2 b 2 r 2, which means
that (a, b) is on the circle with the
equation x 2 y 2 r 2. This is the
circle that has A
B
as a diameter.
a r
2. Suppose PQ
B, Q is between A and B,
A
and PQ
is the geometric mean between A
Q
and Q
B
. Prove that P is on the circle that
has A
B
as a diameter. Use the figure at the
right.
a r
2
b
r a
2 2 1.
b 2 r 2 a 2, and 2
2
a 2 b 2 r 2, since (a, b) is on the
graph of x 2 y 2 r 2. Therefore
ar
b
2
b b 2
2
(slope of AP
PB
) (slope of )
slope of PB
a ( r)
ar
b0
b
ar
ar
b0
slope of AP
b
1. Prove that an angle inscribed in a
semicircle is a right angle. Use the figure at
right. (Hint: Write an equation for the
circle. Use your equation to help show that
P) (slope of PB
(slope of A
) 1).
A (r, O)
A(r, O)
O
O
y
y
Q (a, O)
P (a, b)
P (a, b)
x
B (r, O)
x
B (r, O)
PERIOD
Geometry: Concepts and Applications
You can prove many theorems about circles by using coordinate
geometry. Whenever possible locate the circle so that its center
is at the origin.
Coordinate Proofs with Circles
15–6
Answers
(Lesson 15-6)
Geometry: Concepts and Applications
Chapter 15 Answer Key
Form 1A
Page 659
Page 660
1.
Form 1B
Page 661
Page 662
D
8.
9.
2.
A
3.
C
4.
D
10.
11.
1.
D
B
C
A
8.
A
9.
C
B
D
D
10.
2.
A
3.
B
4.
C
11.
12.
5.
5.
12.
C
13.
6.
16.
B
D
C
D
Bonus
A
7.
B
13.
B
A
B
14.
6.
B
7.
C
© Glencoe/McGraw-Hill
15.
16.
C
C
D
Bonus
D
14.
A17
15.
A
C
Geometry: Concepts and Applications
Chapter 15 Answer Key
Form 2A
Page 663
Page 664
A and B are not
complementary
angles.
1.
2.
p ∨ q
3.
false
14.
Substitution
15.
Subtraction
16.
17.
true
4.
18.
5.
no valid
conclusion
6.
The sum of the
measures of 1
and 2 is 90.
The ratio of the
volumes of two
cubes is equal to
the cube of the ratio
of their heights.
7.
9.
10.
20.
Transitive or
Substitution, 21.
Subtraction, 12.
SAS
13.
CPCTC
© Glencoe/McGraw-Hill
C(b, c)
Sample answer:
PQ RS a, QR 25.
BEA
AA Similarity
Sample answer:
P(0, 0), Q(a, 0),
R(a b, c), S(b, c)
24.
The congruence of
angles is reflexive.
11.
The acute angles in
a right triangle are
complementary.
19.
23.
The sides opposite
congruent angles
are congruent.
Definition of
rectangle
Angle Addition
Postulate
22.
B
A
8.
Symmetric
Bonus
A18
b2 c2
PS ABD CAD
Geometry: Concepts and Applications
Chapter 15 Answer Key
Form 2B
Page 665
Page 666
The perimeter of
rectangle ABCD is
not 30 inches.
1.
14.
Reflexive
15.
16.
Subtraction
Transitive or
Substitution
Definition of
rectangle
p ∧ r
2.
3.
true
17.
4.
false
All right angles are
congruent.
18.
Opposite sides of
a parallelogram
are congruent.
19.
5.
ABC has two
acute angles.
20.
6.
no valid
conclusion
7.
The measure of
a diameter of
B is 2 AB.
21.
When two parallel lines
are cut by a transversal,
corresponding angles
are congruent.
22.
AAS
23.
D(a, 0)
8.
C
M
9.
Definition of
midpoint
24.
10.
The congruence
of segments is
reflexive.
25.
11.
CBM
12.
SSS
13.
CPCTC
© Glencoe/McGraw-Hill
When two parallel lines
are cut by a transversal,
alternate interior angles
are congruent.
Sample answer:
X(a, 0), Y(0, b),
Z(a, 0)
Sample answer:
XY YZ a2 b2
ABD and ACD
are isosceles
triangles.
Bonus
A19
Geometry: Concepts and Applications
Chapter 15 Assessment Answer Key
Page 667, Extended Response Assessment
Scoring Rubric
Score
General Description
Specific Criteria
4
Superior
A correct solution that
is supported by welldeveloped, accurate
explanations
•
Satisfactory
A generally correct solution,
but may contain minor flaws
in reasoning or computation
•
Nearly Satisfactory
A partially correct
interpretation and/or
solution to the problem
•
Nearly Unsatisfactory
A correct solution with no
supporting evidence or
explanation
•
•
Unsatisfactory
An incorrect solution
indicating no mathematical
understanding of the
concept or task, or no
solution is given
•
3
2
1
0
© Glencoe/McGraw-Hill
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Shows thorough understanding of the concepts of paragraph
proofs, and two-column proofs.
Uses appropriate strategies to solve problems.
Computations are correct.
Written explanations are exemplary.
Graphs are accurate and appropriate.
Goes beyond requirements of some or all problems.
Shows an understanding of the concepts of paragraph proofs, and
two-column proofs.
Uses appropriate strategies to solve problems.
Computations are mostly correct.
Written explanations are effective.
Graphs are mostly accurate and appropriate.
Satisfies all requirements of problems.
Shows an understanding of most of the concepts of paragraph
proofs, and two-column proofs.
May not use appropriate strategies to solve problems.
Computations are mostly correct.
Written explanations are satisfactory.
Graphs are mostly accurate.
Satisfies the requirements of most of the problems.
Final computation is correct.
No written explanations or work is shown to substantiate
the final computation.
Graphs may be accurate but lack detail or explanation.
Satisfies minimal requirements of some of the problems.
Shows little or no understanding of the concepts of paragraph
proofs, and two-column proofs.
Does not use appropriate strategies to solve problems.
Computations are incorrect.
Written explanations are unsatisfactory.
Graphs are inaccurate or inappropriate.
Does not satisfy requirements of problems.
No answer may be given.
A20
Geometry: Concepts and Applications
Chapter 15 Answer Key
Extended Response Assessment
Sample Answers
Page 667
⊥ B
the definition of perpendicular, AX
C
.
is the perpendicular
Therefore, AX
bisector of B
C
.
1. Given: Triangle ABC is isosceles with
bisects BAC.
right angle at A; AX
is the perpendicular bisector
Prove: AX
of B
C
.
2. a. The sum of the measures of the central
angles of a circle is 360.
b. All radii of a circle have equal
measures.
c. If two sides of a triangle are
congruent, then the angles opposite
those sides are congruent.
d. Angle Sum Theorem
e. Substitution Property of Equality
f. Subtraction Property of Equality
g. Substitution Property of Equality
h. Subtraction Property of Equality
i. Addition Property of Equality
j. Distributive Property
k. Angle Addition Postulate
l. Substitution Property of Equality
m. Definition of arc measure
n. Substitution Property of Equality
o. Division Property of Equality
B
X
A
C
bisects BAC, so
You know that AX
CAX BAX by the definition of angle
bisector. Since ABC is isosceles,
AC
and ACB ABC. Then,
AB
ACX ABX by ASA. By CPCTC,
is a bisector of the
C
X
XB
. So AX
hypotenuse, B
C
, by the definition of
bisector. Also by CPCTC, CXA BXA.
CXA and BXA are supplementary,
because they form a linear pair and linear
pairs of angles are supplementary. Since
CXA and BXA are congruent and
supplementary, their measures are 90. By
definition, they are right angles. So, by
3. Given: rhombus ABCD; W, X, Y, and Z are the midpoints of
A
B
, B
C
, C
D, and A
D, respectively.
Prove: Quadrilateral WXYZ is a parallelogram.
Statements
Reasons
1. Quadrilateral ABCD is a rhombus.
2. AB BC CD AD
3. W, X, Y, and Z are the midpoints of
A
B
, B
C
, C
D, and A
D, respectively.
1. Given
2. Definition of rhombus
3. Given
1
1
4. AW BW AB; BX CX BC;
4. Definition of midpoint
CY DY 2
1
CD;
2
DZ AZ 2
1
AD
2
5. AW CY and AZ CX; BW DY
and BX DZ
6. A C; B D
7. AWZ CYX; BWX DYZ
8. WZ
Y
X; ZY
X
W
9. WXYZ is a parallelogram.
© Glencoe/McGraw-Hill
5. Substitution Property of Equality
6. In a parallelogram, opposite angles are
congruent.
7. SAS
8. CPCTC
9. If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
A21
Geometry: Concepts and Applications
Chapter 15 Answer Key
Mid-Chapter Test
Page 668
Quiz A
Page 669
1. x 100 or x 120
1.
false
any whole number
whose
ones digit is 5
2.
2.
false
3.
true
3.
true
4.
p → q, r → p
5.
If both pairs of
opposite sides in a
quadrilateral are
congruent, then they
are parallel.
6.
ABCDEF UVWXYZ
7.
no valid
conclusion
8.
inductive
9.
If a quadrilateral is a
square, then it has
4. four right angles.
5.
no valid
conclusion
Quiz B
Page 669
1.
SSS
2.
CDB
deductive
Definition of angle
bisector
3.
EC
10.
C
Vertical angles are
congruent.
11.
12.
DEC
13.
SAS
14.
CPCTC
© Glencoe/McGraw-Hill
4.
A22
Sample
answer: A
Y
X
Z
5.
Substitution
6.
C(a c, b)
B
Geometry: Concepts and Applications
Chapter 15 Answer Key
Cumulative Review
Page 670
1.
Standardized Test Practice
Page 671
Page 672
If x and y are both
odd numbers, then
x y is even.
2.
70
3.
2
3
4.
A
C
DC
5.
isosceles;
vertex; base
6.
30
7.
50
8.
AA Similarity; 9
9.
140
10.
chord
11.
65
12.
565.5 ft2
1.
C
2.
A
3.
D
4.
5.
11.
C
12.
B
13.
A
14.
D
15.
A
16.
B
17.
C
18.
B
19.
D
20.
C
B
B
6.
C
7.
D
8.
C
9.
A
10.
C
Quadrilateral CDEF
is a rectangle.
13.
14.
E
C
DF
© Glencoe/McGraw-Hill
A23
Geometry: Concepts and Applications