Geometry
Chapter 1
Resource Masters
Consumable Workbooks
Many of the worksheets contained in the Chapter Resource Masters booklets
are available as consumable workbooks.
Study Guide and Intervention Workbook
Skills Practice Workbook
Practice Workbook
Reading to Learn Mathematics Workbook
0-07-860191-6
0-07-860192-4
0-07-860193-2
0-07-861061-3
ANSWERS FOR WORKBOOKS The answers for Chapter 1 of these workbooks
can be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.
Printed in the United States of America. Permission is granted to reproduce the
material contained herein on the condition that such material be reproduced only
for classroom use; be provided to students, teachers, and families without charge;
and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction,
for use or sale, is prohibited without prior written permission of the publisher.
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ISBN: 0-07-846589-3
1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03
Geometry
Chapter 1 Resource Masters
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 1-6
Study Guide and Intervention . . . . . . . . . 31–32
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 33
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Reading to Learn Mathematics . . . . . . . . . . . 35
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Lesson 1-1
Study Guide and Intervention . . . . . . . . . . . 1–2
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . . 3
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Reading to Learn Mathematics . . . . . . . . . . . . 5
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Chapter 1 Assessment
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Lesson 1-2
Study Guide and Intervention . . . . . . . . . . . 7–8
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . . 9
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Reading to Learn Mathematics . . . . . . . . . . . 11
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Lesson 1-3
Study Guide and Intervention . . . . . . . . . 13–14
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 15
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Reading to Learn Mathematics . . . . . . . . . . . 17
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1 Test, Form 1 . . . . . . . . . . . . . . 37–38
1 Test, Form 2A . . . . . . . . . . . . . 39–40
1 Test, Form 2B . . . . . . . . . . . . . 41–42
1 Test, Form 2C . . . . . . . . . . . . . 43–44
1 Test, Form 2D . . . . . . . . . . . . . 45–46
1 Test, Form 3 . . . . . . . . . . . . . . 47–48
1 Open-Ended Assessment . . . . . . . 49
1 Vocabulary Test/Review . . . . . . . . 50
1 Quizzes 1 & 2 . . . . . . . . . . . . . . . . 51
1 Quizzes 3 & 4 . . . . . . . . . . . . . . . . 52
1 Mid-Chapter Test . . . . . . . . . . . . . 53
1 Cumulative Review . . . . . . . . . . . . 54
1 Standardized Test Practice . . . 55–56
Standardized Test Practice
Student Recording Sheet . . . . . . . . . . . . . . A1
Lesson 1-4
Study Guide and Intervention . . . . . . . . . 19–20
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 21
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Reading to Learn Mathematics . . . . . . . . . . . 23
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 24
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A29
Lesson 1-5
Study Guide and Intervention . . . . . . . . . 25–26
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 27
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Reading to Learn Mathematics . . . . . . . . . . . 29
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 30
©
Glencoe/McGraw-Hill
iii
Glencoe Geometry
Teacher’s Guide to Using the
Chapter 1 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resources
you use most often. The Chapter 1 Resource Masters includes the core materials needed
for Chapter 1. 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 TeacherWorks CD-ROM.
Vocabulary Builder
Practice
Pages vii–viii
include a student study tool that presents
up to twenty of 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.
There is one master for each
lesson. These problems more closely follow
the structure of the Practice and Apply
section of the Student Edition exercises.
These exercises are of average difficulty.
WHEN TO USE These provide additional
practice options or may be used as
homework for second day teaching of the
lesson.
WHEN TO USE Give these pages to
students before beginning Lesson 1-1.
Encourage them to add these pages to their
Geometry 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 asks questions
about the opening paragraph of the lesson
in the Student Edition. Additional
questions 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 and Intervention
Each lesson in Geometry addresses two
objectives. There is one Study Guide and
Intervention master for each objective.
WHEN TO USE Use these masters as
WHEN TO USE This master can be used
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 students
who have been absent.
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 Learner)
students.
Skills Practice
There is one master for
each lesson. These provide computational
practice at a basic level.
Enrichment
There is one extension
master for each lesson. These activities may
extend the concepts in the lesson, offer an
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.
WHEN TO USE These masters can be
used with students who have weaker
mathematics backgrounds or need
additional reinforcement.
WHEN TO USE These may be used as
extra credit, short-term projects, or as
activities for days when class periods are
shortened.
©
Glencoe/McGraw-Hill
iv
Glencoe Geometry
Assessment Options
Intermediate Assessment
The assessment masters in the Chapter 1
Resources Masters offer a wide range of
assessment tools for intermediate and final
assessment. The following lists describe each
assessment master and its intended use.
• Four free-response quizzes are included
to offer assessment at appropriate
intervals in the chapter.
• A Mid-Chapter Test provides an option
to assess the first half of the chapter. It is
composed of both multiple-choice and
free-response questions.
Chapter Assessment
CHAPTER TESTS
Continuing Assessment
• Form 1 contains multiple-choice questions
and is intended for use with basic level
students.
• 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. This master includes
free-response questions.
• Forms 2A and 2B contain multiple-choice
questions aimed at the average level
student. These tests are similar in format
to offer comparable testing situations.
• The Standardized Test Practice offers
continuing review of geometry concepts
in various formats, which may appear on
the standardized tests that they may
encounter. This practice includes multiplechoice, grid-in, and short-response
questions. Bubble-in and grid-in answer
sections are provided on the master.
• Forms 2C and 2D are composed of freeresponse questions aimed at the average
level student. These tests are similar in
format to offer comparable testing
situations. Grids with axes are provided
for questions assessing graphing skills.
• Form 3 is an advanced level test with
free-response questions. Grids without
axes are provided for questions assessing
graphing skills.
Answers
• Page A1 is an answer sheet for the
Standardized Test Practice questions
that appear in the Student Edition on
pages 58–59. This improves students’
familiarity with the answer formats they
may encounter in test taking.
All of the above tests include a freeresponse Bonus question.
• The Open-Ended 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.
• The answers for the lesson-by-lesson
masters are provided as reduced pages
with answers appearing in red.
• A Vocabulary Test, suitable for all
students, includes a list of the vocabulary
words in the chapter and ten questions
assessing students’ knowledge of those
terms. This can also be used in conjunction with one of the chapter tests or as a
review worksheet.
©
Glencoe/McGraw-Hill
• Full-size answer keys are provided for
the assessment masters in this booklet.
v
Glencoe Geometry
NAME ______________________________________________ DATE
1
____________ PERIOD _____
Reading to Learn Mathematics
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 1.
As you study the chapter, complete each term’s definition or description.
Remember to add the page number where you found the term. Add these pages to
your Geometry Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term
Found
on Page
Definition/Description/Example
acute angle





adjacent angles
uh·JAY·suhnt
angle
angle bisector





collinear
koh·LIN·ee·uhr
complementary angles





congruent
kuhn·GROO·uhnt





coplanar
koh·PLAY·nuhr
line segment
linear pair
(continued on the next page)
©
Glencoe/McGraw-Hill
vii
Glencoe Geometry
Vocabulary Builder
Vocabulary Builder
NAME ______________________________________________ DATE
1
____________ PERIOD _____
Reading to Learn Mathematics
Vocabulary Builder
Vocabulary Term
(continued)
Found
on Page
Definition/Description/Example
midpoint
obtuse angle
perimeter
perpendicular lines





polygon
PAHL·ee·gahn
ray
right angle
segment bisector
supplementary angles
vertical angles
©
Glencoe/McGraw-Hill
viii
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Study Guide and Intervention
Points, Lines, and Planes
Name Points, Lines, and Planes
In geometry, a point is a location, a line contains
points, and a plane is a flat surface that contains points and lines. If points are on the same
line, they are collinear. If points on are the same plane, they are coplanar.
Example
Use the figure to name each of the following.
A
a. a line containing point A
D
B
The line can be named as . Also, any two of the three
points on the line can be used to name it.
AB , AC , or BC
C
Lesson 1-1
N
b. a plane containing point D
The plane can be named as plane N or can be named using three
noncollinear points in the plane, such as plane ABD, plane ACD, and so on.
Exercises
Refer to the figure.
A
1. Name a line that contains point A.
C
m
2. What is another name for line
D
B
E
P
m?
3. Name a point not on AC .
4. Name the intersection of AC and DB .
5. Name a point not on line or line
Draw and label a plane
is in plane
6. AB
m.
Q for each relationship.
S
Q.
X
A
at P.
7. ST intersects AB
P
T
Q
B
Y
8. Point X is collinear with points A and P.
9. Point Y is not collinear with points T and P.
10. Line contains points X and Y.
©
Glencoe/McGraw-Hill
1
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Study Guide and Intervention
(continued)
Points, Lines, and Planes
Points, Lines, and Planes in Space
Space is a boundless, three-dimensional set of
all points. It contains lines and planes.
Example
a. How many planes appear in the figure?
There are three planes: plane
O
P
N
B
N , plane O, and plane P.
A
b. Are points A, B, and D coplanar?
Yes. They are contained in plane
D
O.
C
Exercises
Refer to the figure.
A
1. Name a line that is not contained in plane
N.
B
C
2. Name a plane that contains point B.
N
D
E
3. Name three collinear points.
Refer to the figure.
A
B
4. How many planes are shown in the figure?
D
G
C
H
I
5. Are points B, E, G, and H coplanar? Explain.
F
E
J
6. Name a point coplanar with D, C, and E.
Draw and label a figure for each relationship.
7. Planes
9. Line t contains point H and line
plane N.
Glencoe/McGraw-Hill
M
s
8. Line r is in plane N , line s is in plane
intersect at point J.
©
t
M andN intersect in HJ .
M , and lines r and s
N
H
J
r
t does not lie in plane M or
2
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Skills Practice
Points, Lines, and Planes
Refer to the figure.
A
1. Name a line that contains point D.
B
p
D
n
C
G
2. Name a point contained in line n.
4. Name the plane containing lines
Lesson 1-1
3. What is another name for line p ?
n and p.
Draw and label a figure for each relationship.
5. Point K lies on RT .
K
6. Plane
J contains line s.
T
R
s
J
lies in plane B and contains
7. YP
point C, but does not contain point H.
Y
C
8. Lines q and
in plane U.
H
f
q
P
U
B
Refer to the figure.
f intersect at point Z
Z
F
9. How many planes are shown in the figure?
D
E
A
10. How many of the planes contain points F and E?
C
W
B
11. Name four points that are coplanar.
12. Are points A, B, and C coplanar? Explain.
©
Glencoe/McGraw-Hill
3
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Practice
Points, Lines, and Planes
Refer to the figure.
j
M
1. Name a line that contains points T and P.
P
S
T
R
Q
N
h
g
2. Name a line that intersects the plane containing
points Q, N, and P.
.
3. Name the plane that contains TN and QR
Draw and label a figure for each relationship.
and CG
intersect at point M
4. AK
in plane T.
A
T
C
M
5. A line contains L(4, 4) and M(2, 3). Line
q is in the same coordinate plane but does
. Line q contains point N.
not intersect LM
y
G
K
M
q
x
O
N
L
Refer to the figure.
T
Q
6. How many planes are shown in the figure?
W
7. Name three collinear points.
A
8. Are points N, R, S, and W coplanar? Explain.
S
X
M
P
R
N
VISUALIZATION Name the geometric term(s) modeled by each object.
9.
10.
11.
tip of pin
STOP
12. a car antenna
©
Glencoe/McGraw-Hill
strings
13. a library card
4
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Reading to Learn Mathematics
Points, Lines, and Planes
Pre-Activity
Why do chairs sometimes wobble?
Read the introduction to Lesson 1-1 at the top of page 6 in your textbook.
• How many ways can you do this if you keep the pencil points in the same
position?
• How will your answer change if there are four pencil points?
Reading the Lesson
1. Complete each sentence.
a. Points that lie on the same lie are called
points.
b. Points that do not lie in the same plane are called
points.
c. There is exactly one
through any two points.
d. There is exactly one
through any three noncollinear points.
2. Refer to the figure at the right. Indicate whether each
statement is true or false.
D
U
a. Points A, B, and C are collinear.
C
b. The intersection of plane ABC and line
c. Line and line
m is point P.
B
P
A
m do not intersect.
m
d. Points A, P,and B can be used to name plane
U.
e. Line lies in plane ACB.
3. Complete the figure at the right to show the following
relationship: Lines , m, and n are coplanar and lie in
plane Q. Lines and m intersect at point P. Line n
intersects line m at R, but does not intersect line .
Q
n
P
R
m
Helping You Remember
4. Recall or look in a dictionary to find the meaning of the prefix co-. What does this prefix
mean? How can it help you remember the meaning of collinear?
©
Glencoe/McGraw-Hill
5
Glencoe Geometry
Lesson 1-1
• Find three pencils of different lengths and hold them upright on your
desk so that the three pencil points do not lie along a single line. Can you
place a flat sheet of paper or cardboard so that it touches all three pencil
points?
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Enrichment
Points and Lines on a Matrix
A matrix is a rectangular array of rows and columns. Points and
lines on a matrix are not defined in the same way as in Euclidean
geometry. A point on a matrix is a dot, which can be small or
large. A line on a matrix is a path of dots that “line up.” Between
two points on a line there may or may not be other points. Three
examples of lines are shown at the upper right. The broad line can
be thought of as a single line or as two narrow lines side by side.
Dot-matrix printers for computers used dots to form characters.
The dots are often called pixels. The matrix at the right shows
how a dot-matrix printer might print the letter P.
Draw points on each matrix to create the given figures.
1. Draw two intersecting lines that have
four points in common.
2. Draw two lines that cross but have
no common points.
3. Make the number 0 (zero) so that it
extends to the top and bottom sides
of the matrix.
4. Make the capital letter O so that it
extends to each side of the matrix.
5. Using separate grid paper, make dot designs for several other letters. Which were the
easiest and which were the most difficult?
©
Glencoe/McGraw-Hill
6
Glencoe Geometry
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Study Guide and Intervention
Linear Measure and Precision
Measure Line Segments
A part of a line between two endpoints is called a line
segment. The lengths of M
N
and R
S
are written as MN and RS. When you measure a
segment, the precision of the measurement is half of the smallest unit on the ruler.
Example 2
Find the length of M
N
.
M
N
cm
1
2
3
Find the length of R
S
.
R
4
S
in.
The long marks are centimeters, and the
shorter marks are millimeters. The length of
N
M
is 3.4 centimeters. The measurement is
accurate to within 0.5 millimeter, so M
N
is
between 3.35 centimeters and 3.45
centimeters long.
1
2
The long marks are inches and the short
marks are quarter inches. The length of R
S
3
4
is about 1 inches. The measurement is
accurate to within one half of a quarter inch,
1
8
5
8
S
is between 1 inches and
or inch, so R
7
8
Lesson 1-2
Example 1
1 inches long.
Exercises
Find the length of each line segment or object.
1. A
cm
2. S
B
1
2
3
T
in.
3.
1
4.
in.
1
2
cm
1
2
3
Find the precision for each measurement.
©
5. 10 in.
6. 32 mm
7. 44 cm
8. 2 ft
9. 3.5 mm
10. 2 yd
Glencoe/McGraw-Hill
1
2
7
Glencoe Geometry
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Study Guide and Intervention
(continued)
Linear Measure and Precision
On PQ, to say that point M is
between points P and Q means P, Q, and M are collinear
and PM MQ PQ.
On AC, AB BC 3 cm. We can say that the segments are
congruent, or A
B
B
C
. Slashes on the figure indicate which
segments are congruent.
Calculate Measures
Example 1
1.2 cm
D
B
A
Example 2
Find EF.
Q
C
Find x and AC.
2x 5
1.9 cm
E
M
P
F
x
A
2x
B
C
Calculate EF by adding ED and DF.
B is between A and C.
ED DF EF
1.2 1.9 EF
3.1 EF
AB BC AC
x 2x 2x 5
3x 2x 5
x5
AC 2x 5 2(5) 5 15
Therefore, E
F
is 3.1 centimeters long.
Exercises
Find the measurement of each segment. Assume that the art is not drawn to scale.
1. R
T
2.0 cm
R
2. B
C
2.5 cm
S
3. X
Z
T
3 –21 in.
3
–
4
X
Y
in.
6 in.
A
2 –43
in. B
4. W
X
6 cm
W
Z
C
X
Y
Find x and RS if S is between R and T.
5. RS 5x, ST 3x, and RT 48.
6. RS 2x, ST 5x 4, and RT 32.
7. RS 6x, ST 12, and RT 72.
8. RS 4x, R
S
S
T
, and RT 24.
Use the figures to determine whether each pair of segments is congruent.
9. A
B
and C
D
10. X
Y
and Y
Z
11 cm
A
5 cm
B
X
D
5 cm
11 cm
3x 5
C
Y
©
Glencoe/McGraw-Hill
8
5x 1
9x
2
Z
Glencoe Geometry
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Skills Practice
Linear Measure and Precision
Find the length of each line segment or object.
1.
2.
cm
1
2
3
4
5
in.
1
2
Find the precision for each measurement.
1
2
5. 9 inches
4. 12 centimeters
Lesson 1-2
3. 40 feet
Find the measurement of each segment.
6. N
Q
7. A
C
1–41 in.
1in.
Q
P
8. G
H
4.9 cm
A
N
5.2 cm
B
F
9.7 mm
C
G
H
15 mm
Find the value of the variable and YZ if Y is between X and Z.
9. XY 5p, YZ p, and XY 25
10. XY 12, YZ 2g, and XZ 28
11. XY 4m, YZ 3m, and XZ 42
12. XY 2c 1, YZ 6c, and XZ 81
Use the figures to determine whether each pair of segments is congruent.
13. B
E
, C
D
14. M
P
, N
P
B 2m C
3m
E
©
12 yd
3m
5m
D
Glencoe/McGraw-Hill
M
12 yd
15. W
X
, W
Z
P
Y
10 yd
5 ft
N
X
9
9 ft
Z
5 ft
W
Glencoe Geometry
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Practice
Linear Measure and Precision
Find the length of each line segment or object.
1. E
2.
F
in.
1
2
cm
1
2
3
4
5
Find the precision for each measurement.
1
4
4. 7 inches
3. 120 meters
5. 30.0 millimeters
Find the measurement of each segment.
6. P
S
7. A
D
18.4 cm
P
2–83 in.
4.7 cm
Q
8. W
X
S
A
1–41 in.
C
W
X
Y
89.6 cm
100 cm
D
Find the value of the variable and KL if K is between J and L.
9. JK 6r, KL 3r, and JL 27
10. JK 2s, KL s 2, and JL 5s 10
Use the figures to determine whether each pair of segments is congruent.
11. T
U
, S
W
12. A
D
, B
C
T 2 ft S
2 ft
A
13. G
F
, F
E
12.7 in.
B
G
5x
3 ft
U
3 ft
W
H
6x
D
12.9 in.
C
14. CARPENTRY Jorge used the figure at the right to make a pattern
for a mosaic he plans to inlay on a tabletop. Name all of the
congruent segments in the figure.
F
E
A
F
B
E
C
D
©
Glencoe/McGraw-Hill
10
Glencoe Geometry
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Reading to Learn Mathematics
Linear Measure and Precision
Pre-Activity
Why are units of measure important?
Read the introduction to Lesson 1-2 at the top of page 13 in your textbook.
• The basic unit of length in the metric system is the meter. How many
meters are there in one kilometer?
• Do you think it would be easier to learn the relationships between the
different units of length in the customary system (used in the United
States) or in the metric system? Explain your answer.
Reading the Lesson
Lesson 1-2
1. Explain the difference between a line and a line segment and why one of these can be
measured, while the other cannot.
2. What is the smallest length marked on a 12-inch ruler?
What is the smallest length marked on a centimeter ruler?
3. Find the precision of each measurement.
a. 15 cm
b. 15.0 cm
4. Refer to the figure at the right. Which one of the following
statements is true? Explain your answer.
B
A
C
D
B
A
C
D
A
4.5 cm
D
C
4.5 cm
B
5. Suppose that S is a point on V
W
and S is not the same point as V or W. Tell whether
each of the following statements is always, sometimes, or never true.
a. VS SW
b. S is between V and W.
c. VS VW SW
Helping You Remember
6. A good way to remember terms used in mathematics is to relate them to everyday words
you know. Give three words that are used outside of mathematics that can help you
remember that there are 100 centimeters in a meter.
©
Glencoe/McGraw-Hill
11
Glencoe Geometry
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Enrichment
Points Equidistant from Segments
The distance from a point to a segment is zero if the point is on the
segment. Otherwise, it is the length of the shortest segment from the
point to the segment.
A figure is a locus if it is the set of all points that satisfy
1
4
a set of conditions. The locus of all points that are inch
A
B
from the segment AB is shown by two dashed segments
with semicircles at both ends.
1. Suppose A, B, C, and D are four different points, and consider the locus
of all points x units from A
B
and x units from C
D
. Use any unit you find
convenient. The locus can take different forms. Sketch at least three
possibilities. List some of the things that seem to affect the form of
the locus.
A
C
B
X
Y
R
B
D
A
Y
A
X
C
P
C
S
D
B
Q
D
2. Conduct your own investigation of the locus of points
equidistant from two segments. Describe your results on a
separate sheet of paper.
©
Glencoe/McGraw-Hill
12
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Study Guide and Intervention
Distance and Midpoints
Distance Between Two Points
Distance on a Number Line
A
Pythagorean Theorem:
B
a
Distance in the Coordinate Plane
y
a2 b2 c2
b
B(1, 3)
Distance Formula:
AB | b a | or | a b |
d (x2 x1)2 (y2 y1)2
A(–2, –1)
x
O
C (1, –1)
Find AB.
A
5 4 3 2 1
B
0
1
2
AB | (4) 2 |
| 6 |
6
3
Example 2
Find the distance between
A(2, 1) and B(1, 3).
Pythagorean Theorem
(AB)2 (AC)2 (BC)2
(AB)2 (3)2 (4)2
(AB)2 25
AB 25
5
Distance Formula
d (x2 x1)2 (y2 y1)2
AB (1 (
2))2 (3 (1))2
AB (3)2 (4)2
25
5
Exercises
Use the number line to find each measure.
1. BD
2. DG
3. AF
4. EF
5. BG
6. AG
7. BE
8. DE
A
B
C
–10 –8 –6 –4 –2
DE
0
F
2
G
4
6
8
Use the Pythagorean Theorem to find the distance between each pair of points.
9. A(0, 0), B(6, 8)
11. M(1, 2), N(9, 13)
10. R(2, 3), S(3, 15)
12. E(12, 2), F(9, 6)
Use the Distance Formula to find the distance between each pair of points.
13. A(0, 0), B(15, 20)
14. O(12, 0), P(8, 3)
15. C(11, 12), D(6, 2)
16. E(2, 10), F(4, 3)
©
Glencoe/McGraw-Hill
13
Glencoe Geometry
Lesson 1-3
Example 1
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Study Guide and Intervention
(continued)
Distance and Midpoints
Midpoint of a Segment
If the coordinates of the endpoints of a segment are a and b,
Midpoint on a
Number Line
a b.
then the coordinate of the midpoint of the segment is 2
If a segment has endpoints with coordinates (x1, y1) and (x2, y2),
Midpoint on a
Coordinate Plane
Example 1
P
x x
2
y y
2
1
2
1
2
then the coordinates of the midpoint of the segment are ,
.
Find the coordinate of the midpoint of P
Q
.
Q
–3 –2 –1
0
1
2
The coordinates of P and Q are 3 and 1.
3 1
2
2
2
Q
, then the coordinate of M is or 1.
If M is the midpoint of P
Example 2
M is the midpoint of P
Q
for P(2, 4) and Q(4, 1). Find the
coordinates of M.
x x
2
y y
2
22 4
41
2
1
2
1
2
M ,
, or (1, 2.5)
Exercises
Use the number line to find the coordinate of
the midpoint of each segment.
A
B
C
–10 –8 –6 –4 –2
1. C
E
2. D
G
3. A
F
4. E
G
5. A
B
6. B
G
7. B
D
8. D
E
D
EF
0
2
G
4
6
8
Find the coordinates of the midpoint of a segment having the given endpoints.
9. A(0, 0), B(12, 8)
10. R(12, 8), S(6, 12)
11. M(11, 2), N(9, 13)
12. E(2, 6), F(9, 3)
13. S(10, 22), T(9, 10)
14. M(11, 2), N(19, 6)
©
Glencoe/McGraw-Hill
14
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Skills Practice
Distance and Midpoints
Use the number line to find each measure.
1. LN
2. JL
3. KN
4. MN
J
–6
K
–4
L
–2
0
2
M
4
6
N
8
10
Use the Pythagorean Theorem to find the distance between each pair of points.
5.
6.
y
y
S
G
O
x
O
x
F
D
8. C(3, 1), Q(2, 3)
7. K(2, 3), F(4, 4)
Use the Distance Formula to find the distance between each pair of points.
10. W(2, 2), R(5, 2)
11. A(7, 3), B(5, 2)
Lesson 1-3
9. Y(2, 0), P(2, 6)
12. C(3, 1), Q(2, 6)
Use the number line to find the coordinate
of the midpoint of each segment.
13. D
E
14. B
C
15. B
D
16. A
D
A
–6
–4
B
–2
C
0
2
D
4
6
E
8
10
12
Find the coordinates of the midpoint of a segment having the given endpoints.
17. T(3, 1), U(5, 3)
18. J(4, 2), F(5, 2)
Find the coordinates of the missing endpoint given that P is the midpoint of N
Q
.
19. N(2, 0), P(5, 2)
©
Glencoe/McGraw-Hill
20. N(5, 4), P(6, 3)
15
21. Q(3, 9), P(1, 5)
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Practice
Distance and Midpoints
Use the number line to find each measure.
1. VW
2. TV
3. ST
4. SV
S
–10
–8
–6
T
U
–4
–2
V
0
W
2
4
6
8
Use the Pythagorean Theorem to find the distance between each pair of points.
5.
6.
y
y
S
Z
O
O
x
x
M
E
Use the Distance Formula to find the distance between each pair of points.
7. L(7, 0), Y(5, 9)
8. U(1, 3), B(4, 6)
Use the number line to find the coordinate
of the midpoint of each segment.
9. R
T
10. Q
R
11. S
T
12. P
R
P
–10
Q
–8
–6
R
–4
–2
S
0
T
2
4
6
Find the coordinates of the midpoint of a segment having the given endpoints.
13. K(9, 3), H(5, 7)
14. W(12, 7), T(8, 4)
Find the coordinates of the missing endpoint given that E is the midpoint of D
F
.
15. F(5, 8), E(4, 3)
16. F(2, 9), E(1, 6)
17. D(3, 8), E(1, 2)
18. PERIMETER The coordinates of the vertices of a quadrilateral are R(1, 3), S(3, 3),
T(5, 1), and U(2, 1). Find the perimeter of the quadrilateral. Round to the
nearest tenth.
©
Glencoe/McGraw-Hill
16
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Reading to Learn Mathematics
Distance and Midpoints
Pre-Activity
How can you find the distance between two points without a ruler?
Read the introduction to Lesson 1-3 at the top of page 21 in your textbook.
• Look at the triangle in the introduction to this lesson. What is the special
B
in this triangle?
name for A
• Find AB in this figure. Write your answer both as a radical and as a
decimal number rounded to the nearest tenth.
Reading the Lesson
1. Match each formula or expression in the first column with one of the names in the
second column.
a. d (x2 x1)2 ( y2 y1)2
i. Pythagorean Theorem
ab
2
b. ii. Distance Formula in the Coordinate Plane
c. XY | a b |
iii. Midpoint of a Segment in the Coordinate Plane
d. c2 a2 b2
iv. Distance Formula on a Number Line
x x
2
y y
2
1
2
1
2
, e. v. Midpoint of a Segment on a Number Line
2. Fill in the steps to calculate the distance between the points M(4, 3) and N(2, 7).
,
d
(
)2 (
)2
MN (
)2 (
)2
MN (
)2 (
)2
MN MN ).
Lesson 1-3
Let (x1, y1) (4, 3). Then (x2, y2) (
Find a decimal approximation for MN to the nearest hundredth.
Helping You Remember
3. A good way to remember a new formula in mathematics is to relate it to one you already
know. If you forget the Distance Formula, how can you use the Pythagorean Theorem to
find the distance d between two points on a coordinate plane?
©
Glencoe/McGraw-Hill
17
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Enrichment
Lengths on a Grid
Evenly-spaced horizontal and vertical lines form a grid.
You can easily find segment lengths on
a grid if the endpoints are grid-line
intersections. For horizontal or vertical
segments, simply count squares. For
diagonal segments, use the Pythagorean
Theorem (proven in Chapter 7). This
theorem states that in any right triangle,
if the length of the longest side (the side
opposite the right angle) is c and the two
shorter sides have lengths a and b, then
c2 a2 b2.
R
A
C
S
D
B
I
Q
E
Example
L
J
Find the measure of
EF
on the grid at the right. Locate
a right triangle with E
F
as its
longest side.
F
K
N
M
E
2
5
EF 22 52
F
29
5.4 units
Find each measure to the nearest tenth of a unit.
1. IJ
2. M
N
3. RS
4. Q
S
5. I
K
6. J
K
7. L
M
8. L
N
Use the grid above. Find the perimeter of each triangle to the nearest tenth
of a unit.
9. ABC
10. QRS
11. DEF
12. LMN
13. Of all the segments shown on the
grid, which is longest? What is its
length?
14. On the grid, 1 unit 0.5 cm. How can the
answers above be used to find the measures
in centimeters?
15. Use your answer from exercise 8 to
calculate the length of segment LN
in centimeters. Check by measuring
with a centimeter ruler.
16. Use a centimeter ruler to find the perimeter
of triangle IJK to the nearest tenth of a
centimeter.
©
Glencoe/McGraw-Hill
18
Glencoe Geometry
NAME ______________________________________________ DATE
1-4
____________ PERIOD _____
Study Guide and Intervention
Angle Measure
Measure Angles If two noncollinear rays have a common
endpoint, they form an angle. The rays are the sides of the angle.
The common endpoint is the vertex. The angle at the right can be
named as A, BAC, CAB, or 1.
B
1
A
A right angle is an angle whose measure is 90. An acute angle
has measure less than 90. An obtuse angle has measure greater
than 90 but less than 180.
Example 1
S
R
1 2
C
Example 2
Measure each angle and
classify it as right, acute, or obtuse.
T
3
Q
P
E
D
a. Name all angles that have R as a
vertex.
Three angles are 1, 2, and 3. For
other angles, use three letters to name
them: SRQ, PRT, and SRT.
A
B
C
a. ABD
Using a protractor, mABD 50.
50 90, so ABD is an acute angle.
b. Name the sides of 1.
, RP
RS
b. DBC
Using a protractor, mDBC 115.
180 115 90, so DBC is an obtuse
angle.
c. EBC
Using a protractor, mEBC 90.
EBC is a right angle.
Exercises
A
B
4
1. Name the vertex of 4.
1
D
2. Name the sides of BDC.
3
2
C
3. Write another name for DBC.
Measure each angle in the figure and classify it as right,
acute, or obtuse.
N
M
S
4. MPR
P
5. RPN
R
6. NPS
©
Glencoe/McGraw-Hill
19
Glencoe Geometry
Lesson 1-4
Refer to the figure.
NAME ______________________________________________ DATE
1-4
____________ PERIOD _____
Study Guide and Intervention
(continued)
Angle Measure
Congruent Angles
Angles that have the same measure are
congruent angles. A ray that divides an angle into two congruent
angles is called an angle bisector. In the figure, PN is the angle
bisector of MPR. Point N lies in the interior of MPR and
MPN NPR.
M
N
P
R
Q
R
Example
Refer to the figure above. If mMPN 2x 14 and
mNPR x 34, find x and find mMPR.
Since PN bisects MPR, MPN NPR, or mMPN mNPR.
2x 14 x 34
2x 14 x x 34 x
x 14 34
x 14 14 34 14
x 20
mNPR (2x 14) (x 34)
54 54
108
Exercises
bisects PQT, and QP
and QR
are opposite rays.
QS
1. If mPQT 60 and mPQS 4x 14, find the value of x.
S
T
P
2. If mPQS 3x 13 and mSQT 6x 2, find mPQT.
and BC
are opposite rays, BF
bisects CBE, and
BA
bisects ABE.
BD
E
D
3. If mEBF 6x 4 and mCBF 7x 2, find mEBC.
F
1
A
2 3
B
4
C
4. If m1 4x 10 and m2 5x, find m2.
5. If m2 6y 2 and m1 8y 14, find mABE.
6. Is DBF a right angle? Explain.
©
Glencoe/McGraw-Hill
20
Glencoe Geometry
NAME ______________________________________________ DATE
1-4
____________ PERIOD _____
Skills Practice
Angle Measure
For Exercises 1–12, use the figure at the right.
U
Name the vertex of each angle.
4
1. 4
S
2. 1
T
3
5
1
3. 2
W
4. 5
2V
Name the sides of each angle.
5. 4
6. 5
7. STV
8. 1
Write another name for each angle.
9. 3
10. 4
12. 2
Measure each angle and classify it as right, acute,
or obtuse.
13. NMP
14. OMN
15. QMN
16. QMO
P
Q
O
L
M
N
and BC
are opposite rays,
ALGEBRA In the figure, BA
bisects EBC, and BF
bisects ABE.
BD
E
F
D
17. If mEBD 4x 16 and mDBC 6x 4,
find mEBD.
A
B
C
18. If mABF 7x 8 and mEBF 5x 10,
find mEBF.
©
Glencoe/McGraw-Hill
21
Glencoe Geometry
Lesson 1-4
11. WTS
NAME ______________________________________________ DATE
1-4
____________ PERIOD _____
Practice
Angle Measure
For Exercises 1–10, use the figure at the right.
6
Name the vertex of each angle.
1. 5
2. 3
3. 8
4. NMP
7 O
8
1 P
Q
2 3
5
4
M
N
R
Name the sides of each angle.
5. 6
6. 2
7. MOP
8. OMN
Write another name for each angle.
9. QPR
10. 1
Measure each angle and classify it as right, acute,
or obtuse.
11. UZW
12. YZW
13. TZW
14. UZT
V
W
X
U
T
Z
Y
and CD
are opposite rays,
ALGEBRA In the figure, CB
bisects DCF, and CG
bisects FCB.
CE
15. If mDCE 4x 15 and mECF 6x 5,
find mDCE.
16. If mFCG 9x 3 and mGCB 13x 9,
find mGCB.
17. TRAFFIC SIGNS The diagram shows a sign used to warn
drivers of a school zone or crossing. Measure and classify
each numbered angle.
D
E
C
F
G
B
2
1
©
Glencoe/McGraw-Hill
22
Glencoe Geometry
NAME ______________________________________________ DATE
1-4
____________ PERIOD _____
Reading to Learn Mathematics
Angle Measure
Pre-Activity
How big is a degree?
Read the introduction to Lesson 1-4 at the top of page 29 in your textbook.
• A semicircle is half a circle. How many degrees are there in a
semicircle?
• How many degrees are there in a quarter circle?
1. Match each description in the first column with one of the terms in the second column.
Some terms in the second column may be used more than once or not at all.
a. a figure made up of two noncollinear rays with a
1. vertex
common endpoint
2. angle bisector
b. angles whose degree measures are less than 90
3. opposite rays
c. angles that have the same measure
4. angle
d. angles whose degree measures are between 90 and 180
5. obtuse angles
e. a tool used to measure angles
6. congruent angles
f. the common endpoint of the rays that form an angle
7. right angles
g. a ray that divides an angle into two congruent angles
8. acute angles
9. compass
10. protractor
2. Use the figure to name each of the following.
E
a. a right angle
F
D
b. an obtuse angle
28
28 C
c. an acute angle
d. a point in the interior of EBC
A
B
G
e. a point in the exterior of EBA
f. the angle bisector of EBC
g. a point on CBE
h. the sides of ABF
i. a pair of opposite rays
j. the common vertex of all angles shown in the figure
k. a pair of congruent angles
l. the angle with the greatest measure
Helping You Remember
3. A good way to remember related geometric ideas is to compare them and see how they
are alike and how they are different. Give some similarities and differences between
congruent segments and congruent angles.
©
Glencoe/McGraw-Hill
23
Glencoe Geometry
Lesson 1-4
Reading the Lesson
NAME ______________________________________________ DATE
1-4
____________ PERIOD _____
Enrichment
Angle Relationships
Angles are measured in degrees (). Each degree of an angle is divided into 60 minutes (), and each minute of an angle is
divided into 60 seconds ( ).
60 1
60 1
1
2
67 6730
70.4 70°24
90 89°60
Two angles are complementary if the sum of their measures is 90.
Find the complement of each of the following angles.
1. 3515
2. 2716
3. 1554
4. 291822
5. 342945
6. 8723
Two angles are supplementary if the sum of their measures is 180.
Find the supplement of each of the following angles.
7. 12018
8. 8412
10. 451624
11. 392154
12. 1291836
13. 985259
14. 9232
15. 123
©
Glencoe/McGraw-Hill
9. 1102
24
Glencoe Geometry
NAME ______________________________________________ DATE
1-5
____________ PERIOD _____
Study Guide and Intervention
Angle Relationships
Pairs of Angles Adjacent angles are angles in the same plane that have a common
vertex and a common side, but no common interior points. Vertical angles are two
nonadjacent angles formed by two intersecting lines. A pair of adjacent angles whose
noncommon sides are opposite rays is called a linear pair.
Example
Identify each pair of angles as adjacent angles, vertical angles,
and/or as a linear pair.
a.
b.
S
T
U
R
M
4
P
1
3N
2
S
R
SRT and TRU have a common
vertex and a common side, but no
common interior points. They are
adjacent angles.
c.
d.
D
5
A
1 and 3 are nonadjacent angles formed
by two intersecting lines. They are vertical
angles. 2 and 4 are also vertical angles.
60
6
B
C
30
6 and 5 are adjacent angles whose
noncommon sides are opposite rays.
The angles form a linear pair.
B
A
120
F
60
G
A and B are two angles whose measures
have a sum of 90. They are complementary.
F and G are two angles whose measures
have a sum of 180. They are supplementary.
Exercises
Identify each pair of angles as adjacent, vertical, and/or
as a linear pair.
2. 1 and 6
V
2
1
3. 1 and 5
4. 3 and 2
3 4
6Q
R
R
S
P
For Exercises 5–7, refer to the figure at the right.
5. Identify two obtuse vertical angles.
S
5
V
N
U
6. Identify two acute adjacent angles.
T
7. Identify an angle supplementary to TNU.
8. Find the measures of two complementary angles if the difference in their measures is 18.
©
Glencoe/McGraw-Hill
25
Glencoe Geometry
Lesson 1-5
1. 1 and 2
T
U
NAME ______________________________________________ DATE
1-5
____________ PERIOD _____
Study Guide and Intervention
(continued)
Angle Relationships
Perpendicular Lines
Lines, rays, and segments that form four right
angles are perpendicular. The right angle symbol indicates that the lines
is perpendicular to are perpendicular. In the figure at the right, AC
BD ,
or AC ⊥ BD .
A
B
C
D
Example
Find x so that D
Z
⊥P
Z
.
If DZ
⊥P
Z
, then mDZP 90.
mDZQ mQZP
(9x 5) (3x 1)
12x 6
12x
x
mDZP
90
90
84
7
D
Q
(9x 5)
(3x 1)
Sum of parts whole
Substitution
Z
Simplify.
P
Subtract 6 from each side.
Divide each side by 12.
Exercises
⊥ MQ
.
1. Find x and y so that NR
N
P
2. Find mMSN.
5x M
x
(9y 18) S
Q
R
⊥ BF
. Find x.
3. mEBF 3x 10, mDBE x, and BD
E
4. If mEBF 7y 3 and mFBC 3y 3, find y so
⊥ that EB
BC .
D
F
B
A
C
5. Find x, mPQS, and mSQR.
P
S
3x (8x 2)
Q
R
6. Find y, mRPT, and mTPW.
T
(4y 5)
(2y 5)
R
P
W
V
S
©
Glencoe/McGraw-Hill
26
Glencoe Geometry
NAME ______________________________________________ DATE
1-5
____________ PERIOD _____
Skills Practice
Angle Relationships
For Exercises 1–6, use the figure at the right and
a protractor.
E
F
1. Name two acute vertical angles.
K
H
2. Name two obtuse vertical angles.
G
J
3. Name a linear pair.
4. Name two acute adjacent angles.
5. Name an angle complementary to EKH.
6. Name an angle supplementary to FKG.
7. Find the measures of an angle and its complement if one angle measures 18 degrees
more than the other.
8. The measure of the supplement of an angle is 36 less than the measure of the angle.
Find the measures of the angles.
ALGEBRA For Exercises 9–10, use the figure at the right.
R
.
9. If mRTS 8x 18, find x so that TR ⊥ TS
10. If mPTQ 3y 10 and mQTR y, find y so that
PTR is a right angle.
Q
P
T
Determine whether each statement can be assumed
from the figure. Explain.
S
W
V
11. WZU is a right angle.
X
Y
Z
U
Lesson 1-5
12. YZU and UZV are supplementary.
13. VZU is adjacent to YZX.
©
Glencoe/McGraw-Hill
27
Glencoe Geometry
NAME ______________________________________________ DATE
1-5
____________ PERIOD _____
Practice
Angle Relationships
For Exercises 1–4, use the figure at the right and
a protractor.
G
H
F
1. Name two obtuse vertical angles.
C
B
E
2. Name a linear pair whose vertex is B.
A
D
3. Name an angle not adjacent to but complementary to FGC.
4. Name an angle adjacent and supplementary to DCB.
5. Two angles are complementary. The measure of one angle is 21 more than twice the
measure of the other angle. Find the measures of the angles.
6. If a supplement of an angle has a measure 78 less than the measure of the angle, what
are the measures of the angles?
ALGEBRA For Exercises 7–8, use the figure at
the right.
A
B
7. If mFGE 5x 10, find x so that
FC
⊥ AE .
C
G
F
8. If mBGC 16x 4 and mCGD 2x 13,
find x so that BGD is a right angle.
D
E
Determine whether each statement can be
assumed from the figure. Explain.
N
O
9. NQO and OQP are complementary.
P
Q
M
10. SRQ and QRP is a linear pair.
R
S
12. STREET MAPS Darren sketched a map of the cross streets nearest
to his home for his friend Miguel. Describe two different angle
relationships between the streets.
©
Glencoe/McGraw-Hill
28
Be
aco
n
11. MQN and MQR are vertical angles.
Olive
Ma
in
Glencoe Geometry
NAME ______________________________________________ DATE
1-5
____________ PERIOD _____
Reading to Learn Mathematics
Angle Relationships
Pre-Activity
What kinds of angles are formed when streets intersect?
Read the introduction to Lesson 1-5 at the top of page 37 in your textbook.
• How many separate angles are formed if three lines intersect at a common
point? (Do not use an angle whose interior includes part of another angle.)
• How many separate angles are formed if n lines intersect at a common
point? (Do not count an angle whose interior includes part of another angle.)
Reading the Lesson
1. Name each of the following in the figure at the right.
65 2 3 4
1
a. two pairs of congruent angles
b. a pair of acute vertical angles
c. a pair of obtuse vertical angles
d. four pairs of adjacent angles
e. two pairs of vertical angles
f. four linear pairs
g. four pairs of supplementary angles
2. Tell whether each statement is always, sometimes, or never true.
a. If two angles are adjacent angles, they form a linear pair.
b. If two angles form a linear pair, they are complementary.
c. If two angles are supplementary, they are congruent.
d. If two angles are complementary, they are adjacent.
e. When two perpendicular lines intersect, four congruent angles are formed.
f. Vertical angles are supplementary.
g. Vertical angles are complementary.
h. The two angles in a linear pair are both acute.
i. If two angles form a linear pair, one is acute and the other is obtuse.
3. Complete each sentence.
a. If two angles are supplementary and x is the measure of one of the angles, then the
measure of the other angle is
.
Helping You Remember
4. Look up the nonmathematical meaning of supplementary in your dictionary. How can
this definition help you to remember the meaning of supplementary angles?
©
Glencoe/McGraw-Hill
29
Glencoe Geometry
Lesson 1-5
b. If two angles are complementary and x is the measure of one of the angles, then the
measure of the other angle is
.
NAME ______________________________________________ DATE
1-5
____________ PERIOD _____
Enrichment
Curve Stitching
The star design at the right was created by a
method known as curve stitching. Although the
design appears to contain curves, it is made up
entirely of line segments.
To begin the star design, draw a 60° angle. Mark
eight equally-spaced points on each ray, and
number the points as shown below. Then connect
pairs of points that have the same number.
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
To make a complete star, make the same design in
six 60° angles that have a common central vertex.
1. Complete the section of the star design above by connecting
pairs of points that have the same number.
2. Complete the following design.
11 12 13 14 15 16 17 18 19
1
11
2
12
3
13
4
14
5
15
6
16
7
17
8
18
9
19
1
2
3
4
5
6
7
8
9
3. Create your own design. You may use several angles, and
the angles may overlap.
©
Glencoe/McGraw-Hill
30
Glencoe Geometry
NAME ______________________________________________ DATE
1-6
____________ PERIOD _____
Study Guide and Intervention
Polygons A polygon is a closed figure formed by a finite number of coplanar line
segments. The sides that have a common endpoint must be noncollinear and each side
intersects exactly two other sides at their endpoints. A polygon is named according to its
number of sides. A regular polygon has congruent sides and congruent angles. A polygon
can be concave or convex.
Example
Name each polygon by its number of sides. Then classify it as
concave or convex and regular or irregular.
a. D
E
b.
F
H
L
I
G
J
The polygon has 4 sides, so it is a quadrilateral.
It is concave because part of D
E
or E
F
lies in the
interior of the figure. Because it is concave, it
cannot have all its angles congruent and so it is
irregular.
K
The figure is not closed, so it is
not a polygon.
d.
c.
The polygon has 5 sides, so it is a pentagon. It is
convex. All sides are congruent and all angles are
congruent, so it is a regular pentagon.
The figure has 8 congruent sides
and 8 congruent angles. It is
convex and is a regular octagon.
Exercises
Name each polygon by its number of sides. Then classify it as concave or convex
and regular or irregular.
©
1.
2.
3.
4.
5.
6.
Glencoe/McGraw-Hill
31
Glencoe Geometry
Lesson 1-6
Polygons
NAME ______________________________________________ DATE
1-6
____________ PERIOD _____
Study Guide and Intervention
(continued)
Polygons
Perimeter The perimeter of a polygon is the sum of the lengths of all the sides of the
polygon. There are special formulas for the perimeter of a square or a rectangle.
Example
Write an expression or formula for the perimeter of each polygon.
Find the perimeter.
a.
b.
4 in.
b
c
3 in. a
5 cm s
5 in.
c.
5 cm
s
3 ft
s 5 cm
2 ft w
s
5 cm
Pabc
345
12 in.
w
P 2 2w
2(3) 2(2)
10 ft
P 4s
4(5)
20 cm
Exercises
Find the perimeter of each figure.
1.
2.
3 cm
2.5 cm
5.5 ft
square
3.5 cm
3.
4.
27 yd
19 yd
1 cm
14 yd
12 yd
24 yd
Find the length of each side of the polygon for the given perimeter.
5. P 96
6. P 48
x
2x
rectangle
©
x
x2
x
Glencoe/McGraw-Hill
2x
32
Glencoe Geometry
NAME ______________________________________________ DATE
1-6
____________ PERIOD _____
Skills Practice
Name each polygon by its number of sides and then classify it as convex or
concave and regular or irregular.
1.
2.
3.
4.
5.
6.
Find the perimeter of each figure.
7.
8.
20 yd
9.
6m
4m
20 yd
18 yd
40 yd
3m
2m
5m
2 in.
2 in.
2 in.
10 in.
10 in.
2 in.
2 in.
2 in.
COORDINATE GEOMETRY Find the perimeter of each polygon.
10. triangle ABC with vertices A(3, 5), B(3, 1), and C(0, 1)
11. quadrilateral QRST with vertices Q(3, 2), R(1, 2), S(1, 4), and T(3, 4)
12. quadrilateral LMNO with vertices L(1, 4), M(3, 4), N(2, 1), and O(2, 1)
ALGEBRA Find the length of each side of the polygon for the given perimeter.
13. P 104 millimeters
14. P 84 kilometers
15. P 88 feet
4w 1
w
©
Glencoe/McGraw-Hill
33
Glencoe Geometry
Lesson 1-6
Polygons
NAME ______________________________________________ DATE
1-6
____________ PERIOD _____
Practice
Polygons
Name each polygon by its number of sides and then classify it as convex or
concave and regular or irregular.
1.
2.
3.
Find the perimeter of each figure.
4.
7 mm
5.
18 mm
6.
21 mi
10 mm
18 mm
14 cm
2 cm
33 mi
6 cm
4 cm 6 cm
6 cm
4 cm
14 cm
32 mi
COORDINATE GEOMETRY Find the perimeter of each polygon.
7. quadrilateral OPQR with vertices O(3, 2), P(1, 5), Q(6, 4), and R(5, 2)
8. pentagon STUVW with vertices S(0, 0), T(3, 2), U(2, 5), V(2, 5), and W(3, 2)
ALGEBRA Find the length of each side of the polygon for the given perimeter.
9. P 26 inches
10. P 39 centimeters
11. P 89 feet
3x 5
6n 8
2x 2
2x 3
n
x9
5x 4
SEWING For Exercises 12–13, use the following information.
Jasmine plans to sew fringe around the scarf shown in the diagram.
12. How many inches of fringe does she need to purchase?
16 in.
4 in.
4 in.
16 in.
13. If Jasmine doubles the width of the scarf, how many inches of fringe will she need?
©
Glencoe/McGraw-Hill
34
Glencoe Geometry
NAME ______________________________________________ DATE
1-6
____________ PERIOD _____
Reading to Learn Mathematics
Pre-Activity
How are polygons related to toys?
Read the introduction to Lesson 1-6 at the top of page 45 in your textbook.
Name four different shapes that can each be formed by four sticks connected to
form a closed figure. Assume you have sticks with a good variety of lengths.
Reading the Lesson
1. Tell why each figure is not a polygon.
a.
b.
c.
2. Name each polygon by its number of sides. Then classify it as convex or concave and
regular or not regular.
a.
b.
c.
3. What is another name for a regular quadrilateral?
4. Match each polygon in the first column with the formula in the second column that can
be used to find its perimeter. (s represents the length of each side of a regular polygon.)
a. regular dodecagon
b. square
i. P 8s
ii. P 6s
c. regular hexagon
iii. P a b c
d. rectangle
iv. P 12s
e. regular octagon
f. triangle
v. P 2 2w
vi. P 4s
Helping You Remember
5. One way to remember the meaning of a term is to explain it to another person.
How would you explain to a friend what a regular polygon is?
©
Glencoe/McGraw-Hill
35
Glencoe Geometry
Lesson 1-6
Polygons
NAME ______________________________________________ DATE
1-6
____________ PERIOD _____
Enrichment
Perimeter and Area of Irregular Shapes
Two formulas that are used frequently in mathematics are perimeter and
area of a rectangle.
Perimeter: P 2 2w
Area: A w, where is the length and w is the width
However, many figures are combinations of two or more rectangles creating
irregular shapes. To find the area of an irregular shape, it helps to separate
the shape into rectangles, calculate the formula for each rectangle, then find
the sum of the areas.
Example
Find the area of the
figure at the right.
Separate the figure into two rectangles.
A w
A1 9 2
18
9m
2m
5m
3m
A2 3 3
9
9m
18 9 27
1
2m
The area of the irregular shape is 27 m2.
5m
2
3m
Find the area and perimeter of each irregular shape.
1.
2.
1 in.
12 m
4 in.
4 in.
9m
7m
13 m
6m
2 in.
26 m
3.
6 cm
2 cm
4.
4 cm
7 ft
3 ft
4 cm
6 ft
2 ft
2 cm
4 cm
9 ft
4 ft
4 cm
8 cm
For Exercises 5–8, find the perimeter of the figures in Exercises 1–4.
5.
6.
7.
8.
9. Describe the steps you used to find the perimeter in Exercise 1.
©
Glencoe/McGraw-Hill
36
Glencoe Geometry
NAME
1
DATE
PERIOD
Chapter 1 Test, Form 1
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Name the geometric shape modeled by a pinhole in a wall.
A. line segment
B. plane
C. line
D. point
B
2. Which is another name for line ?
A. AB
C. C
3. Name the intersection of lines and
A. A
C. C
P
A
F
B. BD
D. P
2.
m
D
E
C
G
m.
3.
B. B
D. P
4. Name three points coplanar with point A.
A. B, C, F
B. E, F, G
C. B, C, E
D. B, D, G
5. Find the length of R
S
.
A. 33 mm
C. 35 mm
4.
R
B. 34 mm
D. 36 mm
5.
S
cm
1
2
3
4
6. Find the precision for a measurement of 72 centimeters.
A. 0.5 cm
B. 0.1 cm
C. 1 mm
D. 0.5 mm
7. Find the length of B
C
.
A. 12 cm
C. 25 cm
6.
B. 13 cm
D. 38 cm
8. Use the number line to find MN.
A. 5
C. 5
A
13 cm
B
C
M
B. 1
D. 10
9. Find the distance between points P and Q.
A. 5
B. 7
C. 9
D. 25
5 4 3 2 1
0
1
©
Glencoe/McGraw-Hill
C. (0, 3)
37
2
3
y
P
9.
Q
x
O
10. Find the coordinates of the midpoint of P
Q
.
1
B. 0, 3
2
8.
N
For Questions 9 and 10, use the figure given at
the right.
1
A. 2, 3
2
7.
25 cm
10.
1
D. 3, 0
2
Glencoe Geometry
Assessments
For Questions 2–4, use the figure given at
the right.
1.
NAME
1
DATE
Chapter 1 Test, Form 1
PERIOD
(continued)
For Questions 11–13, use the figure at the right.
E
11. Which point is the vertex of all the angles in this figure?
A. A
B. B
C. C
D. E
11.
F
C
50
B
D
A
12. What type of angle is ABC?
A. acute angle
B. right angle
C. obtuse angle
13. Which is true?
A. mEBF 140 B. mEBF 90
12.
D. straight angle
13.
C. mEBF 50
D. mEBF 40
14. For what value of x is ATK MJS if mATK 5x 4 and
mMJS 8x 11?
A. 29
B. 15
C. 10
D. 5
For Questions 15–17, use the figure at the right.
15. Which pair of angles are vertical angles?
A. RST, TSU B. RSX, TSU
C. TSU, USV D. RSX, XSW
T
15.
(10y 10) 5x U
S
4x R
X
16. Which angle is supplementary to USV ?
A. TSU
B. VSW
C. RSV
17. Find x and y.
A. x 10, y 12
14.
W
V
16.
D. WSR
17.
B. x 20, y 7
C. x 10, y 8
D. x 50, y 40
For Questions 18–20, use the figures below.
8 cm
25 cm
12 cm
18 cm
8 cm
15 cm
4 cm
8 cm
15 cm
12 cm
Figure A
15 cm
25 cm
8 cm
15 cm
Figure B
18. Which figure is not a polygon?
A. Figure A
B. Figure B
8 cm
18 cm
15 cm
Figure C
Figure D
18.
C. Figure C
19. Find the perimeter of the convex pentagon.
A. 46 cm
B. 50 cm
C. 61 cm
D. Figure D
19.
D. 72 cm
20. Suppose the length and width of the rectangle are doubled. What is its
perimeter?
A. 120 cm
B. 92 cm
C. 76 cm
D. 46 cm
20.
Bonus Each side of a square is 2x 6 yards long. If the
B:
perimeter of the square is 72 yards what is the value of x?
©
Glencoe/McGraw-Hill
38
Glencoe Geometry
NAME
1
DATE
PERIOD
Chapter 1 Test, Form 2A
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. How many planes can be drawn through any three noncollinear points?
A. 0
B. 1
C. 2
D. 3
For Questions 2 and 3, use the figure at the right.
D
2. Which three points in the figure are collinear?
A. A, B, D
B. E, C, A
C. A, B, C
D. F, E, G
A
B
3. Name the intersection of the plane P and the
plane that contains points B, C, and D.
A. point B
B. B
D
C. BC
D. triangle BCD
5
A. 1 in.
16
7
C. 1 in.
16
G
R
3
B. 1 in.
8
5
D. 1 in.
8
in.
1
1
2
1
C. ft
8
1
2
1
4
B. ft
6. Find the length of P
Q
.
A. 50.9 cm
B. 46.3 cm
C. 25.7 cm
D. 21.3 cm
2
5.
D. 1 in.
6.
38.3 cm
P
Q 12.6 cm R
7. Find y if B is between A and C, AB is 2y, BC is 6y, and AC is 48.
A. 24
B. 8
C. 6
D. 4
7.
8. Find the distance between P(2, 8) and Q(5, 3).
A. 9
B. 18
C. 34
8.
D. 170
9. Find the coordinates of the midpoint of L
B
if L(8, 5) and B(6, 2).
1
A. 1, 3
2
1
B. 2, 1
2
1
C. 7, 3
2
9.
1
D. 7, 1
2
10. Find the coordinates of T given that S is the midpoint of RT
, R(4, 2),
and S(6, 8).
A. (14, 4)
B. (16, 14)
C. (2, 10)
D. (1, 5)
©
3.
4.
S
5. Find the precision for a measurement of 18 feet.
A. ft
P
E
Glencoe/McGraw-Hill
39
10.
Glencoe Geometry
Assessments
2.
F
C
4. Find the length of R
S
.
1.
NAME
1
DATE
Chapter 1 Test, Form 2A
PERIOD
(continued)
For Questions 11 and 12, use the figure at the right.
G
11. What type of angle is ABC?
A. acute angle
B. right angle
C. obtuse angle
D. straight angle
A B
11.
E
D
F
C
12. Use a protractor to measure the angles in the figure.
Which segment is an angle bisector?
A. G
E
B. B
C
C. ED
12.
D. E
F
For Questions 13–17, use the figure at the right.
13. Find mFBD if FBD and DBE are complementary
and mFBD is twice mDBE.
A. 30
B. 45
C. 60
D. 90
14. Which pair of angles are supplementary?
A. ABE, CBD B. ABC, ABD C. ABC, CBD
15. Which angle is a vertical angle to ABE?
A. DBE
B. CBD
C. ABC
16. If mCBF 6x 18, find x so that CB ⊥ BF.
A. 90
B. 45
C. 18
C
A
13.
F
B
D
E
14.
D. ABC, EBD
15.
D. EBA
16.
D. 12
17. Find mABC if mABC 4x 9 and mEBD 7x 9.
A. 6
B. 33
C. 45
D. 73
17.
For Questions 18 and 19, use the figure at the right.
(x 3) km
18. Which describes this figure?
A. hexagon, concave, not regular
B. pentagon, concave, regular
C. hexagon, convex, not regular
D. not a polygon
19. What is x for a perimeter of 108 kilometers?
A. 53
B. 15
C. 18
19.
D. 105
20. A rectangle has a length of 1.4 feet and a width of 1.2 feet. What is the effect
on the perimeter of this rectangle if the length and width are doubled?
A. The perimeter is doubled.
B. The perimeter is increased by 8.
C. The perimeter is multiplied by 4.
D. The perimeter is tripled.
Bonus Find mA if A is complementary to B, B is
supplementary to C, mB 15x 2, and
mC 25x 22.
©
Glencoe/McGraw-Hill
40
18.
20.
B:
Glencoe Geometry
NAME
1
DATE
PERIOD
Chapter 1 Test, Form 2B
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Suppose A and B are points. How many lines contain both A and B?
A. 0
B. 1
C. 2
D. 3
1.
A
2. Which three points in the figure are collinear?
A. C, D, F
B. B, C, D
C. A, E, F
D. A, D, E
C
11
16
9
C. 1 in.
16
A. 1 in.
B
F
3. Name the intersection of the plane that contains
points A, B, and D and the plane P.
A. point D
B. A
D
C. triangle BCD
D. BD
4. Find the length of X
Y
.
2.
3.
D
E
X
5
8
1
D. 1 in.
2
4.
Y
B. 1 in.
in.
1
2
5. Find the precision for a measurement of 34.0 centimeters.
A. 0.5 cm
B. 1 mm
C. 0.5 mm
D. 1 cm
5.
6. Find the length of H
J
.
A. 11.3 cm
C. 13.7 cm
6.
29.1 cm
B. 12.3 cm
D. 45.9 cm
G
16.8 cm H
J
7. Find x if S is between R and T, RS is x 3, ST is 5x, and RT is 57.
A. 9
B. 10
C. 10.8
D. 12
7.
8. Find the distance between M(2, 3) and N(8, 2).
8.
A. 8
B. 61
D. 101
C. 10
9. Find the coordinates of the midpoint of A
S
if A(4, 7) and S(5, 3).
A. (1, 10)
1
B. 4, 2
2
1
C. , 5
2
9.
1
D. , 5
2
10. Find the coordinates of T given that S is the midpoint of RT
, R(2, 6),
and S(2, 0).
A. (6, 12)
B. (6, 6)
C. (0, 3)
D. (2, 3)
©
P
Glencoe/McGraw-Hill
41
10.
Glencoe Geometry
Assessments
For Questions 2 and 3, use the figure at the right.
NAME
1
DATE
Chapter 1 Test, Form 2B
PERIOD
(continued)
For Questions 11 and 12, use the figure at the right.
D
B
11. What type of angle is BAC?
A. acute angle
B. right angle
C. obtuse angle
D. straight angle
C
E
12. Use a protractor to measure the angles in the figure.
Which segment is an angle bisector?
A. A
B
B. C
D
C. CB
11.
F
A
12.
D. A
E
For Questions 13–17, use the figure at the right.
13. Find mVSW if WSR and VSW are complementary
and mWSR is four times mVSW.
A. 72
B. 36
C. 22.5
D. 18
14. Which pair of angles are supplementary?
A. USV, VSW B. VSW, WSR C. TSV, VSW
15. Which angle is a vertical angle to UST?
A. VSW
B. USV
C. TSR
16. If mVSR 8x 18, find x so that US
⊥V
S
.
A. 9
B. 12.25
C. 72
V
W
S
U
13.
R
T
14.
D. TSR, USW
15.
D. WSR
16.
D. 90
17. Find mUSW if mUSW 7x 34 and mTSR 4x 29.
A. 147
B. 113
C. 84
D. 21
17.
For Questions 18 and 19, use the figure at the right.
y5
18. Which describes this figure?
A. hexagon, convex, regular
B. pentagon, concave, regular
C. pentagon, convex, not regular
D. not a polygon
19. What is y for a perimeter of 100 feet?
A. 5
B. 15
y
19.
C. 17
D. 23
20. A square has sides with a length of 5.8 inches. What is the effect on the
perimeter of this square if the sides are tripled?
A. The perimeter stays the same.
B. The perimeter is increased by 12.
C. The perimeter is multiplied by 3.
D. The perimeter is multiplied by 9.
Bonus Find mA if A is supplementary to B, B is
supplementary to C, mB 12x 8, and
mC 8x 8.
©
Glencoe/McGraw-Hill
18.
42
20.
B:
Glencoe Geometry
NAME
PERIOD
Chapter 1 Test, Form 2C
For Questions 1–4, use the figure at
the right.
P
C
A
1. What is another name for line ?
SCORE
B
1.
D
E
2. Name three points on plane P.
2.
F
3. Name the intersection of planes
N
P and N.
3.
4. Name three noncoplanar points.
For Questions 5 and 6,
use the figure at the
right.
4.
A
cm
Assessments
1
DATE
B
1
2
3
4
5
6
5. What is the length of
B
A
?
5.
6. What is the precision of your measurement of A
B
?
6.
7. Find the length of D
E
if D is between points C and E,
CD 6.5 centimeters, and CE 13.8 centimeters.
7.
8. Find the length of X
Z
.
8.
4x 3
2x 7
X 8 cm Y
9. Find x if R
S
S
T
.
Z
9.
52 in.
R
6x 8 S
For Questions 10–12, use the
coordinate grid.
T
y
B
10. Find the distance between A and B.
10.
A
x
O
11. Find the coordinates of the midpoint
of C
D
.
C
11.
D
12. Find the coordinates of a point E if C
is the midpoint of A
E
.
12.
13. The vertices of a triangle are located at P(0, 0), Q(8, 6), and
R(3, 4). What is the perimeter of this triangle?
13.
14. Find x and y if U
V
bisects T
W
and
UV 40.
14.
U
3y 1
3x 2
3y 1
Z
T
2y 6
W
V
©
Glencoe/McGraw-Hill
43
Glencoe Geometry
NAME
1
DATE
Chapter 1 Test, Form 2C
15. Measure PQR. Then classify PQR
as right, acute, or obtuse.
(continued)
15.
P
Q
R
and EB
are opposite
In the figure, EA
bisects FEG.
rays and EC
F
16. Find x if mFEG 82, and
mFEC 5x 11.
PERIOD
C
G
16.
A
E
B
D
17. If mAED 16y 10, find y so that
D
E
⊥A
B
.
17.
For Questions 18–21, use the
figure at the right.
1
72 (8y 16)
2
40
18. Find y.
(11x 24)
19. Find m1.
18.
19.
20. Find m2.
20.
21. Find x .
21.
For Questions 22–25, use the
polygons at the right.
22. Name polygon ABCDEF by
its sides. Then classify it as
convex or concave and
regular or not regular.
B
C
R
6x 5
A
3y 1
D
F
E
T
2y 11
22.
S
23. Find the perimeter of polygon ABCDEF for x 4.
23.
24. Find the length of each side of polygon RST.
24.
25. Find the length of one side of a regular pentagon whose
perimeter is the same as the perimeter of RST.
25.
Bonus Find the dimensions of a rectangle whose length is
3 more than twice its width and has a perimeter of
30 centimeters.
B:
©
Glencoe/McGraw-Hill
44
Glencoe Geometry
NAME
1
DATE
PERIOD
Chapter 1 Test, Form 2D
SCORE
For Questions 1–4, use the figure at
the right.
2. Name three points on plane
R
X
m?
Y
B.
1.
T
U B
S
Z
2.
V
A
m
A and B.
3. Name the intersection of planes
3.
4.
4. Name three noncollinear points.
For Questions 5 and 6, use
the figure at the right.
R
?
5. What is the length of Q
Assessments
1. What is another name for line
Q
R
in.
1
2
5.
6. What is the precision of
your measurement of Q
R
?
6.
7. Find the length of L
O
if O is between points L and M,
LM 18.6 centimeters, and OM 12.9 centimeters.
7.
8. Find the length of D
E
.
8.
24 cm
3x 5
D
E 5x 13 F
9. Find y if X
Y
Y
Z
.
X
Y
23 in.
For Questions 10–12, use the
coordinate grid.
10. Find the distance between L
and M.
9.
5y 11
Z
y
M
10.
x
O
P
L
N
11.
12. Find the coordinates of a point Q if P is the midpoint of N
Q
.
12.
13. The vertices of a triangle are located at P(0, 6), Q(8, 12), and
R(3, 3). What is the perimeter of this triangle?
13.
14. Find x if R
S
bisects A
B
and RS 36.
14.
11. Find the coordinates of the
midpoint of M
N
.
R
18
25 3x
A
2y 5
T
B
2y 6
S
©
Glencoe/McGraw-Hill
45
Glencoe Geometry
NAME
1
DATE
Chapter 1 Test, Form 2D
15. Measure ABC. Then classify
ABC as right, acute, or obtuse.
PERIOD
(continued)
15.
A
B
and RD
are opposite
In the figure, RC
bisects WRV.
rays and RQ
C
P
(13x 12)
C
16. Find y if mWRQ 48 and
mQRV 7y 6.
R
16.
D
W
V
Q
17. Find x so that C
R
⊥P
R
.
17.
For Questions 18–21, use the figure at
the right.
2
(9x 5)
18. Find x.
(7y 27)
1
58
18.
19. Find m1.
19.
20. Find m2.
20.
21. Find y.
21.
For Questions 22–25, use
the polygons at the right.
S
R
A
B
27 4x
T
22. Name polygon RSTUV
5y 8
by its sides. Then
V
U
classify it as convex or
concave and regular or not regular.
D
8x 3
22.
C
23. Find the perimeter of polygon RSTUV for y 9.
23.
24. Find the length of each side of polygon ABCD.
24.
25. Find the length of the sides of a regular triangle whose
perimeter is the same as the perimeter of ABCD.
25.
Bonus Find the lengths of the sides of a triangle whose
perimeter is 37. The measure of the first side of the
triangle is 8 less than the second side, and the second
side is twice the length of the third side.
B:
©
Glencoe/McGraw-Hill
46
Glencoe Geometry
NAME
PERIOD
Chapter 1 Test, Form 3
For Questions 1–3, use the figure
at the right.
SCORE
B
C
1. Name five planes shown in the
figure.
1.
F
A
E
D
P
2.
2. Name a line that is coplanar with
.
AD and AB
3. Name the intersection of plane
points A, B, and E.
P and the plane that contains
For Questions 4 and 5, use
the figure at the right.
B
.
4. Find the length of A
A
in.
Assessments
1
DATE
3.
B
6
4.
7
5. Find the precision for the measurement of A
B
.
5.
6. Find two possible lengths for C
D
if C, D, and E are collinear,
CE 15.8 centimeters, and DE 3.5 centimeters.
6.
7. Find the length of R
S
if S is between R and T, the length of
1
S
R
is the length of R
T
, RS 3x 3, and ST 2x 6.
7.
8. Find y if AC 3y 5, CB 4y 1, AB 9y 12, and point
C lies between A and B.
8.
3
For Questions 9–11, use the coordinate
grid at the right.
y
B
9.
9. Find the distance between A and B.
O
x
A
10. Find two possible coordinates of a point
1
D on a line containing A
B
so that AD AB.
10.
11. Find two values of y for C located at (1, y) and AC 5.
11.
4
©
Glencoe/McGraw-Hill
47
Glencoe Geometry
NAME
1
DATE
Chapter 1 Test, Form 3
PERIOD
(continued)
12. Find y if S is the midpoint of R
T
, T is the midpoint of R
U
,
RS 6x 5, ST 8x 1, and TU 11y 13.
12.
13. Find all values of x that will make A an obtuse angle given
mA 12x 6.
13.
bisects RSU and
14. Find mRST if ST
bisects TSV.
SU
14.
R
(x 2y 1)
T
(6x 9)
S
(2y 5)
U
V
15. Find m1 if 1 is complementary to 2, 2 is supplementary 15.
to 3, and m3 126.
⊥ XZ
, Y is in the interior of WXZ,
16. Find y if XW
mWXY 6y 3, and mYXZ 4y 13.
16.
is the
17. Find the length of L
M
if ON
bisector of L
M
and LN 3x 2.
17.
O
7x 1
L
N
M
For Questions 18 and 19, use the coordinate grid.
18. Graph polygon ABCD with vertices A(4, 3), B(0, 3), C(2, 2),
and D(5, 6). Then name polygon ABCD by its number of
sides and classify it as convex or concave and regular or
irregular.
18.
19. Find the perimeter of polygon ABCD.
19.
20. Find the perimeter of regular triangle DEF if DE 28 3y
and EF 2y 3.
20.
Bonus Suppose a regular quadrilateral and a regular triangle have B:
the same perimeter. The sides of the triangle are 3 inches
longer than the sides of the quadrilateral. Find the lengths
of the sides of the quadrilateral and the triangle.
©
Glencoe/McGraw-Hill
48
Glencoe Geometry
NAME
1
DATE
Chapter 1 Open-Ended Assessment
PERIOD
SCORE
Demonstrate your knowledge by giving a clear, concise solution to
each problem. Be sure to include all relevant drawings and justify
your answers. You may show your solution in more than one way or
investigate beyond the requirements of the problem.
1. Draw and label a figure that shows that plane R contains both lines s and
that intersect at point B. Name three collinear points in plane R .
AC
Assessments
2. Draw a line on a coordinate plane so that you can determine at least two
points on the graph. Label those two points D and G.
a. Find the distance between points D and G.
b. Find the coordinates of E, the midpoint of D
G
.
c. Find the coordinates of point H given that G is the midpoint of D
H
.
3. Rectangle WXYZ has a length that is 5 more than three times its width.
a. Draw and label a figure for rectangle WXYZ.
b. Write an algebraic expression for the perimeter of the rectangle.
c. Find the width if the perimeter is 58 millimeters. Explain how you can
check that your answer is correct.
d. Use a ruler to draw and label P
Q
, which is congruent to the segment
representing the length of rectangle WXYZ. What is the measure
of P
Q
?
e. Explain how to find the precision of the measurement of P
Q
.
4. Draw an acute angle, ABC. Let mABC 6x 1.
a. Use a protractor to determine the measure of ABC. Use this
measure to determine the value of x.
b. Explain how you would determine the measure of an angle that is
complementary to ABC.
c. Explain how you would determine the measure of an angle that is
supplementary to ABC.
is in the interior of TRU, mTRS 4x 6, and mSRU 8x 6.
5. RS
.
a. Draw TRU and RS
an angle bisector. Explain
b. Determine the value of x that will make RS
your steps.
and RT
when x 7.5.
c. Describe the relationship between RU
©
Glencoe/McGraw-Hill
49
Glencoe Geometry
NAME
1
DATE
PERIOD
Chapter 1 Vocabulary Test/Review
convex
coplanar
degree
distance
exterior
interior
line
line segment
linear pair
midpoint
acute angle
adjacent angles
angle
angle bisector
betweenness
collinear
complementary
concave
congruent
construction
SCORE
regular polygon
relative error
right angle
segment bisector
sides
space
supplementary
undefined terms
vertex
vertical angles
n-gon
obtuse angle
opposite rays
perimeter
perpendicular
plane
point
polygon
precision
ray
Choose from the terms above to complete each sentence.
1. Two lines are
?
2. Two angles are
if they intersect to form a right angle.
1.
if their measures have a sum of 90°.
2.
?
?
3. When two rays intersect with a common endpoint a(n)
is formed.
?
4. The
is the point located halfway between the
endpoints of a segment.
5.
4.
?
are nonadjacent angles formed by the intersection of
two lines.
6. A(n)
?
divides an angle into two congruent angles.
7. Two angles are
?
9. A(n)
?
?
if
is an angle whose measure is less than 90°.
10. Two segments are
?
5.
6.
if their measures have a sum of 180°.
8. Two angles that lie in the same plane are called
they share a common side and a common vertex.
3.
if they have the same measure.
7.
8.
9.
10.
In your own words—
11. Explain how to find the precision of a measurement of
1
5 inches on a ruler marked in half inches.
11.
12. Describe what is meant by betweenness of points using
collinear points M, P, and Q.
12.
2
©
Glencoe/McGraw-Hill
50
Glencoe Geometry
NAME
1
DATE
PERIOD
Chapter 1 Quiz
SCORE
(Lessons 1–1 and 1–2)
For Questions 1–3, use the figure at
the right.
1. What is another name for line ?
S
V
2. Name the intersection of lines and
m
T
R
U
m.
2.
3. Name three collinear points.
3.
A
B
.
4. Find the length of A
B
in.
4.
1
5. Find the precision of the
measurement of A
B
.
5.
6. Find the length of U
W
if W is between U and V,
UV 16.8 centimeters, and VW 7.9 centimeters.
6.
7. Find x if RS 24 centimeters.
6x 4
R
7.
10 cm
T
S
8. Find the length of L
O
if M is between L and O, LM 7x 9,
MO 14 inches, and LO 10x 7.
8.
9. Find x if P
Q
R
S
, PQ 9x 7, and RS 29.
9.
10. STANDARDIZED TEST PRACTICE Which of the following is
not an undefined term in geometry?
A. plane
B. point
C. bisector
D. line
NAME
1
Assessments
For Questions 4 and 5, use the
figure at the right.
1.
10.
DATE
PERIOD
Chapter 1 Quiz
SCORE
(Lesson 1–3)
A
B for A(2, 5) and B(6, 9).
1. Find the coordinates of the midpoint of 1.
2. Find the coordinates of D if E is the midpoint of C
D
, for
C(3, 4) and E(0, 1).
2.
3. What is the length of F
H
if G is the midpoint, FG 12x 5,
and GH 7x?
W
4. What is the length of U
V
if WX is the
segment bisector of U
V
at point Z?
V
6x 8
3.
4.
Z 9x 2
U
X
5. STANDARDIZED TEST PRACTICE Find the distance between
A(2, 1) and B(4, 3).
A. 52
B. 52
C. 20
D. 8
©
Glencoe/McGraw-Hill
51
5.
Glencoe Geometry
NAME
1
DATE
PERIOD
Chapter 1 Quiz
SCORE
(Lessons 1–4 and 1–5)
For Questions 1–6, use the figure at the right.
U
1. Name the vertex of 1.
2. Classify TSV as right, acute, or obtuse.
W
S
T
3. Name a point in the exterior of RTS.
V
1
2.
P
R
3.
4. Find mTSU if S
U
bisects TSV,
mTSU 4y 11, and mUSV 6y 5.
4.
5. Name a pair of adjacent angles.
5.
6. Name a pair of vertical angles.
A
7. Find mDBC if mABC 5x 3 and
ABD DBC.
For Questions 8 and 9, lines
adjacent angles 1 and 2.
(3x 2)
B
D
C
6.
7.
p and q intersect to form
8. If m1 7x 6 and m2 8x 6, find x so that
perpendicular to q.
p is
8.
9. If m1 4x 3 and m2 3x 8, find x so that 1 is
supplementary to 2.
9.
10. STANDARDIZED TEST PRACTICE The difference between
two complementary angles is 14. Which is the measure of one
of those angles?
A. 14
B. 52
C. 83
D. 90
NAME
1
1.
10.
DATE
PERIOD
Chapter 1 Quiz
SCORE
(Lesson 1–6)
1. Draw a concave pentagon.
1.
2. Find the length of each side of a regular hexagon whose
perimeter is 84 meters.
2.
3. If x 5, find the perimeter of the rectangle whose length is
6x 4 and whose width is 3x 2.
3.
4. The perimeter of a convex pentagon is 15 feet. What is the
effect on its perimeter if each side is doubled?
4.
5. For what value of y is triangle ABC a regular
triangle?
5.
B
9y 4
A
©
Glencoe/McGraw-Hill
52
5y 20
7y 12
C
Glencoe Geometry
NAME
1
DATE
PERIOD
Chapter 1 Mid-Chapter Test
SCORE
(Lessons 1–1 through 1–3)
Part I Write the letter for the correct answer in the blank at the right of each question.
For Questions 1 and 2, use the figure at the right.
E
1. Which point is coplanar with points A and C?
A. A
B. B
C. C
D. D
A
1.
M
B
C
For Questions 3 and 4, use the figure at
the right.
D.
M
A
B
?
3. What is the length of A
1
A. about 1 in.
4
3
C. about 1 in.
4
2.
B
in.
1
2
D. about 2 in.
4. What is the precision for the measurement of A
B
?
A. 1 in.
3.
1
B. about 1 in.
2
1
B. in.
2
5. What is the length of T
S
?
A. 9.4 cm
C. 4.7 cm
4.
1
C. in.
4
1
D. in.
8
5.
8.9 cm
B. 8.9 cm
D. 4.2 cm
R
4.7 cm
T
S
Part II
For Questions 6–8, use the coordinate grid.
y
R
U
S
6. Find the distance between R and S.
6.
x
O
7. Find the coordinates of the midpoint of T
U
.
7.
T
8. Find the coordinates of a point M given
that U is the midpoint of M
S
.
8.
9. Find y if M is the midpoint of L
N
.
L
10. In the figure, WZ bisects X
Y
. Find the
length of X
Y
.
9.
6y 5
9y 4
M
N
10.
W
X
V 6x 11
4x 5
Y
Z
©
Glencoe/McGraw-Hill
53
Glencoe Geometry
Assessments
D
.
2. Name the point of intersection of plane M and DE
A. D
B. E
C. B
NAME
1
DATE
PERIOD
Chapter 1 Cumulative Review
SCORE
(Chapter 1)
For Questions 1 and 2, use the figure
at the right.
A
1. Name three points that are collinear.
C
B
D
H
G
(Lesson 1-1)
1.
F
2. Name the intersection of AE and CG .
E
2.
(Lesson 1-1)
Find the measurement of each segment. Assume that the
art is not drawn to scale. (Lesson 1-2)
3. A
B
A
3.
B 2 cm C
5 cm
4. K
N
J
K
4 mm
L
M
4.
N
3 mm
5. Use the Pythagorean Theorem to find the distance between
A(12, 13) and B(2, 11). (Lesson 1-3)
5.
6. Find the coordinates of B if A has coordinates (3, 5) and
Y(2, 3) is the midpoint of A
B
. (Lesson 1-3)
6.
For Questions 7 and 8, use the
figure to name the vertex and
sides of each angle. Then measure
and classify each angle. (Lesson 1-4)
7. JNK
K
L
77
J
N 157
H
26
7.
M
8. HNK
For Questions 9–11, use the
figure at the right. (Lesson 1-5)
9. Name a pair of supplementary
nonadjacent angles.
8.
B
C
120
60
A
30
F
D
G
9.
E
10. Name two obtuse vertical angles.
10.
11. Name an angle complementary to CFD.
11.
12. If mHJK 7y 2 and mPQR 133, find y so that HJK
is supplementary to PQR. (Lesson 1-5)
12.
13. Name this polygon by its number of sides and
then classify it as convex or concave and regular
or irregular. (Lesson 1-6)
13.
14. Find the perimeter of ABC if A(1, 1), B(4, 3), and C(3, 2). 14.
(Lesson 1-6)
15. Find the length of each side of a regular pentagon whose
perimeter is 90 centimeters. (Lesson 1-6)
©
Glencoe/McGraw-Hill
54
15.
Glencoe Geometry
NAME
1
DATE
PERIOD
Standardized Test Practice
SCORE
(Chapter 1)
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
1.
A
B
C
D
2.
E
F
G
H
3. What is the precision of a measurement of 49.5 centimeters on a
ruler with millimeter marks? (Lesson 1-2)
A. 49 cm to 50 cm
B. 49.0 cm to 50.0 cm
C. 490 mm to 500 mm
D. 494.5 mm to 495.5 mm
3.
A
B
C
D
4. When segments have the same measure, they are said to be
?
. (Lesson 1-2)
E. accurate
F. congruent
G. precise
H. constructed
4.
E
F
G
H
5. Find the distance between A(3, 5) and B(4, 2), to the nearest
hundredth. (Lesson 1-3)
A. 6.75
B. 7.62
C. 8.06
D. 10
5.
A
B
C
D
6. Find EF if E is the midpoint of DF
, DE 15 3x, and
EF x 3. (Lesson 1-3)
E. 1
F. 3
G. 6
H. 9
6.
E
F
G
H
7.
A
B
C
D
8.
E
F
G
H
9.
A
B
C
D
10.
E
F
G
H
(Lesson 1-1)
C. a meter stick D. a diskette
2. Which figure shows AB and point G contained in plane R ? (Lesson 1-1)
E.
F.
A
B G
R
G
A
G.
B
R
H.
G
A
A
B R
G
R
B
For Questions 7–9, use the figure.
7. What is another name for 2?
A. WYX
B. WXY
C. 3
D. Y
W
5
2
(Lesson 1-4)
8. Which angles form a linear pair?
E. 1 and 3
F. 2 and 5
4
3
Y
1
X
(Lesson 1-5)
G. 2 and 3
9. Name the angle that is vertical to 3. (Lesson 1-5)
A. 1
B. 2
C. 3
H. 1 and 4
D. 4
10. Find the length of one side of a regular hexagon whose perimeter
is 75 feet. (Lesson 1-6)
E. 25 ft
F. 18.75 ft
G. 15 ft
H. 12.5 ft
©
Glencoe/McGraw-Hill
55
Glencoe Geometry
Assessments
1. Which object models a line?
A. a fly
B. a wall
NAME
1
DATE
Standardized Test Practice
PERIOD
(continued)
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column box
and then shading in the appropriate oval that corresponds to that entry.
11. How many points name a line?
12. What is the measure of A
C
?
3.7
A
11.
(Lesson 1-1)
(Lesson 1-2)
B
C
K
G
2
1
.
/
.
/
.
.
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
5.2
bisects KHI and
For Questions 13–15, HL
and HI
are opposite rays.
HG
J
12.
2
L
13.
3
4
H
I
13. If 1 2, mKHG 70, and m1 3d 2,
find d. (Lesson 1-4)
14. If m2 a 15 and m3 a 35, find a so
⊥ HJ
. (Lesson 1-5)
that HL
15. Find m4, if mGHL 125. (Lesson 1-5)
15.
14.
1 1
.
/
.
/
.
.
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
8 . 9
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
2 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
0
1
2
3
4
5
6
7
8
9
5 5
.
/
.
/
.
.
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
Part 3: Short Response
Instructions: Show your work or explain in words how you found your answer.
16. Find the length of X
Z
if Y(4, 4) is the midpoint of X
Z
and
X has coordinates (2, 4). (Lesson 1-3)
17. Find the perimeter of this hexagon.
16.
17.
30 m
7m
(Lesson 1-6)
6m
8.5 m
18. Find the measure of W
X
, if the perimeter
of pentagon UVWXY is 48 units. (Lesson 1-6)
V
20 m
10 m
18.
U 10 a
Y
2a
W
©
Glencoe/McGraw-Hill
56
4a 7
X
Glencoe Geometry
NAME
1
DATE
PERIOD
Standardized Test Practice
Student Record Sheet
(Use with pages 58–59 of the Student Edition.)
Part 1 Multiple Choice
Select the best answer from the choices given and fill in the corresponding oval.
1
A
B
C
D
4
A
B
C
D
7
A
B
C
D
2
A
B
C
D
5
A
B
C
D
8
A
B
C
D
3
A
B
C
D
6
A
B
C
D
9
A
B
C
D
Part 2 Short Response/Grid In
Solve the problem and write your answer in the blank.
For Questions 14 and 15, also enter your answer by writing each number or
symbol in a box. Then fill in the corresponding oval for that number or symbol.
14
11
12
13
14
(grid in)
15
(grid in)
15
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
Answers
10
Part 3 Open-Ended
Record your answers for Questions 16–17 on the back of this paper.
©
Glencoe/McGraw-Hill
A1
Glencoe Geometry
©
____________ PERIOD _____
Points, Lines, and Planes
Study Guide and Intervention
Glencoe/McGraw-Hill
Use the figure to name each of the following.
A2
Exercises
D or E
m ? ៭៮៬
BD
Q.
Q for each relationship.
m. E
B
Glencoe/McGraw-Hill
10. Line ᐉ contains points X and Y.
1
9. Point Y is not collinear with points T and P.
8. Point X is collinear with points A and P.
៮៬ at P.
7. ៭៮៬
ST intersects ៭AB
៮៬ is in plane
6. ៭AB
Draw and label a plane
5. Name a point not on line ᐉ or line
4. Name the intersection of ៭៮៬
AC and ៭៮៬
DB .
3. Name a point not on ៭៮៬
AC .
2. What is another name for line
៭៮៬, AC
៭៮៬, ៭៮៬
BC , or
1. Name a line that contains point A. AB
Refer to the figure.
©
N
ᐉ
A
Q
A
T
P
S
m
A
X
E
B
D
C
Y
C
ᐉ
B
B
D
P
Glencoe Geometry
Answers for Exercises 6–10
ᐉ
ᐉ
The plane can be named as plane N or can be named using three
noncollinear points in the plane, such as plane ABD, plane ACD, and so on.
b. a plane containing point D
The line can be named as ᐉ. Also, any two of the three
points on the line can be used to name it.
៭៮៬
AC , or ៭៮៬
BC
AB , ៭៮៬
a. a line containing point A
Example
In geometry, a point is a location, a line contains
points, and a plane is a flat surface that contains points and lines. If points are on the same
line, they are collinear. If points on are the same plane, they are coplanar.
Name Points, Lines, and Planes
1-1
NAME ______________________________________________ DATE
O.
F or J
©
M andN intersect in ៭៮៬
HJ .
Glencoe/McGraw-Hill
9. Line t contains point H and line
plane N.
M , and lines r and s
2
N
F
G
N
P
t
J
C
D
H
A
I
D
J
D
M
E
A
H
E
A
B
B
s
C
O
N
r
C
B
Glencoe Geometry
Answers for Exercises 7–9
t does not lie in plane M or
8. Line r is in plane N , line s is in plane
intersect at point J.
7. Planes
Draw and label a figure for each relationship.
6. Name a point coplanar with D, C, and E.
No; B, G, and H lie in plane BGH, but E does not.
5. Are points B, E, G, and H coplanar? Explain.
4. How many planes are shown in the figure? 6
Refer to the figure.
plane N , plane ABC, plane ABD, plane EBC,
plane EBD
3. Name three collinear points. A, B, E
2. Name a plane that contains point B.
1. Name a line that is not contained in plane
Refer to the figure.
Exercises
Yes. They are contained in plane
N. ៭៮៬
AB
N , plane O, and plane P.
b. Are points A, B, and D coplanar?
There are three planes: plane
a. How many planes appear in the figure?
Example
all points. It contains lines and planes.
Space is a boundless, three-dimensional set of
Points, Lines, and Planes
(continued)
____________ PERIOD _____
Study Guide and Intervention
Points, Lines, and Planes in Space
1-1
NAME ______________________________________________ DATE
Answers
(Lesson 1-1)
Glencoe Geometry
Lesson 1-1
©
Points, Lines, and Planes
Skills Practice
Glencoe/McGraw-Hill
n and p.
C
D
p
A
A3
R
T
C
P
H
U
©
Glencoe/McGraw-Hill
3
Yes; points A, B, and C lie in plane W.
12. Are points A, B, and C coplanar? Explain.
A, B, E, F or B, C, D, E or A, C, D, F
11. Name four points that are coplanar.
2
q
Z
f
A
F
B
E
W
Glencoe Geometry
C
D
f intersect at point Z
s
J contains line s.
8. Lines q and
in plane U.
J
6. Plane
10. How many of the planes contain points F and E?
5
9. How many planes are shown in the figure?
Refer to the figure.
B
Y
៮៬ lies in plane B and contains
7. ៭YP
point C, but does not contain point H.
K
5. Point K lies on ៭៮៬
RT .
n
G
B
____________ PERIOD _____
Draw and label a figure for each relationship. Sample answers are given.
Sample answer: plane G
4. Name the plane containing lines
៭៮៬ or ៭DC
៮៬
CD
3. What is another name for line p ?
A or B
2. Name a point contained in line n.
៭៮៬
p or CD
1. Name a line that contains point D.
Refer to the figure.
1-1
NAME ______________________________________________ DATE
(Average)
Points, Lines, and Planes
Practice
S
R
M
T
N
Q
g
C
M
K
G
S, X, M
L
O
y
N
M
q
A
x
M
W
S
X
T
plane and line
STOP
Glencoe Geometry
Glencoe/McGraw-Hill
Answers
©
line and point
12. a car antenna
9.
10.
point
tip of pin
4
plane
13. a library card
11.
lines
R
Q
N
P
Glencoe Geometry
strings
VISUALIZATION Name the geometric term(s) modeled by each object.
No; sample answer: points N, R, and S lie
in plane A, but point W does not.
8. Are points N, R, S, and W coplanar? Explain.
7. Name three collinear points.
h
5. A line contains L(⫺4, ⫺4) and M(2, 3). Line
q is in the same coordinate plane but does
៮៬. Line q contains point N.
not intersect ៭LM
6. How many planes are shown in the figure? 6
Refer to the figure.
T
A
៮៬ and ៭CG
៮៬ intersect at point M
4. ៭AK
in plane T.
Draw and label a figure for each relationship. Sample answers are given.
S
P
j
____________ PERIOD _____
៮៬. Sample answer: plane
3. Name the plane that contains ៭៮៬
TN and ៭QR
៭៮៬
j or MT
2. Name a line that intersects the plane containing
points Q, N, and P.
៭៮៬, TN
៭៮៬, NP
៭៮៬
g, TP
1. Name a line that contains points T and P.
Refer to the figure.
1-1
NAME ______________________________________________ DATE
Answers
(Lesson 1-1)
Lesson 1-1
©
Glencoe/McGraw-Hill
A4
plane
c. There is exactly one
d. There is exactly one
Q
P
A
R
ᐉ
m
P
D
m
B
n
ᐉ
Glencoe/McGraw-Hill
5
Glencoe Geometry
Sample answer: The prefix co- means together. The word collinear
contains the word line, so collinear means together on a line.
4. Recall or look in a dictionary to find the meaning of the prefix co-. What does this prefix
mean? How can it help you remember the meaning of collinear?
Helping You Remember
3. Complete the figure at the right to show the following
relationship: Lines ᐉ, m, and n are coplanar and lie in
plane Q. Lines ᐉ and m intersect at point P. Line n
intersects line m at R, but does not intersect line ᐉ.
e. Line ᐉ lies in plane ACB. true
U. false
m is point P. true
m do not intersect. false
C
U
through any three noncollinear points.
d. Points A, P,and B can be used to name plane
c. Line ᐉ and line
b. The intersection of plane ABC and line
a. Points A, B, and C are collinear. false
noncoplanar points.
points.
through any two points.
2. Refer to the figure at the right. Indicate whether each
statement is true or false.
line
b. Points that do not lie in the same plane are called
a. Points that lie on the same lie are called
1. Complete each sentence.
collinear
answer: It may not be possible to place the paper to touch
all four points.
• How will your answer change if there are four pencil points? Sample
• How many ways can you do this if you keep the pencil points in the same
position? one
• Find three pencils of different lengths and hold them upright on your
desk so that the three pencil points do not lie along a single line. Can you
place a flat sheet of paper or cardboard so that it touches all three pencil
points? yes
Read the introduction to Lesson 1-1 at the top of page 6 in your textbook.
Why do chairs sometimes wobble?
Points, Lines, and Planes
Reading the Lesson
©
____________ PERIOD _____
Reading to Learn Mathematics
Pre-Activity
1-1
NAME ______________________________________________ DATE
Enrichment
©
4. Make the capital letter O so that it
extends to each side of the matrix.
2. Draw two lines that cross but have
no common points.
____________ PERIOD _____
Glencoe/McGraw-Hill
6
Glencoe Geometry
5. Using separate grid paper, make dot designs for several other letters. Which were the
easiest and which were the most difficult? See students’ work.
3. Make the number 0 (zero) so that it
extends to the top and bottom sides
of the matrix.
1. Draw two intersecting lines that have
four points in common.
Draw points on each matrix to create the given figures.
Answers may vary. Sample answers are shown.
Dot-matrix printers for computers used dots to form characters.
The dots are often called pixels. The matrix at the right shows
how a dot-matrix printer might print the letter P.
A matrix is a rectangular array of rows and columns. Points and
lines on a matrix are not defined in the same way as in Euclidean
geometry. A point on a matrix is a dot, which can be small or
large. A line on a matrix is a path of dots that “line up.” Between
two points on a line there may or may not be other points. Three
examples of lines are shown at the upper right. The broad line can
be thought of as a single line or as two narrow lines side by side.
Points and Lines on a Matrix
1-1
NAME ______________________________________________ DATE
Answers
(Lesson 1-1)
Glencoe Geometry
Lesson 1-1
©
____________ PERIOD _____
Linear Measure and Precision
Study Guide and Intervention
Glencoe/McGraw-Hill
1
2
3
N
4
N
.
Find the length of M
in.
A5
in.
1
2
1
B
3
2
2.5 cm
©
Glencoe/McGraw-Hill
1
ᎏᎏ ft or 6 in.
2
8. 2 ft
1
ᎏᎏ in.
2
5. 10 in.
0.05 mm
9. 3.5 mm
0.5 mm
6. 32 mm
7
5
8
cm
in.
1
2
1ᎏᎏ inches long.
7
8
3
1
4
1ᎏᎏ in.
1.7 cm
Glencoe Geometry
1
ᎏᎏ yd or 9 in.
4
10. 2ᎏᎏ yd
1
2
0.5 cm
7. 44 cm
1
T
苶S
苶 is between 1ᎏᎏ inches and
or ᎏᎏ inch, so R
1
8
accurate to within one half of a quarter inch,
2ᎏᎏ in. 4.
1
4
2
is about 1ᎏᎏ inches. The measurement is
2. S
Find the precision for each measurement.
3.
cm
1. A
3
4
1
S
Find the length of R
S
.
The long marks are inches and the short
marks are quarter inches. The length of 苶
RS
苶
R
Example 2
Find the length of each line segment or object.
Exercises
The long marks are centimeters, and the
shorter marks are millimeters. The length of
M
苶N
苶 is 3.4 centimeters. The measurement is
accurate to within 0.5 millimeter, so 苶
MN
苶 is
between 3.35 centimeters and 3.45
centimeters long.
cm
M
Example 1
A part of a line between two endpoints is called a line
segment. The lengths of M
苶N
苶 and R
苶S
苶 are written as MN and RS. When you measure a
segment, the precision of the measurement is half of the smallest unit on the ruler.
Measure Line Segments
1-2
NAME ______________________________________________ DATE
Linear Measure and Precision
A
x
B
2x
C
C
Q
X
R
S
3 1–2 in.
2.0 cm
Y
2.5 cm
3
–
4
Z
in.
T
1
4
4 ᎏᎏ in.
4.5 cm
Y
C
3 cm
Glencoe Geometry
11 cm
11 cm
Glencoe/McGraw-Hill
Answers
©
B
5 cm
A
9. A
苶B
苶 and C
苶D
苶 yes
C
5 cm
D
8
Y
3x ⫹ 5
9x
2
X
10. X
苶Y
苶 and Y
苶Z
苶
Z
5x ⫺ 1
no
Glencoe Geometry
Use the figures to determine whether each pair of segments is congruent.
8. RS ⫽ 4x, 苶
RS
苶⬵苶
ST
苶, and RT ⫽ 24. 3, 12
X
6 cm
2 3–4 in. B
1
4
3 ᎏᎏ in.
7. RS ⫽ 6x, ST ⫽12, and RT ⫽ 72. 10, 60
W
A
6 in.
6. RS ⫽ 2x, ST ⫽ 5x ⫹ 4, and RT ⫽ 32. 4, 8
4. W
苶X
苶
2. B
苶C
苶
5. RS ⫽ 5x, ST ⫽ 3x, and RT ⫽ 48. 6, 30
Find x and RS if S is between R and T.
3. X
苶Z
苶
1. R
苶T
苶
Find the measurement of each segment. Assume that the art is not drawn to scale.
Exercises
Therefore, 苶
EF
苶 is 3.1 centimeters long.
AB ⫹ BC ⫽ AC
x ⫹ 2x ⫽ 2x ⫹ 5
3x ⫽ 2x ⫹ 5
x⫽5
AC ⫽ 2x ⫹ 5 ⫽ 2(5) ⫹ 5 ⫽ 15
F
Find x and AC.
ED ⫹ DF ⫽ EF
1.2 ⫹ 1.9 ⫽ EF
3.1 ⫽ EF
D
2x ⫹ 5
Example 2
B
M
B is between A and C.
E
Find EF.
1.9 cm
A
P
Calculate EF by adding ED and DF.
1.2 cm
Example 1
Calculate Measures
(continued)
____________ PERIOD _____
Study Guide and Intervention
On ៭៮៬
PQ, to say that point M is
between points P and Q means P, Q, and M are collinear
and PM ⫹ MQ ⫽ PQ.
On ៭៮៬
AC, AB ⫽ BC ⫽ 3 cm. We can say that the segments are
congruent, or 苶
AB
苶⬵苶
BC
苶. Slashes on the figure indicate which
segments are congruent.
1-2
NAME ______________________________________________ DATE
Answers
(Lesson 1-2)
Lesson 1-2
©
Linear Measure and Precision
Skills Practice
Glencoe/McGraw-Hill
1
2
about 55 mm
cm
3
4
5
2.
0.5 cm
A6
1
4
1in.
P
11–4 in.
N
B
10.1 cm
A
4.9 cm
7. A
苶C
苶
1
5.2 cm
C
©
5m
D
3m
Glencoe/McGraw-Hill
yes
E
3m
B 2m C
13. 苶
BE
苶, 苶
CD
苶
12 yd
no
M
12 yd
14. M
苶P
苶, 苶
NP
苶
H
10; 60
12. XY ⫽ 2c ⫹ 1, YZ ⫽ 6c, and XZ ⫽ 81
8; 16
10. XY ⫽ 12, YZ ⫽ 2g, and XZ ⫽ 28
9
N
10 yd
P
no
X
5 ft
Y
9 ft
15. W
苶X
苶, 苶
WZ
苶
Glencoe Geometry
W
5 ft
Z
Use the figures to determine whether each pair of segments is congruent.
6; 18
11. XY ⫽ 4m, YZ ⫽ 3m, and XZ ⫽ 42
5; 5
9. XY ⫽ 5p, YZ ⫽ p, and XY ⫽ 25
G
15 mm
9.7 mm
5.3 mm
F
8. G
苶H
苶
1
ᎏᎏ in.
4
1
5. 9ᎏᎏ inches
2
2
____________ PERIOD _____
Find the value of the variable and YZ if Y is between X and Z.
2ᎏᎏ in.
Q
6. N
苶Q
苶
in.
1
about 2ᎏᎏ in.
4
4. 12 centimeters
Find the measurement of each segment.
0.5 ft
3. 40 feet
Find the precision for each measurement.
1.
Find the length of each line segment or object.
1-2
NAME ______________________________________________ DATE
(Average)
1
F
2
0.5 m
18.4 cm
Q
S
4.7 cm
5
8
23–8 in.
3ᎏᎏ in.
A
7. A
苶D
苶
C
D
2
5
X
89.6 cm
100 cm
10.4cm
W
8. W
苶X
苶
0.5 mm
6; 8
3 ft
W
3 ft
yes
D
A
12. A
苶D
苶, 苶
BC
苶
12.9 in.
12.7 in.
B
C
©
Glencoe/McGraw-Hill
C
B
F
E
, A
B
C
D
D
E
F
A
10
5x
F
no
G
13. G
苶F
苶, 苶
FE
苶
14. CARPENTRY Jorge used the figure at the right to make a pattern
for a mosaic he plans to inlay on a tabletop. Name all of the
congruent segments in the figure.
no
U
2 ft
T 2 ft S
11. 苶
TU
苶, 苶
SW
苶
Y
H
D
A
C
Glencoe Geometry
E
F
E
6x
B
10. JK ⫽ 2s, KL ⫽ s ⫹ 2, and JL ⫽ 5s ⫺ 10
Use the figures to determine whether each pair of segments is congruent.
3; 9
9. JK ⫽ 6r, KL ⫽ 3r, and JL ⫽ 27
4
5. 30.0 millimeters
3
____________ PERIOD _____
Find the value of the variable and KL if K is between J and L.
23.1 cm
P
苶S
苶
6. P
Find the measurement of each segment.
1
ᎏᎏ in.
8
1
4
4. 7ᎏᎏ inches
3. 120 meters
42 mm
cm
11–4 in.
2.
Find the precision for each measurement.
11
16
in.
1ᎏᎏ in.
1. E
1
Linear Measure and Precision
Practice
Find the length of each line segment or object.
1-2
NAME ______________________________________________ DATE
Answers
(Lesson 1-2)
Glencoe Geometry
Lesson 1-2
©
Glencoe/McGraw-Hill
A7
4.5 cm
A
B
C
4.5 cm
Glencoe/McGraw-Hill
century, centennial
11
Glencoe Geometry
6. A good way to remember terms used in mathematics is to relate them to everyday words
you know. Give three words that are used outside of mathematics that can help you
remember that there are 100 centimeters in a meter. Sample answer: cent,
Helping You Remember
5. Suppose that S is a point on V
苶W
苶 and S is not the same point as V or W. Tell whether
each of the following statements is always, sometimes, or never true.
a. VS ⫽ SW sometimes
b. S is between V and W. always
c. VS ⫹ VW ⫽ SW never
AB
C
D
; Sample answer: The two segments are
congruent because they have the same measure or
length. They are not equal because they are not the
same segment.
4. Refer to the figure at the right. Which one of the following
statements is true? Explain your answer.
A
CD
苶B
苶⫽苶
CD
苶
苶B
A
苶⬵苶
苶
3. Find the precision of each measurement.
a. 15 cm 0.5 cm
b. 15.0 cm 0.05 cm
2. What is the smallest length marked on a 12-inch ruler? Sample answer: ᎏᎏ in.
16
What is the smallest length marked on a centimeter ruler? 1 mm
1
Sample answer: A line is infinite. Since it has no endpoints, a line does
not have a definite length and cannot be measured. A line segment has
two endpoints, so it has a definite length and can be measured.
1. Explain the difference between a line and a line segment and why one of these can be
measured, while the other cannot.
D
The metric system is easier because you can change
between the different units by just moving the decimal point.
• Do you think it would be easier to learn the relationships between the
different units of length in the customary system (used in the United
States) or in the metric system? Explain your answer. Sample answer:
• The basic unit of length in the metric system is the meter. How many
meters are there in one kilometer? 1000
Read the introduction to Lesson 1-2 at the top of page 13 in your textbook.
Why are units of measure important?
Linear Measure and Precision
Reading the Lesson
©
____________ PERIOD _____
Reading to Learn Mathematics
Pre-Activity
1-2
NAME ______________________________________________ DATE
Enrichment
A
X
Y
D
B
X
C
D
The locus is a set of
2 points, X and Y.
A
Y
B
Glencoe Geometry
Answers
Glencoe/McGraw-Hill
12
2. Conduct your own investigation of the locus of points
equidistant from two segments. Describe your results on a
separate sheet of paper. See students’ work.
©
P
C
D
Q
S
The locus is a pair
of line segments,
S
R
and P
Q
.
A
R
B
Glencoe Geometry
B
and C
D
depends on the
The locus of points x units from A
B
and C
D
are situated relative to one another.
distance x and how A
The locus is 1
segment X
Y
midway
B
and C
D
.
between A
C
A
1. Suppose A, B, C, and D are four different points, and consider the locus
苶B
苶 and x units from 苶
CD
苶. Use any unit you find
of all points x units from A
convenient. The locus can take different forms. Sketch at least three
possibilities. List some of the things that seem to affect the form of
the locus. Sample answers are shown.
from the segment AB is shown by two dashed segments
with semicircles at both ends.
a set of conditions. The locus of all points that are ᎏᎏ inch
1
4
A figure is a locus if it is the set of all points that satisfy
B
____________ PERIOD _____
The distance from a point to a segment is zero if the point is on the
segment. Otherwise, it is the length of the shortest segment from the
point to the segment.
Points Equidistant from Segments
1-2
NAME ______________________________________________ DATE
Answers
(Lesson 1-2)
Lesson 1-2
©
Distance and Midpoints
Glencoe/McGraw-Hill
b
a
0
A8
1
2
B
3
Find AB.
A(–2, –1)
O
x
C (1, –1)
B(1, 3)
苶
AB ⫽ 兹25
⫽5
(AB)2 ⫽ (AC)2 ⫹ (BC)2
(AB)2 ⫽ (3)2 ⫹ (4)2
(AB)2 ⫽ 25
Pythagorean Theorem
4. EF 3
6. AG 17
8. DE 1
5. BG 15
7. BE 7
–10 –8
–6
B
–4
C
–2
0
DE
2
F
4
6
12. E(⫺12, 2), F(⫺9, 6) 5
10. R(⫺2, 3), S(3, 15) 13
©
Glencoe/McGraw-Hill
15. C(11, ⫺12), D(6, 2)
221
14.9
13. A(0, 0), B(15, 20) 25
13
16. E(⫺2, 10), F(⫺4, 3)
8
G
Glencoe Geometry
53
7.3
14. O(⫺12, 0), P(⫺8, 3) 5
Use the Distance Formula to find the distance between each pair of points.
11. M(1, ⫺2), N(9, 13) 17
9. A(0, 0), B(6, 8) 10
Use the Pythagorean Theorem to find the distance between each pair of points.
2. DG 9
1. BD 6
3. AF 12
A
⫽ 兹25
苶
⫽5
AB ⫽ 兹苶
(3)2 ⫹苶
(4)2
AB ⫽ 兹苶
(1 ⫺ (苶
⫺2))2 苶
⫹ (3 ⫺苶
(⫺1))2苶
d ⫽ 兹苶
(x2 ⫺ 苶
x1)2 ⫹苶
(y2 ⫺苶
y1)2
Distance Formula
Find the distance between
A(⫺2, ⫺1) and B(1, 3).
Example 2
(x2 ⫺ 苶
x1)2 ⫹苶
(y2 ⫺苶
y1)2
d ⫽ 兹苶
Distance Formula:
a2 ⫹ b2 ⫽ c2
Pythagorean Theorem:
Distance in the Coordinate Plane
Use the number line to find each measure.
Exercises
AB ⫽ | (⫺4) ⫺ 2 |
⫽ |⫺ 6 |
⫽6
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
A
Example 1
AB ⫽ | b ⫺ a | or | a ⫺ b |
B
A
Distance on a Number Line
y
____________ PERIOD _____
Study Guide and Intervention
Distance Between Two Points
1-3
NAME ______________________________________________ DATE
Distance and Midpoints
0
1
–1
–3
2
y ⫹y
2
冣
⫺3 ⫹ 1
2
⫺2
2
x ⫹x
2
y ⫹y
2
冣 冢 ⫺22⫹ 4
4⫹1
2
冣
1
2
8. D
苶E
苶 1
A
–10 –8
–6
B
C
–4
–2
0
D
2
EF
4
6
©
Glencoe/McGraw-Hill
13. S(10, ⫺22), T(9, 10) (9.5, ⫺6)
11. M(11, ⫺2), N(⫺9, 13) (1, 5.5)
9. A(0, 0), B(12, 8) (6, 4)
14
8
G
Glencoe Geometry
14. M(⫺11, 2), N(⫺19, 6) (⫺15, 4)
12. E(⫺2, 6), F(⫺9, 3) (⫺5.5, 4.5)
10. R(⫺12, 8), S(6, 12) (⫺3, 10)
Find the coordinates of the midpoint of a segment having the given endpoints.
7. B
苶D
苶 ⫺3ᎏᎏ
1
2
5. A
苶B
苶 ⫺8
6. B
苶G
苶 ᎏᎏ
4. E
苶G
苶 5
2. D
苶G
苶 4
3. A
苶F
苶 ⫺3
1. C
苶E
苶 ⫺1
Use the number line to find the coordinate of
the midpoint of each segment.
Exercises
1
2
1
2
ᎏ
ᎏ
M⫽ ᎏ
,ᎏ
⫽ ᎏᎏ, ᎏᎏ or (1, 2.5)
冢
Example 2 M is the midpoint of PQ
for P(⫺2, 4) and Q(4, 1). Find the
coordinates of M.
苶, then the coordinate of M is ᎏᎏ ⫽ ᎏᎏ or ⫺1.
If M is the midpoint of 苶
PQ
The coordinates of P and Q are ⫺3 and 1.
–2
x ⫹x
2
1
2
1
2
ᎏ
ᎏ
,ᎏ
.
then the coordinates of the midpoint of the segment are ᎏ
冢
If a segment has endpoints with coordinates (x1, y1) and (x2, y2),
2
a ⫹ᎏ
b.
then the coordinate of the midpoint of the segment is ᎏ
If the coordinates of the endpoints of a segment are a and b,
Find the coordinate of the midpoint of P
Q
.
Q
P
Example 1
Midpoint on a
Coordinate Plane
Midpoint on a
Number Line
(continued)
____________ PERIOD _____
Study Guide and Intervention
Midpoint of a Segment
1-3
NAME ______________________________________________ DATE
Answers
(Lesson 1-3)
Glencoe Geometry
Lesson 1-3
©
Distance and Midpoints
Skills Practice
Glencoe/McGraw-Hill
4. MN 3
3. KN 11
–6
J
–4
K
–2
0
2
L
4
6
M
8
N
10
____________ PERIOD _____
F
O
y
G
x
5
D
O
y
x
S
17
4.1
8. C(⫺3, ⫺1), Q(⫺2, 3)
6.
10
A9
–6
–4
–2
B
0
2
C
4
6
D
8
10
E
1
ᎏᎏ, 0
2
18. J(⫺4, 2), F(5, ⫺2)
12
©
Glencoe/McGraw-Hill
Q (8, 4)
19. N(2, 0), P(5, 2)
Q (7, 2)
15
20. N(5, 4), P(6, 3)
N (⫺5, 1)
Glencoe Geometry
21. Q(3, 9), P(⫺1, 5)
Find the coordinates of the missing endpoint given that P is the midpoint of N
Q
.
(4, 2)
17. T(3, 1), U(5, 3)
Find the coordinates of the midpoint of a segment having the given endpoints.
16. A
苶D
苶 1ᎏᎏ
15. 苶
BD
苶 3
1
2
14. B
苶C
苶 1
13. 苶
DE
苶 9
A
50
7.1
12. C(⫺3, 1), Q(2, 6)
7
10. W(⫺2, 2), R(5, 2)
Use the number line to find the coordinate
of the midpoint of each segment.
13
11. A(⫺7, ⫺3), B(5, 2)
6
9. Y(2, 0), P(2, 6)
Use the Distance Formula to find the distance between each pair of points.
5
2.2
7. K(2, 3), F(4, 4)
5.
Use the Pythagorean Theorem to find the distance between each pair of points.
2. JL 8
1. LN 6
Use the number line to find each measure.
1-3
NAME ______________________________________________ DATE
(Average)
Distance and Midpoints
Practice
4. SV 8
2. TV 5
–10
–8
S
–6
T
–4
–2
U
0
V
2
4
W
6
8
____________ PERIOD _____
O
65
8.1
M
y
Z
x
6.
113
10.6
E
O
y
x
S
1
2
P
–8
Q
–6
–4
R
–2
0
S
2
4
T
(⫺10, ⫺5.5)
14. W(⫺12, ⫺7), T(⫺8, ⫺4)
6
D (⫺4, 3)
16. F(2, 9), E(⫺1, 6)
F (5, 4)
17. D(⫺3, ⫺8), E(1, ⫺2)
Glencoe Geometry
Glencoe/McGraw-Hill
Answers
©
16
Glencoe Geometry
18. PERIMETER The coordinates of the vertices of a quadrilateral are R(⫺1, 3), S(3, 3),
T(5, ⫺1), and U(⫺2, ⫺1). Find the perimeter of the quadrilateral. Round to the
nearest tenth. 19.6 units
D (3, ⫺2)
15. F(5, 8), E(4, 3)
Find the coordinates of the missing endpoint given that E is the midpoint of D
F
.
(⫺2, 5)
13. K(⫺9, 3), H(5, 7)
Find the coordinates of the midpoint of a segment having the given endpoints.
12. P
苶R
苶 ⫺5 ᎏᎏ
11. 苶
ST
苶 2ᎏᎏ
1
2
10. Q
苶R
苶 ⫺4
9. R
苶T
苶 1
–10
18
4.2
8. U(1, 3), B(4, 6)
Use the number line to find the coordinate
of the midpoint of each segment.
15
7. L(⫺7, 0), Y(5, 9)
Use the Distance Formula to find the distance between each pair of points.
5.
Use the Pythagorean Theorem to find the distance between each pair of points.
3. ST 3
1. VW 4
Use the number line to find each measure.
1-3
NAME ______________________________________________ DATE
Answers
(Lesson 1-3)
Lesson 1-3
©
Glencoe/McGraw-Hill
A10
ii. Distance Formula in the Coordinate Plane
⫹
i
冣 iii
v. Midpoint of a Segment on a Number Line
iv. Distance Formula on a Number Line
兹苶苶苶苶
( ⫺2 ⫺ 4 )2 ⫹ ( 7 ⫺ ⫺3 )2
兹苶苶
( ⫺6 )2 ⫹ ( 10 )2
36 ⫹ 100
兹苶苶
136
兹苶
MN ⫽
MN ⫽
MN ⫽
MN ⫽
7 ).
Glencoe/McGraw-Hill
17
Glencoe Geometry
segment determined by the points is neither horizontal nor vertical, draw
a right triangle that has the segment as its hypotenuse. The horizontal
side will have length |x2 ⫺ x1| and the vertical side will have length
|y2 ⫺ y1|. By the Pythagorean Theorem, d 2 ⫽ |x2 ⫺ x1| 2 ⫹ | y2 ⫺ y1| 2 ⫽
(x2 ⫺ x1) 2 ⫹ (y2 ⫺ y1) 2.
3. A good way to remember a new formula in mathematics is to relate it to one you already
know. If you forget the Distance Formula, how can you use the Pythagorean Theorem to
find the distance d between two points on a coordinate plane? Sample answer: If the
Helping You Remember
Find a decimal approximation for MN to the nearest hundredth. 11.66
兹苶苶苶苶
( x2 ⫺ x1 )2 ⫹ ( y2 ⫺ y1 )2
d⫽
Let (x1, y1) ⫽ (4, ⫺3). Then (x2, y2) ⫽ ( ⫺2 ,
2. Fill in the steps to calculate the distance between the points M(4, ⫺3) and N(⫺2, 7).
冢
e.
⫽
x1 ⫹ x2 y1 ⫹ y2
ᎏᎏ, ᎏᎏ
2
2
d.
b2
c2
a2
c. XY ⫽ | a ⫺ b | iv
iii. Midpoint of a Segment in the Coordinate Plane
a⫹b
b. ᎏᎏ
2
v
i. Pythagorean Theorem
a. d ⫽ 兹苶
(x2 ⫺ 苶
x1)2 ⫹苶
( y2 ⫺苶
y1)2 ii
1. Match each formula or expression in the first column with one of the names in the
second column.
5
F
J
6. J
苶苶
K 5
5. 苶I苶
K 7.6
7. L
苶M
苶 4.1
3. 苶
RS
苶 4.2
E
K
D
S
8. L
苶苶
N 7.2
4. Q
苶S
苶 5.8
N
F
B
A
Glencoe/McGraw-Hill
15. Use your answer from exercise 8 to
calculate the length of segment LN
in centimeters. Check by measuring
with a centimeter ruler. 3.6 cm
13. Of all the segments shown on the
grid, which is longest? What is its
length? BC ⴝ 8.1
©
M
18
Glencoe Geometry
16. Use a centimeter ruler to find the perimeter
of triangle IJK to the nearest tenth of a
centimeter. 7.8 cm
Divide by 2 or multiply by 0.5.
14. On the grid, 1 unit ⫽ 0.5 cm. How can the
answers above be used to find the measures
in centimeters?
10. 䉭QRS 18
11. 䉭 DEF 16.6
12. 䉭 LMN 18.3
Answers shown are found by rounding segment lengths before adding.
9. 䉭 ABC 20.2
L
Q
R
____________ PERIOD _____
Use the grid above. Find the perimeter of each triangle to the nearest tenth
of a unit.
2. M
苶苶
N 7
苶 3
1. 苶IJ
Find each measure to the nearest tenth of a unit.
EF ⫽ 兹苶
22 ⫹ 52苶 ⫽ 兹29
苶 ⬇ 5.4 units
2
E
Example
Find the measure of
E
F
on the grid at the right. Locate
a right triangle with EF
as its
longest side.
I
C
You can easily find segment lengths on
a grid if the endpoints are grid-line
intersections. For horizontal or vertical
segments, simply count squares. For
diagonal segments, use the Pythagorean
Theorem (proven in Chapter 7). This
theorem states that in any right triangle,
if the length of the longest side (the side
opposite the right angle) is c and the two
shorter sides have lengths a and b, then
c2 ⫽ a2 ⫹ b2.
• Find AB in this figure. Write your answer both as a radical and as a
decimal number rounded to the nearest tenth. 61
units; 7.8 units
Evenly-spaced horizontal and vertical lines form a grid.
• Look at the triangle in the introduction to this lesson. What is the special
苶B
苶 in this triangle? hypotenuse
name for A
Enrichment
Lengths on a Grid
1-3
NAME ______________________________________________ DATE
Read the introduction to Lesson 1-3 at the top of page 21 in your textbook.
Lesson 1-3
How can you find the distance between two points without a ruler?
Distance and Midpoints
Reading the Lesson
©
____________ PERIOD _____
Reading to Learn Mathematics
Pre-Activity
1-3
NAME ______________________________________________ DATE
Answers
(Lesson 1-3)
Glencoe Geometry
©
Angle Measure
Glencoe/McGraw-Hill
S
P
1 2
R
3
Q
T
A11
B
©
Glencoe/McGraw-Hill
6. ⬔NPS 45; acute
5. ⬔RPN 90; right
19
C
B
E
C
D
1
M
2
A
B
R
S
3
C
Glencoe Geometry
P
N
4
c. ⬔EBC
Using a protractor, m⬔EBC ⫽ 90.
⬔EBC is a right angle.
b. ⬔DBC
Using a protractor, m⬔DBC ⫽ 115.
180 ⬎ 115 ⬎ 90, so ⬔DBC is an obtuse
angle.
a. ⬔ABD
Using a protractor, m⬔ABD ⫽ 50.
50 ⬍ 90, so ⬔ABD is an acute angle.
A
D
Measure each angle in the figure and classify it as right,
acute, or obtuse.
4. ⬔MPR 120; obtuse
A
Measure each angle and
classify it as right, acute, or obtuse.
Example 2
3. Write another name for ⬔DBC. ⬔3 or ⬔CBD
៮៮៬, DC
៮
៮ ៬
2. Name the sides of ⬔BDC. DB
1. Name the vertex of ⬔4.
Refer to the figure.
Exercises
b. Name the sides of ⬔1.
៮៮៬, RP
៮៮៬
RS
a. Name all angles that have R as a
vertex.
Three angles are ⬔1, ⬔2, and ⬔3. For
other angles, use three letters to name
them: ⬔SRQ, ⬔PRT, and ⬔SRT.
Example 1
A right angle is an angle whose measure is 90. An acute angle
has measure less than 90. An obtuse angle has measure greater
than 90 but less than 180.
1
B
____________ PERIOD _____
Study Guide and Intervention
Measure Angles If two noncollinear rays have a common
endpoint, they form an angle. The rays are the sides of the angle.
The common endpoint is the vertex. The angle at the right can be
named as ⬔A, ⬔BAC, ⬔CAB, or ⬔1.
1-4
NAME ______________________________________________ DATE
m⬔NPR ⫽ (2x ⫹ 14) ⫹ (x ⫹ 34)
⫽ 54 ⫹ 54
⫽ 108
Glencoe Geometry
Glencoe/McGraw-Hill
Answers
©
A
P
D
S
M
1
T
B
4
E
2 3
Q
P
N
C
F
R
R
20
Glencoe Geometry
៮៮៬ and BF
៮៮៬ are bisectors, m⬔2 ⫹ m⬔3 must equal half the
Yes; since BD
total angle measure, and half of 180 is 90.
6. Is ⬔DBF a right angle? Explain.
100
5. If m⬔2 ⫽ 6y ⫹ 2 and m⬔1 ⫽ 8y ⫺ 14, find m⬔ABE.
50
4. If m⬔1 ⫽ 4x ⫹ 10 and m⬔2 ⫽ 5x, find m⬔2.
80
3. If m⬔EBF ⫽ 6x ⫹ 4 and m⬔CBF ⫽ 7x ⫺ 2, find m⬔EBC.
៮៮៬ and BC
៮៮៬ are opposite rays, BF
៮៮៬ bisects ⬔CBE, and
BA
៮៮៬ bisects ⬔ABE.
BD
56
2. If m⬔PQS ⫽ 3x ⫹ 13 and m⬔SQT ⫽ 6x ⫺ 2, find m⬔PQT.
4
1. If m⬔PQT ⫽ 60 and m⬔PQS ⫽ 4x ⫹ 14, find the value of x.
៮៮៬ bisects ⬔PQT, and QP
៮៮៬ and QR
៮៮៬ are opposite rays.
QS
Exercises
2x ⫹ 14 ⫽ x ⫹ 34
2x ⫹ 14 ⫺ x ⫽ x ⫹ 34 ⫺ x
x ⫹ 14 ⫽ 34
x ⫹ 14 ⫺ 14 ⫽ 34 ⫺ 14
x ⫽ 20
Example
Refer to the figure above. If m⬔MPN ⫽ 2x ⫹ 14 and
m⬔NPR ⫽ x ⫹ 34, find x and find m⬔MPR.
Since ៮៮៬
PN bisects ⬔MPR, ⬔MPN ⬵ ⬔NPR, or m⬔MPN ⫽ m⬔NPR.
Angles that have the same measure are
congruent angles. A ray that divides an angle into two congruent
angles is called an angle bisector. In the figure, ៮៮៬
PN is the angle
bisector of ⬔MPR. Point N lies in the interior of ⬔MPR and
⬔MPN ⬵ ⬔NPR.
Angle Measure
(continued)
____________ PERIOD _____
Study Guide and Intervention
Congruent Angles
1-4
NAME ______________________________________________ DATE
Answers
(Lesson 1-4)
Lesson 1-4
©
Angle Measure
Skills Practice
Glencoe/McGraw-Hill
V
3. ⬔2
A12
T
W
100⬚, obtuse
16. ⬔QMO
40⬚, acute
14. ⬔OMN
©
Glencoe/McGraw-Hill
55
21
18. If m⬔ABF ⫽ 7x ⫺ 8 and m⬔EBF ⫽ 5x ⫹ 10,
find m⬔EBF.
40
17. If m⬔EBD ⫽ 4x ⫹ 16 and m⬔DBC ⫽ 6x ⫹ 4,
find m⬔EBD.
៮៮៬ bisects ⬔EBC, and BF
៮៮៬ bisects ⬔ABE.
BD
៮៮៬ are opposite rays,
៮៮៬ and BC
ALGEBRA In the figure, BA
140⬚, obtuse
15. ⬔QMN
90⬚, right
13. ⬔NMP
S
W
1
5
U
T
2V
3
4
L
Q
A
F
B
E
C
D
N
Glencoe Geometry
M
P
O
____________ PERIOD _____
⬔WVT, ⬔TVW, ⬔WVU, ⬔UVW
12. ⬔2
⬔UTS, ⬔STU
10. ⬔4
៮៮៬, WV
៮៮៬
WT
8. ⬔1
៮៮៬, TW
៮៮៬
TS
6. ⬔5
4. ⬔5
2. ⬔1
Measure each angle and classify it as right, acute,
or obtuse.
⬔STW, ⬔5
11. ⬔WTS
⬔WTV, ⬔VTW
9. ⬔3
Write another name for each angle.
៮៮៬, TV
៮៮៬
TS
7. ⬔STV
៮៮៬, TS
៮៮៬
TU
5. ⬔4
Name the sides of each angle.
T
1. ⬔4
Name the vertex of each angle.
For Exercises 1–12, use the figure at the right.
1-4
NAME ______________________________________________ DATE
(Average)
Angle Measure
Practice
O
M
P
20⬚, acute
14. ⬔UZT
70⬚, acute
12. ⬔YZW
©
Glencoe/McGraw-Hill
22
m⬔1 ⫽ 90, right angle; m⬔2 ⫽ 130, obtuse
17. TRAFFIC SIGNS The diagram shows a sign used to warn
drivers of a school zone or crossing. Measure and classify
each numbered angle.
16. If m⬔FCG ⫽ 9x ⫹ 3 and m⬔GCB ⫽ 13x ⫺ 9,
find m⬔GCB. 30
15. If m⬔DCE ⫽ 4x ⫹ 15 and m⬔ECF ⫽ 6x ⫺ 5,
find m⬔DCE. 55
៮៮៬ bisects ⬔DCF, and CG
៮៮៬ bisects ⬔FCB.
CE
M
5
4
6
R
T
U
F
E
G
V
Y
Glencoe Geometry
1
2
B
C
D
Z
W
X
7 O
8
1 P
Q
2 3
N
____________ PERIOD _____
⬔MPO, ⬔OPM, ⬔MPN, ⬔NPM
10. ⬔1
៮៮៬, MN
៮៮៬
MO
8. ⬔OMN
៮៮៬ are opposite rays,
៮៮៬ and CD
ALGEBRA In the figure, CB
110⬚, obtuse
13. ⬔TZW
90⬚, right
11. ⬔UZW
M
៮៮៬, PM
៮៮៬
PR
6. ⬔2
4. ⬔NMP
2. ⬔3
Measure each angle and classify it as right, acute,
or obtuse.
⬔3, ⬔RPQ
9. ⬔QPR
Write another name for each angle.
៮៮៬, OP
៮៮៬ or OR
៮៮៬
OM
7. ⬔MOP
៮៮៬, NO
៮៮៬ or NP
៮៮៬ or NR
៮៮៬
NM
5. ⬔6
Name the sides of each angle.
3. ⬔8
1. ⬔5
Name the vertex of each angle.
For Exercises 1–10, use the figure at the right.
1-4
NAME ______________________________________________ DATE
Answers
(Lesson 1-4)
Glencoe Geometry
Lesson 1-4
©
Glencoe/McGraw-Hill
A13
Glencoe/McGraw-Hill
23
Glencoe Geometry
Sample answer: Congruent segments and congruent angles are alike
because they both involve a pair of figures with the same measure. They
are different because congruent segments have the same length, which
can be measured in units such as inches or centimeters, while congruent
angles have the same degree measure.
3. A good way to remember related geometric ideas is to compare them and see how they
are alike and how they are different. Give some similarities and differences between
congruent segments and congruent angles.
Helping You Remember
1. Match each description in the first column with one of the terms in the second column.
Some terms in the second column may be used more than once or not at all.
a. a figure made up of two noncollinear rays with a
1. vertex
common endpoint 4
2. angle bisector
b. angles whose degree measures are less than 90 8
3. opposite rays
c. angles that have the same measure 6
4. angle
d. angles whose degree measures are between 90 and 180 5
5. obtuse angles
e. a tool used to measure angles 10
6. congruent angles
f. the common endpoint of the rays that form an angle 1
7. right angles
g. a ray that divides an angle into two congruent angles 2
8. acute angles
9. compass
10. protractor
2. Use the figure to name each of the following.
E
a. a right angle ⬔ABE or ⬔EBG
F
D
b. an obtuse angle ⬔ABF or ⬔ABC
28⬚
28⬚ C
c. an acute angle ⬔EBF, ⬔FBC, ⬔CBG, ⬔EBC, or ⬔FBG
d. a point in the interior of ⬔EBC F
A
B
G
e. a point in the exterior of ⬔EBA F, C, or G
៮៮៬
f. the angle bisector of ⬔EBC BF
g. a point on ⬔CBE C, B, or E
៮៮៬ and BF
៮៮៬
h. the sides of ⬔ABF BA
៮៮៬ and BG
៮៮៬
i. a pair of opposite rays BA
j. the common vertex of all angles shown in the figure B
k. a pair of congruent angles ⬔EBF and ⬔FBC, or ⬔ABE and ⬔EBG
l. the angle with the greatest measure ⬔ABG
55⬚30⬘15⬙
5. 34⬚29⬘45⬙
62⬚44⬘
2. 27⬚16⬘
2⬚57⬘57⬙
6. 87⬚2⬘3⬙
74⬚06⬘
3. 15⬚54⬘
Glencoe Geometry
Glencoe/McGraw-Hill
Answers
©
81⬚7⬘1⬙
13. 98⬚52⬘59⬙
24
170⬚57⬘28⬙
14. 9⬚2⬘32⬙
140⬚38⬘6⬙
11. 39⬚21⬘54⬙
10. 45⬚16⬘24⬙
134⬚43⬘36⬙
95⬚48⬘
8. 84⬚12⬘
59⬚42⬘
7. 120⬚18⬘
Glencoe Geometry
178⬚57⬘57⬙
15. 1⬚2⬘3⬙
50⬚41⬘24⬙
12. 129⬚18⬘36⬙
69⬚58⬘
9. 110⬚2⬘
Two angles are supplementary if the sum of their measures is 180⬚.
Find the supplement of each of the following angles.
60⬚41⬘38⬙
4. 29⬚18⬘22⬙
54⬚45⬘
1. 35⬚15⬘
Two angles are complementary if the sum of their measures is 90⬚.
Find the complement of each of the following angles.
90⬚ ⫽ 89°60⬘
70.4⬚ ⫽ 70°24⬘
67ᎏᎏ⬚ ⫽ 67⬚30⬘
1
2
60⬙ ⫽ 1⬘
60⬘ ⫽ 1⬚
Angles are measured in degrees (⬚). Each degree of an angle is divided into 60 minutes (⬘), and each minute of an angle is
divided into 60 seconds (⬙).
Enrichment
____________ PERIOD _____
Angle Relationships
1-4
NAME ______________________________________________ DATE
Read the introduction to Lesson 1-4 at the top of page 29 in your textbook.
• A semicircle is half a circle. How many degrees are there in a
semicircle? 180
• How many degrees are there in a quarter circle? 90
Lesson 1-4
How big is a degree?
Angle Measure
Reading the Lesson
©
____________ PERIOD _____
Reading to Learn Mathematics
Pre-Activity
1-4
NAME ______________________________________________ DATE
Answers
(Lesson 1-4)
©
____________ PERIOD _____
Angle Relationships
Study Guide and Intervention
Glencoe/McGraw-Hill
A14
T
U
5
B
6
C
Exercises
⬔6 and ⬔5 are adjacent angles whose
noncommon sides are opposite rays.
The angles form a linear pair.
A
D
⬔SRT and ⬔TRU have a common
vertex and a common side, but no
common interior points. They are
adjacent angles.
R
S
d.
b.
4
3N
1
2
S
A
60⬚
B
adjacent
vertical
©
G
60⬚
V
P
1
U
2
T
V
R
6Q
3 4
5
T
N
U
S
R
S
Glencoe/McGraw-Hill
36 and 54
25
Glencoe Geometry
8. Find the measures of two complementary angles if the difference in their measures is 18.
and ⬔VNT or
⬔VNT and ⬔TNU
7. Identify an angle supplementary to ⬔TNU. ⬔UNS or ⬔TNR
6. Identify two acute adjacent angles. ⬔RNV
5. Identify two obtuse vertical angles. ⬔RNT and ⬔SNU
For Exercises 5–7, refer to the figure at the right.
4. ⬔3 and ⬔2
F
120⬚
⬔A and ⬔B are two angles whose measures
have a sum of 90. They are complementary.
⬔F and ⬔G are two angles whose measures
have a sum of 180. They are supplementary.
30⬚
linear pair; adjacent
2. ⬔1 and ⬔6
3. ⬔1 and ⬔5
adjacent
1. ⬔1 and ⬔2
P
⬔1 and ⬔3 are nonadjacent angles formed
by two intersecting lines. They are vertical
angles. ⬔2 and ⬔4 are also vertical angles.
M
R
Identify each pair of angles as adjacent, vertical, and/or
as a linear pair.
c.
a.
Example
Identify each pair of angles as adjacent angles, vertical angles,
and/or as a linear pair.
Adjacent angles are angles in the same plane that have a common
vertex and a common side, but no common interior points. Vertical angles are two
nonadjacent angles formed by two intersecting lines. A pair of adjacent angles whose
noncommon sides are opposite rays is called a linear pair.
Pairs of Angles
1-5
NAME ______________________________________________ DATE
©
⫽
⫽
⫽
⫽
⫽
m⬔DZP
90
90
84
7
x ⫽ 15, y ⫽ 8
Divide each side by 12.
Subtract 6 from each side.
Simplify.
Substitution
Sum of parts ⫽ whole
Glencoe/McGraw-Hill
26
y ⫽ 15, m⬔RPT ⫽ 55, m⬔TPW ⫽ 35
6. Find y, m⬔RPT, and m⬔TPW.
x ⫽ 8, m⬔PQS ⫽ 24, m⬔SQR ⫽ 66
5. Find x, m⬔PQS, and m⬔SQR.
4. If m⬔EBF ⫽ 7y ⫺ 3 and m⬔FBC ⫽ 3y ⫹ 3, find y so
៮៬ ⊥ ៮៮៬
that ៮EB
BC . 9
៮៬ ⊥ ៮BF
៮៬. Find x.
3. m⬔EBF ⫽ 3x ⫹ 10, m⬔DBE ⫽ x, and ៮BD
2. Find m⬔MSN. 90
៮៬ ⊥ ៭MQ
៮៬.
1. Find x and y so that ៭NR
Exercises
m⬔DZQ ⫹ m⬔QZP
(9x ⫹ 5) ⫹ (3x ⫹ 1)
12x ⫹ 6
12x
x
Example
Find x so that D
Z
⊥P
Z
.
If 苶
DZ
苶⊥苶
PZ
苶, then m⬔DZP ⫽ 90.
x ⫽ 20
W
D
A
N
5x ⬚
T
S
V
Q
Glencoe Geometry
P
R
R
F
C
B
P
C
P
x⬚
(8x ⫹ 2)⬚
3x ⬚
S
E
B
(4y ⫺ 5)⬚
Q
P
D
R
(9y ⫹ 18)⬚ S
Z
A
Q
(9x ⫹ 5)⬚
(3x ⫹ 1)⬚
(2y ⫹ 5)⬚
M
D
Lines, rays, and segments that form four right
angles are perpendicular. The right angle symbol indicates that the lines
៮៬ is perpendicular to ៭៮៬
are perpendicular. In the figure at the right, ៭AC
BD ,
៮៬ ⊥ ៭៮៬
or ៭AC
BD .
Angle Relationships
(continued)
____________ PERIOD _____
Study Guide and Intervention
Perpendicular Lines
1-5
NAME ______________________________________________ DATE
Answers
(Lesson 1-5)
Glencoe Geometry
Lesson 1-5
A15
Angle Relationships
Skills Practice
Glencoe/McGraw-Hill
27
No; the angles do not share a common side.
13. ⬔VZU is adjacent to ⬔YZX.
Yes; the sum of their measures is 180 since the
angles form a linear pair.
12. ⬔YZU and ⬔UZV are supplementary.
Yes; it is marked with a right angle symbol.
11. ⬔WZU is a right angle.
Determine whether each statement can be assumed
from the figure. Explain.
10. If m⬔PTQ ⫽ 3y ⫺ 10 and m⬔QTR ⫽ y, find y so that
⬔PTR is a right angle. 25
៮៮៬. 9
9. If m⬔RTS ⫽ 8x ⫹ 18, find x so that ៮៮៬
TR ⊥ TS
X
Y
P
Q
W
V
U
S
Glencoe Geometry
Z
T
R
8. The measure of the supplement of an angle is 36 less than the measure of the angle.
Find the measures of the angles. 72, 108
7. Find the measures of an angle and its complement if one angle measures 18 degrees
more than the other. 36, 54
6. Name an angle supplementary to ⬔FKG. ⬔EKF or ⬔GKH
5. Name an angle complementary to ⬔EKH. ⬔GKJ
ALGEBRA For Exercises 9–10, use the figure at the right.
©
2. Name a linear pair whose vertex is B. ⬔GBC, ⬔CBA
Sample answer: ⬔GFH, ⬔CFE
1. Name two obtuse vertical angles.
For Exercises 1–4, use the figure at the right and
a protractor.
A
B
D
C
G
E
H
F
Glencoe Geometry
Answers
Glencoe/McGraw-Hill
28
Sample answer: Beacon ⊥ Main; Olive divides two of the
angles formed by Bacon and Main into pairs of
complementary angles.
12. STREET MAPS Darren sketched a map of the cross streets nearest
to his home for his friend Miguel. Describe two different angle
relationships between the streets.
No; the angles are adjacent.
11. ⬔MQN and ⬔MQR are vertical angles.
angles whose noncommon sides are opposite rays.
10. ⬔SRQ and ⬔QRP is a linear pair. Yes; they are adjacent
No; m⬔NQP is not known to be 90.
9. ⬔NQO and ⬔OQP are complementary.
Determine whether each statement can be
assumed from the figure. Explain.
8. If m⬔BGC ⫽ 16x ⫺ 4 and m⬔CGD ⫽ 2x ⫹ 13,
find x so that ⬔BGD is a right angle. 4.5
7. If m⬔FGE ⫽ 5x ⫹ 10, find x so that
៭FC
៮៬ ⊥ ៭៮៬
AE . 16
M
E
G
A
R
Q
O
D
C
Olive
Ma
in
P
Glencoe Geometry
S
N
B
6. If a supplement of an angle has a measure 78 less than the measure of the angle, what
are the measures of the angles? 129, 51
ALGEBRA For Exercises 7–8, use the figure at
the right.
©
F
5. Two angles are complementary. The measure of one angle is 21 more than twice the
measure of the other angle. Find the measures of the angles. 23, 67
4. Name an angle adjacent and supplementary to ⬔DCB. ⬔BCG or ⬔DCH
J
G
(Average)
Angle Relationships
Practice
____________ PERIOD _____
4. Name two acute adjacent angles. ⬔FKG, ⬔GKJ
K
F
1-5
NAME ______________________________________________ DATE
3. Name an angle not adjacent to but complementary to ⬔FGC. ⬔FED
H
E
____________ PERIOD _____
Lesson 1-5
3. Name a linear pair. Sample answer: ⬔EKH, ⬔EKF
2. Name two obtuse vertical angles. ⬔EKF, ⬔HKG
1. Name two acute vertical angles. ⬔EKH, ⬔FKG
For Exercises 1–6, use the figure at the right and
a protractor.
1-5
NAME ______________________________________________ DATE
n
Glencoe/McGraw-Hill
aco
©
Be
Answers
(Lesson 1-5)
©
Glencoe/McGraw-Hill
A16
2n
Glencoe/McGraw-Hill
29
Glencoe Geometry
answer: Supplementary means something added to complete a thing.
An angle and its supplement can be joined to obtain a linear pair.
4. Look up the nonmathematical meaning of supplementary in your dictionary. How can
this definition help you to remember the meaning of supplementary angles? Sample
Helping You Remember
b. If two angles are complementary and x is the measure of one of the angles, then the
90 ⫺ x
measure of the other angle is
.
a. If two angles are supplementary and x is the measure of one of the angles, then the
180 ⫺ x
measure of the other angle is
.
3. Complete each sentence.
i. If two angles form a linear pair, one is acute and the other is obtuse. sometimes
h. The two angles in a linear pair are both acute. never
g. Vertical angles are complementary. sometimes
f. Vertical angles are supplementary. sometimes
e. When two perpendicular lines intersect, four congruent angles are formed. always
d. If two angles are complementary, they are adjacent. sometimes
c. If two angles are supplementary, they are congruent. sometimes
b. If two angles form a linear pair, they are complementary. never
a. If two angles are adjacent angles, they form a linear pair. sometimes
2. Tell whether each statement is always, sometimes, or never true.
⬔4 and ⬔1
g. four pairs of supplementary angles ⬔1 and ⬔2, ⬔2 and ⬔3, ⬔3 and ⬔4,
f. four linear pairs ⬔1 and ⬔2, ⬔2 and ⬔3, ⬔3 and ⬔4, ⬔4 and ⬔1
e. two pairs of vertical angles ⬔1 and ⬔3, ⬔2 and ⬔4
d. four pairs of adjacent angles ⬔1 and ⬔2, ⬔2 and ⬔3, ⬔3 and ⬔4, ⬔4 and ⬔1
c. a pair of obtuse vertical angles ⬔1 and ⬔3
b. a pair of acute vertical angles ⬔2 and ⬔4
a. two pairs of congruent angles ⬔1 and ⬔3, ⬔2 and ⬔4
1. Name each of the following in the figure at the right.
65⬚ 2 3 4
1
• How many separate angles are formed if n lines intersect at a common
point? (Do not count an angle whose interior includes part of another angle.)
• How many separate angles are formed if three lines intersect at a common
point? (Do not use an angle whose interior includes part of another angle.) 6
7
1
5
2
3
3
1
4
5
6
7
8
©
1
7
2
5
3
3
4
5
6
7
8
9
17
19
18
15
16
13
14
11
12
11 12 13 14 15 16 17 18 19
Glencoe/McGraw-Hill
See students’ work.
30
3. Create your own design. You may use several angles, and
the angles may overlap.
9
8
6
4
2
1
2. Complete the following design.
1. Complete the section of the star design above by connecting
pairs of points that have the same number.
To make a complete star, make the same design in
six 60° angles that have a common central vertex.
8
6
4
2
To begin the star design, draw a 60° angle. Mark
eight equally-spaced points on each ray, and
number the points as shown below. Then connect
pairs of points that have the same number.
The star design at the right was created by a
method known as curve stitching. Although the
design appears to contain curves, it is made up
entirely of line segments.
Enrichment
Curve Stitching
1-5
NAME ______________________________________________ DATE
Read the introduction to Lesson 1-5 at the top of page 37 in your textbook.
Lesson 1-5
What kinds of angles are formed when streets intersect?
Angle Relationships
Reading the Lesson
©
____________ PERIOD _____
Reading to Learn Mathematics
Pre-Activity
1-5
NAME ______________________________________________ DATE
Glencoe Geometry
____________ PERIOD _____
Answers
(Lesson 1-5)
Glencoe Geometry
©
Polygons
Study Guide and Intervention
____________ PERIOD _____
Glencoe/McGraw-Hill
A17
F
Exercises
The polygon has 5 sides, so it is a pentagon. It is
convex. All sides are congruent and all angles are
congruent, so it is a regular pentagon.
d.
b.
J
K
L
The figure has 8 congruent sides
and 8 congruent angles. It is
convex and is a regular octagon.
The figure is not closed, so it is
not a polygon.
I
H
©
triangle; convex;
irregular
hexagon; convex;
regular
Glencoe/McGraw-Hill
4.
1.
5.
2.
31
pentagon; concave;
irregular
quadrilateral; convex;
irregular
6.
3.
Glencoe Geometry
octagon; concave;
irregular
pentagon; concave;
irregular
Name each polygon by its number of sides. Then classify it as concave or convex
and regular or irregular.
c.
G
E
The polygon has 4 sides, so it is a quadrilateral.
It is concave because part of D
苶E
苶 or 苶
EF
苶 lies in the
interior of the figure. Because it is concave, it
cannot have all its angles congruent and so it is
irregular.
a. D
Example
Name each polygon by its number of sides. Then classify it as
concave or convex and regular or irregular.
Polygons A polygon is a closed figure formed by a finite number of coplanar line
segments. The sides that have a common endpoint must be noncollinear and each side
intersects exactly two other sides at their endpoints. A polygon is named according to its
number of sides. A regular polygon has congruent sides and congruent angles. A polygon
can be concave or convex.
1-6
NAME ______________________________________________ DATE
c
5 in.
Exercises
P⫽a⫹b⫹c
⫽3⫹4⫹5
⫽ 12 in.
3 in. a
4 in.
b
b.
96 yd
12 yd
19 yd
9 cm
2.5 cm
24 yd
27 yd
3.5 cm
14 yd
3 cm
4.
2.
square
10 cm
1 cm
22 ft
5.5 ft
s 5 cm
c.
Glencoe Geometry
Glencoe/McGraw-Hill
Answers
©
rectangle
16, 32
x
2x
5. P ⫽ 96
32
x
2x
8, 10, 10, 20
x⫺2
6. P ⫽ 48
ᐉ
3 ft
ᐉ
x
w
Glencoe Geometry
P ⫽ 2ᐉ ⫹ 2w
⫽ 2(3) ⫹ 2(2)
⫽ 10 ft
2 ft w
Find the length of each side of the polygon for the given perimeter.
3.
1.
s
5 cm
5 cm
s
P ⫽ 4s
⫽ 4(5)
⫽ 20 cm
5 cm s
Find the perimeter of each figure.
a.
Example
Write an expression or formula for the perimeter of each polygon.
Find the perimeter.
The perimeter of a polygon is the sum of the lengths of all the sides of the
polygon. There are special formulas for the perimeter of a square or a rectangle.
Polygons
(continued)
____________ PERIOD _____
Study Guide and Intervention
Perimeter
1-6
NAME ______________________________________________ DATE
Answers
(Lesson 1-6)
Lesson 1-6
©
Polygons
Skills Practice
____________ PERIOD _____
Glencoe/McGraw-Hill
heptagon; convex;
regular
quadrilateral; convex;
irregular
5.
2.
A18
40 yd
20 yd
98 yd
18 yd
20 yd
8.
20 m
2m
4m
5m
3m
9.
6.
3.
32 in.
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
©
Glencoe/McGraw-Hill
All are 13 mm.
13. P ⫽ 104 millimeters
33
All are 28 km.
14. P ⫽ 84 kilometers
w
Glencoe Geometry
9 ft, 9 ft, 35 ft, 35 ft
4w ⫺ 1
15. P ⫽ 88 feet
ALGEBRA Find the length of each side of the polygon for the given perimeter.
14.3 units
12. quadrilateral LMNO with vertices L(⫺1, 4), M(3, 4), N(2, 1), and O(⫺2, 1)
20 units
11. quadrilateral QRST with vertices Q(⫺3, 2), R(1, 2), S(1, ⫺4), and T(⫺3, ⫺4)
12 units
10. triangle ABC with vertices A(3, 5), B(3, 1), and C(0, 1)
10 in.
10 in.
dodecagon;
concave; irregular
pentagon; concave;
irregular
COORDINATE GEOMETRY Find the perimeter of each polygon.
7.
6m
quadrilateral; convex;
irregular
triangle; convex;
regular
Find the perimeter of each figure.
4.
1.
Name each polygon by its number of sides and then classify it as convex or
concave and regular or irregular.
1-6
NAME ______________________________________________ DATE
Polygons
Practice
(Average)
____________ PERIOD _____
hexagon; concave;
irregular
2.
53 mm
10 mm
7 mm
18 mm
18 mm
5.
86 mi
21 mi
32 mi
6.
3.
4 cm
n
2x ⫺ 3
17 cm, 17 cm, 5 cm
3x ⫹ 5
10. P ⫽ 39 centimeters
4 in.
16 in.
16 in.
©
Glencoe/McGraw-Hill
48 in.
34
4 in.
Glencoe Geometry
13. If Jasmine doubles the width of the scarf, how many inches of fringe will she need?
40 in.
12. How many inches of fringe does she need to purchase?
5x ⫺ 4
2x ⫹ 2
18 ft, 18 ft, 36 ft, 17 ft
x⫹9
11. P ⫽ 89 feet
SEWING For Exercises 12–13, use the following information.
Jasmine plans to sew fringe around the scarf shown in the diagram.
3 in., 3 in., 10 in., 10 in.
6n ⫺ 8
9. P ⫽ 26 inches
ALGEBRA Find the length of each side of the polygon for the given perimeter.
17.5 units
8. pentagon STUVW with vertices S(0, 0), T(3, ⫺2), U(2, ⫺5), V(⫺2, ⫺5), and W(⫺3, ⫺2)
25.1 units
7. quadrilateral OPQR with vertices O(⫺3, 2), P(1, 5), Q(6, 4), and R(5, ⫺2)
14 cm
4 cm 6 cm
6 cm
14 cm
6 cm
56 cm
2 cm
quadrilateral;
convex; irregular
COORDINATE GEOMETRY Find the perimeter of each polygon.
4.
33 mi
nonagon; convex;
regular
Find the perimeter of each figure.
1.
Name each polygon by its number of sides and then classify it as convex or
concave and regular or irregular.
1-6
NAME ______________________________________________ DATE
Answers
(Lesson 1-6)
Glencoe Geometry
Lesson 1-6
©
Glencoe/McGraw-Hill
A19
Sides intersect at a point
that is not an endpoint.
pentagon, convex,
regular
b.
quadrilateral,
concave, not regular
c.
quadrilateral, convex,
not regular
iv. P ⫽ 12s
v. P ⫽ 2ᐉ ⫹ 2w
vi. P ⫽ 4s
f. triangle iii
iii. P ⫽ a ⫹ b ⫹ c
c. regular hexagon ii
d. rectangle v
ii. P ⫽ 6s
b. square vi
e. regular octagon i
i. P ⫽ 8s
a. regular dodecagon iv
4. Match each polygon in the first column with the formula in the second column that can
be used to find its perimeter. (s represents the length of each side of a regular polygon.)
3. What is another name for a regular quadrilateral? a square
a.
Glencoe/McGraw-Hill
35
Glencoe Geometry
Sample answer: A regular polygon looks the same no matter what
part you look at. The sides are the same length, and the angles are
the same size.
5. One way to remember the meaning of a term is to explain it to another person.
How would you explain to a friend what a regular polygon is?
Helping You Remember
©
curved (not all made
up of segments)
2. Name each polygon by its number of sides. Then classify it as convex or concave and
regular or not regular.
not closed
b.
1. Tell why each figure is not a polygon.
c.
Sample answer: square, rectangle, parallelogram, trapezoid
Name four different shapes that can each be formed by four sticks connected to
form a closed figure. Assume you have sticks with a good variety of lengths.
Read the introduction to Lesson 1-6 at the top of page 45 in your textbook.
How are polygons related to toys?
Polygons
Reading the Lesson
a.
____________ PERIOD _____
Reading to Learn Mathematics
Pre-Activity
1-6
NAME ______________________________________________ DATE
Enrichment
____________ PERIOD _____
A2 ⫽ 3 ⭈ 3
⫽9
m2.
2m
2m
1
4 cm
4 in.
2 cm
2 in.
4 cm
8 cm
4 cm
2 cm
4 cm
6 cm
4 in.
1 in.
A ⫽ 40 cm2
P ⫽ 44 cm
A ⫽ 12 in2
P ⫽ 20 in.
4.
2.
3 ft
7m
7 ft
9 ft
2 ft
13 m
26 m
6m
4 ft
5m
12 m
A ⫽ 320 m2
P ⫽ 96 m
Glencoe Geometry
Glencoe/McGraw-Hill
See students’ work.
Answers
©
6. 96 m
7. 44 cm
36
9. Describe the steps you used to find the perimeter in Exercise 1.
5. 17 in.
8. 48 ft
Glencoe Geometry
A ⫽ 90 ft2
P ⫽ 46 ft
9m
3m
2
3m
5m
For Exercises 5–8, find the perimeter of the figures in Exercises 1–4.
3.
1.
6 ft
9m
9m
Find the area and perimeter of each irregular shape.
The area of the irregular shape is 27
18 ⫹ 9 ⫽ 27
A ⫽ ᐉw
A1 ⫽ 9 ⭈ 2
⫽ 18
Example
Find the area of the
figure at the right.
Separate the figure into two rectangles.
However, many figures are combinations of two or more rectangles creating
irregular shapes. To find the area of an irregular shape, it helps to separate
the shape into rectangles, calculate the formula for each rectangle, then find
the sum of the areas.
Perimeter: P ⫽ 2ᐉ ⫹ 2w
Area: A ⫽ ᐉw, where ᐉ is the length and w is the width
Two formulas that are used frequently in mathematics are perimeter and
area of a rectangle.
Perimeter and Area of Irregular Shapes
1-6
NAME ______________________________________________ DATE
Answers
(Lesson 1-6)
Lesson 1-6
Chapter 1 Assessment Answer Key
Form 1
Page 37
1.
2.
D
B
4.
C
11.
B
12.
A
13.
D
14.
D
15.
B
16.
C
17.
A
A
7.
A
8.
C
18.
D
19.
D
A
20.
10.
1.
B
2.
D
3.
C
4.
A
5.
B
6.
C
7.
C
8.
C
9.
A
10.
B
D
6.
9.
Page 38
A
3.
5.
Form 2A
Page 39
B
B
B:
12 yd
(continued on the next page)
© Glencoe/McGraw-Hill
A20
Glencoe Geometry
Chapter 1 Assessment Answer Key
11.
C
12.
A
Form 2B
Page 41
1.
2.
13.
14.
C
15.
B
16.
D
17.
B
4.
B
20.
A
32
© Glencoe/McGraw-Hill
11.
A
12.
B
13.
D
14.
C
15.
D
16.
A
17.
B
18.
C
D
D
A
5.
C
6.
B
A
19.
B:
B
C
3.
18.
Page 42
Answers
Form 2A (continued)
Page 40
7.
A
8.
D
19.
C
9.
C
20.
C
10.
B
B:
A21
64
Glencoe Geometry
Chapter 1 Assessment Answer Key
Form 2C
Page 43
Page 44
Sample answer:
DE
1.
2.
A, B, C
3.
AB
15.
64°, acute
16.
6
17.
5
18.
11
19.
108
20.
68
21.
6
4. Sample answer:
D, E, C
5.
6.3 cm
6.
0.5 mm
7.
7.3 cm
8.
27 cm
9.
3 in.
10.
11.
12.
58
3
,
2
2
5
22. hexagon, convex,
regular
(1, 3)
13.
15 55
14.
x 8, y 7
© Glencoe/McGraw-Hill
23.
174
24.
35
25.
21
B:
length 11;
width 4
A22
Glencoe Geometry
Chapter 1 Assessment Answer Key
Form 2D
Page 45
Page 46
Sample answer:
TU
1.
Sample answer:
T, U, V
2.
3.
15.
135°, obtuse
16.
6
17.
6
18.
7
19.
122
20.
32
RS
4. Sample answer:
5.
6.
7.
1
2
1
in.
4
1 in.
Answers
X, Y, Z
5.7 cm
8.
17 cm
9.
7
21.
9
22.
10.
65
pentagon,
convex, regular
11.
32, 0
23.
185
24.
51
25.
68
B:
10, 18, 9
1
12.
(1, 1)
250
13. 10 90
or 10 810
35.3 units
14.
2
© Glencoe/McGraw-Hill
A23
Glencoe Geometry
Chapter 1 Assessment Answer Key
Form 3
Page 47
Page 48
planes ABCD,
1. BFCE, FBA, CDE,
and plane P or
ADEF
3
12.
15.5 x 8
13.
2.
BC
DC or 3.
AE
15.
4.
1
1 in.
4
16.
5.
1
in.
8
27
14.
36
8
8.5
17.
6.
12.3 cm and
19.3 cm
7.
6
y
D
8.
18.
8
B
A
C
O
9.
20
or
25
4.5
2, 2,
1
10.
(4, 1.5)
11.
y 2, y 4
© Glencoe/McGraw-Hill
x
quadrilateral,
concave, irregular
19. 9 20.
B:
A24
5
310
20.7
39
square: 9,
triangle: 12
Glencoe Geometry
Chapter 1 Assessment Answer Key
Score
General Description
Specific Criteria
4
Superior
A correct solution that
is supported by welldeveloped, accurate
explanations
• Shows a thorough understanding of concepts involving
special angle relationships, classification of angles,
distance formula, regular polygons, angle bisectors, and
perimeters.
• Uses appropriate strategies to solve problems.
• Computations are correct.
• Written explanations are exemplary.
• Graphs and figures are accurate and appropriate.
• Goes beyond requirements of some or all problems.
3
Satisfactory
A generally correct solution,
but may contain minor flaws
in reasoning or computation
• Shows an understanding of the concepts involving special
angle relationships, classification of angles, distance
formula, regular polygons, angle bisectors, and perimeters.
• Uses appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are effective.
• Graphs and figures are mostly accurate and appropriate.
• Satisfies all requirements of problems.
2
Nearly Satisfactory
A partially correct
interpretation and/or
solution to the problem
• Shows an understanding of most of the concepts involving
special angle relationships, classification of angles,
distance formula, regular polygons, angle bisectors, and
perimeters.
• May not use appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are satisfactory.
• Graphs and figures are mostly accurate.
• Satisfies the requirements of most of the problems.
1
Nearly Unsatisfactory
A correct solution with no
supporting evidence or
explanation
• Final computation is correct.
• No written explanations or work is shown to substantiate
the final computation.
• Graphs and figures may be accurate but lack detail or
explanation.
• Satisfies minimal requirements of some of the problems.
0
Unsatisfactory
An incorrect solution
indicating no mathematical
understanding of the
concept or task, or no
solution is given
• Shows little or no understanding of most of the concepts
involving special angle relationships, classification of
angles, distance formula, regular polygons, angle
bisectors, and perimeters.
• Does not use appropriate strategies to solve problems.
• Computations are incorrect.
• Written explanations are unsatisfactory.
• Graphs and figures are inaccurate or inappropriate.
• Does not satisfy requirements of most problems.
• No answer may be given.
© Glencoe/McGraw-Hill
A25
Glencoe Geometry
Answers
Page 49, Open-Ended Assessment
Scoring Rubric
Chapter 1 Assessment Answer Key
Page 49, Open-Ended Assessment
Sample Answers
In addition to the scoring rubric found on page A25, the following sample answers
may be used as guidance in evaluating open-ended assessment items.
1.
A
4a. After drawing an acute angle, the student
labels the vertex B and point A on one
ray and point C on the other ray. Then
the student uses a protractor to find the
measure of ABC. The student lets the
measure of ABC equal (6x 1) and
solves for x.
s
B
R
C
points A, B, and C
2a. After drawing a line on a coordinate
grid, students should label two points
on the graph D and G.
b. To find the measure of an angle that is
complementary to ABC, you would
subtract mABC from 90.
b. The students should use either the
Pythagorean Theorem, the Distance
Formula, or the Midpoint Formula to
determine the distance between points
D and G.
c. To find the measure of an angle that is
supplementary to ABC, you would
subtract mABC from 180.
5a.
c. Using the Midpoint Formula and the
known coordinates for points D(x1, y1)
and G(x2, y2), the coordinates of point
H(x, y) can be found by solving for x
xx
2
T
(4x 6)
R
yy
2
and y in 1 x2 and 1 y2.
(8x 6)
S
U
is an angle bisector, then mTRS
b. If RS
and mSRU must be equal. Therefore,
solve 4x 6 8x 6 for x.
4x 6 8x 6
6 6 8x 4x Add 6 and subtract 4x from
3a. The student draws a rectangle, labels
the vertices W, X, Y, and Z, labels the
width with a variable, such as x, and
the length in terms of that variable,
3x 5.
each side.
b. An expression for the perimeter, where
x is the width, would be either
2(3x 5) 2x or 8x 10.
12 4x
3x
Combine like terms.
Divide each side by 4.
c. When x 7.5, mTRS 4(7.5) 6 and
mSRU 8(7.5) 6. Simplifying each
expression results in mTRS 36 and
mSRU 8(7.5) 6 54. Since the
and
sum of the two measures is 90, RU
RT
must be perpendicular.
c. Solving 58 8x 10 for x, the width is
found to be 6 mm. To check that this
answer is correct, use the value of the
width to determine the length, 23. The
sum of all four sides, 23 23 6 6,
should equal 58.
d. After using a ruler to draw a segment
that is 23 mm long, students should
label the endpoints P and Q.
e. A measurement of 23 mm for P
Q
is
accurate to within 0.5 mm. So, a
measurement of 23 mm could be 22.5 to
23.5 mm.
© Glencoe/McGraw-Hill
A26
Glencoe Geometry
Chapter 1 Assessment Answer Key
Quiz 1
Page 51
Quiz 3
Page 52
1. RS , or RU or SU
point S
3. points R, S, U or
points T, S, V
2.
4.
1.
perpendicular
2. complementary
angle
3.
4.
midpoint
5. vertical angles
6.
1
4
1 in.
5.
1
inch
8
6.
8.9 cm
7.
3
8.
33 in.
9.
4
10.
C
T
obtuse
U or V
23
1.
2.
3.
4.
Sample answer:
TSU
and USV
5.
TSU and WSP or
6. TSP and USW
7.
19
8.
12
9.
25
10.
B
angle bisector
7. supplementary
8. adjacent angles
9.
acute angle
10.
congruent
Sample answer:
2.
(4, 7)
(3, 2)
3.
14
4.
40
1.
Sample answer: Since
11. the measuring tool is
1
divided into -inch
2
increments, the
measurement is
precise to within
1
inch.
Quiz 4
Page 52
Quiz 2
Page 51
4
12. Sample answer: Point
M is between points P
and Q only if P, Q, and
M are collinear and
PM MQ PQ.
© Glencoe/McGraw-Hill
5.
B
A27
1.
2.
14 m
3.
86 units
The perimeter is
doubled.
4.
5.
4
Glencoe Geometry
Answers
Vocabulary Test/Review
Page 50
Chapter 1 Assessment Answer Key
Mid-Chapter Test
Page 53
Cumulative Review
Page 54
Part I
1.
2.
3.
4.
B
1.
A, C, and E or
B, D, and F
2.
C
3.
3 cm
4.
10 mm
5.
26 units
6.
B(7, 1)
C
B
C
vertex: N; sides: NJ
7. and NK ; 90; right
5. D
8. vertex: N; sides: NK
; 100; obtuse
and NH
AFB and FCD or
9. EFG and FCD or
CFD and CDG
Part II
6.
37
7.
12, 1
8.
(0, 3)
9.
3
10.
74 units
10. AFE and BFD
11. AFB or EFG
12.
7
13.
quadrilateral;
concave; irregular
17.1 units
14. 10 50
15.
© Glencoe/McGraw-Hill
A28
18 cm
Glencoe Geometry
Chapter 1 Assessment Answer Key
Standardized Test Practice
1.
2.
A
E
Page 56
B
F
C
G
D
11.
H
13.
3.
4.
A
E
B
F
C
G
15.
5.
A
B
C
E
F
G
H
7.
A
B
C
D
E
F
G
/
.
.
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
14.
1 1
.
/
.
/
.
.
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
8 . 9
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
2 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
0
1
2
3
4
5
6
7
8
9
5 5
D
6.
8.
.
/
.
D
H
12.
2
.
/
.
/
.
.
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
16.
20 units
17.
81.5 units
18.
21 units
H
9.
A
B
C
D
10.
E
F
G
H
© Glencoe/McGraw-Hill
A29
Glencoe Geometry
Answers
Page 55