– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS – 61. d. It is ironic that in a place pdf

– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS –

61. d. It is ironic that in a place where there are so
many ways to describe one food (indicating that
this food is a central part of the culture),
Thomas is hungry. The passage does not mention the language of the reservation, so choice a
is incorrect. The sentence does not show any
measure of how hungry Thomas is, so choice b
is incorrect. The sentence does not describe fry
bread or make it sound in any way appealing, so
choice c is also incorrect. The passage tells us
that it was Thomas’s hunger, not the number of
ways to say fry bread, that provided his inspiration, so choice e is incorrect.
62. c. The author tells us that the new house was in
“the best neighborhood in town,” and the neighborhood’s “prestige outweighed its deadliness”
(lines 5–8). There is no indication that their old
house was falling apart (choice a) or that they
needed more room (choice b). The neighborhood is clearly not great for children (“it was not
a pleasant place to live [especially for children]”), so choice d is incorrect. The author tells
us that business was going well for his father—
so well, in fact, that he could pay for the house
in cash—but that does not mean the house was
affordable (choice e). In fact, if it was in the
most prestigious neighborhood, it was probably
expensive.
63. a. The author tells us that his father was “always a
man of habit”—so much so that he forgot he’d
moved and went to his old house, into his old
room, and lay down for a nap, not even noticing
that the furniture was different. This suggests
that he has a difficult time accepting and adjusting to change. There is no evidence that he is a

calculating man (choice b). He may be unhappy
with his life (choice c), which could be why he
chose not to notice things around him, but there
is little to support this in the passage, while
there is much to support choice a. We do not
know if he was proud of the house (choice d).
We do know that he was a man of habit, but we
do not know if any of those habits were bad
(choice e).

64. d. That his father would not realize that someone
else was living in the house—that he would not
notice, for example, different furniture arranged
in a different way—suggests that his father did
not pay any attention to things around him and
just went through the motions of his life by
habit. Being habitual is different from being
stubborn, so choice a is incorrect. The author is
writing about his father and seems to know him
quite well, so choice b is incorrect. We do not
know if the author’s father was inattentive to his
needs (choice c), though if he did not pay attention to things around him, he likely did not pay
much attention to his children. Still, there is not
enough evidence in this passage to draw this
conclusion. His father may have been very
attached to the old house (choice e), but the
incident doesn’t just show attachment; it shows
a lack of awareness of the world around him.
65. b. The bulk of this excerpt is the story that the
author finds “pathetic,” so the most logical conclusion regarding his feelings for his father is

that he lived a sad life. We know that his business was going well, but the author does not discuss his father’s methods or approach to
business, so choice a is incorrect. Choice c is
likewise incorrect; there is no discussion of his
father’s handling of financial affairs. Choice d is
incorrect because there is no evidence that his
father was ever cruel. His father may have been
impressive and strong (choice e), but the dominant theme is his habitual nature and the sad
fact that he did not notice things changing
around him.

375


– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS –

Glossar y of Terms: Language
Arts, Reading
the repetition of sounds, especially at the
beginning of words
antagonist the person, force, or idea working against
the protagonist
antihero a character who is pathetic rather than
tragic, who does not take responsibility for his or her
destructive actions
aside in drama, when a character speaks directly to
the audience or another character concerning the
action on stage, but only the audience or character
addressed in the aside is meant to hear
autobiography the true account of a person’s life
written by that person

ballad a poem that tells a story, usually rhyming abcb
blank verse poetry in which the structure is controlled only by a metrical scheme (also called metered
verse)
characters people created by an author to carry the
action, language, and ideas of a story or play
climax the turning point or high point of action and
tension in the plot
closet drama a play that is meant only to be read,
not performed
comedy humorous literature that has a happy
ending
commentary literature written to explain or illuminate other works of literature or art
complication the series of events that “complicate”
the plot and build up to the climax
conflict a struggle or clash between two people,
forces, or ideas
connotation implied or suggested meaning
context the words and sentences surrounding a
word or phrase that help determine the meaning of
that word or phrase
couplet a pair of rhyming lines in poetry
denotation exact or dictionary meaning
denouement the resolution or conclusion of the
action
dialect language that differs from the standard language in grammar, pronunciation, and idioms (natural speech versus standard English); language used by
a specific group within a culture
dialogue the verbal exchange between two or more
people; conversation
alliteration

the particular choice and use of words
drama literature that is meant to be performed
dramatic irony when a character’s speech or actions
have an unintended meaning known to the audience
but not to the character
elegy a poem that laments the loss of someone or
something
exact rhyme the repetition of exactly identical
stressed sounds at the end of words
exposition in plot, the conveyance of background
information necessary to understand the complication of the plot
eye rhyme words that look like they should rhyme
because of spelling, but because of pronunciation,
they do not
falling action the events that take place immediately
after the climax in which “loose ends” of the plot are
tied up
feet in poetry, a group of stressed and unstressed
syllables
fiction prose literature about people, places, and
events invented by the author
figurative language comparisons not meant to be
taken literally but used for artistic effect, including
similes, metaphors, and personification
flashback when an earlier event or scene is inserted
into the chronology of the plot
free verse poetry that is free from any restrictions of
meter and rhyme
functional texts literature that is valued mainly for
the information it conveys, not for its beauty of form,

emotional impact, or message about human experience
genre category or kind; in literature, the different
kinds or categories of texts
haiku a short, imagistic poem of three unrhymed
lines of five, seven, and five syllables, respectively
half-rhyme the repetition of the final consonant at
the end of words
hyperbole extreme exaggeration not meant to be
taken literally, but done for effect
iambic pentameter a metrical pattern in poetry in
which each line has ten syllables (five feet) and the
stress falls on every second syllable
imagery the representation of sensory experiences
through language
inference a conclusion based upon reason, fact, or
evidence
diction

376


– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS –
irony

see dramatic irony, situational irony, or verbal

irony
any written or published text
literary texts literature valued for its beauty of
form, emotional impact, and message(s) about the

human experience
main idea the overall fact, feeling, or thought a writer
wants to convey about his or her subject
melodrama a play that starts off tragic but has a
happy ending
memoir an autobiographical text that focuses on a
limited number of events and explores their impact
metaphor a type of figurative language that compares two things by saying they are equal
meter the number and stress of syllables in a line of
poetry
monologue in drama, a play or part of a play
performed by one character speaking directly to the
audience
narrator in fiction, the character or person who tells
the story
nonfiction prose literature about real people, places,
and events
ode a poem that celebrates a person, place, or thing
omniscient narrator a third-person narrator who
knows and reveals the thoughts and feelings of the
characters
onomatopoeia when the sound of a word echoes its
meaning
paragraph a group of sentences about the same idea
personification figurative language that endows
nonhuman or nonanimal objects with human
characteristics
plot the ordering of events in a story
poetry literature written in verse
point of view the perspective from which something

is told or written
prose literature that is not written in verse or dramatic form
protagonist the “hero” or main character of a story,
the one who faces the central conflict
pun a play on the meaning of a word
quatrain in poetry, a stanza of four lines
readability techniques strategies writers use to
make information easier to process, including the use
of headings and lists
rhyme the repetition of an identical or similar
stressed sound(s) at the end of words
literature

the overall sound or “musical” effect of the
pattern of words and sentences
sarcasm sharp, biting language intended to ridicule
its subject
satire a form of writing that exposes and ridicules its
subject with the hope of bringing about change
setting the time and place in which a story unfolds
simile a type of figurative language that compares two
things using like or as
situational irony the tone that results when there is
incongruity between what is expected to happen and
what actually occurs
soliloquy in drama, a speech made by a character
who reveals his or her thoughts to the audience as if
he or she is alone and thinking aloud
sonnet a poem composed of fourteen lines, usually
in iambic pentameter, with a specific rhyme scheme

speaker in poetry, the voice or narrator of the poem
stage directions in drama, the instructions provided by the playwright that explain how the action
should be staged, including directions for props, costumes, lighting, tone, and character movements
stanza a group of lines in a poem, a poetic paragraph
structure the manner in which a work of literature is
organized; its order of arrangement and divisions
style the manner in which a text is written, composed
of word choice, sentence structure, and level of formality and detail
subgenre a category within a larger category
suspense the state of anxiety caused by an undecided or unresolved situation
symbol a person, place, or object invested with special meaning to represent something else
theme the overall meaning or idea of a literary work
thesis the main idea of a nonfiction text
thesis statement the sentence(s) that express an
author’s thesis
tone the mood or attitude conveyed by writing or
voice
topic sentence the sentence in a paragraph that
expresses the main idea of that paragraph
tragedy a play that presents a character’s fall due to
a tragic flaw
tragic hero the character in a tragedy who falls from
greatness and accepts responsibility for that fall
tragic flaw the characteristic of a hero in a tragedy
that causes his or her downfall
rhythm

377

– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS –
tragicomedy

a tragic play that includes comic

scenes
understatement

restrained

a statement that is deliberately

when the intended meaning of a word
or phrase is the opposite of its expressed meaning
voice in nonfiction, the sound of the author speaking
directly to the reader
verbal irony

378


P A R T

VI

The GED
Mathematics
Exam

T

his section covers the material you need to know to prepare for the GED Mathematics Exam. You will learn how the test is structured so you will know what
to expect on test day. You will also review and practice the fundamental mathematics skills you need to do well on the exam.
Before you begin Chapter 40, take a few minutes to do the pretest that follows. The
questions and problems are the same type you will find on the GED. When you are finished, check the answer key carefully to assess your results. Your pretest score will help you
determine how much preparation you need and in which areas you need the most careful review and practice.

379


– THE GED MATHEMATICS EXAM –

Question 3 is based on the following figure.

Pretest: GED Mathematics
Directions: Read each of the questions below carefully
and determine the best answer.
To practice the timing of the GED exam, please allow
18 minutes for this pretest. Record your answers on the
answer sheet provided here and the answer grids for
questions 9 and 10.
Note: On the GED, you are not permitted to write in
the test booklet. Make any notes or calculations on a separate piece of paper.

a+

3a + b
2a + b

ANSWER SHEET

1.
2.
3.
4.
5.
6.
7.
8.

a
a
a
a
a
a
a
a

b
b
b
b
b
b
b
b

c
c

c
c
c
c
c
c

d
d
d
d
d
d
d
d

3a + 2b

3.

On five successive days, a motorcyclist listed his
mileage as follows: 135, 162, 98, 117, 216.
If his motorcycle averages 14 miles for each
gallon of gas used, how many gallons of gas did
he use during these five days?
a. 42
b. 52
c. 115
d. 147
e. 153

2.

Bugsy has a piece of wood 9 feet 8 inches long.
He wishes to cut it into 4 equal lengths. How far
from the edge should he make the first cut?
a. 2.5 ft.
b. 2 ft 5 in.
c. 2.9 ft.
d. 29 ft.
e. 116 in.

380

What is the perimeter of the figure?
a. 8a + 5b
b. 9a + 7b
c. 7a + 5b
d. 6a + 6b
e. 8a + 6b

4.

e
e
e
e
e
e
e

e

1.

3b

Jossie has $5 more than Siobhan, and Siobhan
has $3 less than Michael. If Michael has $30, how
much money does Jossie have?
a. $30
b. $27
c. $32
d. $36
e. Not enough information is given.


– THE GED MATHEMATICS EXAM –

8.

Mr. DeLandro earns $12 per hour. One week,
Mr. DeLandro worked 42 hours; the following
week, he worked 37 hours. Which of the
following indicates the number of dollars Mr.
DeLandro earned for 2 weeks?
a. 12 × 2 + 37
b. 12 × 42 + 42 × 37
c. 12 × 37 + 42
d. 12 + 42 × 37
e. 12(42 + 37)

9.

Questions 5 and 6 are based on the following graph.

What is the slope of the line that passes through
points A and B on the coordinate graph below?
Mark your answer in the circles in the grid
below.

Personal
Service
12%
Manufacturing
33%

All Others
17%

There are 180,000
employees total.

Trade and
Finance
25%

Food
Service
5%
Professional

8%

5.

The number of persons engaged in Food Service
in the city during this period was
a. 900.
b. 9,000.
c. 14,400.
d. 36,000.
e. 90,000.

6.

y
5
4
3

B (3,5)
A (1,3)

2
1

If the number of persons in trade and finance is
represented by M, then the approximate number
in manufacturing is represented as

x

−5 −4 −3 −2 −1
−1
−2
−3
−4
−5

M
a. ᎏ5ᎏ
b. M + 3
c. 30M

1 2 3 4 5

4M
d. ᎏ3ᎏ
e. Not enough information is given.

/

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/

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9

9

Question 7 is based on the following figure.
•

A
B

E

D

7.

C

In the figure ៮៮៮ | | ៮៮៮, ៮៮៮ bisects ∠BCD, and
AB CD CE
m∠ABC = 112°. Find m∠ECD.
a. 45°
b. 50°
c. 56°

d. 60°
e. Not enough information is given.

381


– THE GED MATHEMATICS EXAM –

10.

What is the value of the expression 3(2x − y) +
(3 + x)2, when x = 4 and y = 5? Mark your
answer in the circles on the grid below.

a+

3b

3a + b

/

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•

0

0

0

0

1

1

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8

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9

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9

9

•

2a + b

3a + 2b

3. b. To find the perimeter of the figure, find the sum
of the lengths of the four sides: 2a + b + a + 3b
+ 3a + b + 3a + 2b = 9b + 7b.
4. c. Michael has $30. Siobhan has $30 − $3 = $27.
Jossie has $27 + $5 = $32.
Personal
Service
12%

Pretest Answers and Explanations

1. b. First, find the total mileage; 135 + 162 + 98 +

117 + 216 = 728 miles. Divide the total mileage
(728) by the number of miles covered for each
gallon of gas used (14) to find the number of
gallons of gas needed; 728 ÷ 14 = 52 gallons.

Manufacturing
33%

Food
Service
5%

All Others
17%

There are 180,000
employees total.

Trade and
Finance
25%

Professional
8%

5. b. To find 5% of a number, multiply the number
by .05: 180,000 × .05 = 9,000. There are 9,000
food service workers in the city.

2. b. 1 ft. = 12 in. 9 ft. 8 in. = 9 × 12 + 8 = 116 in.;

116 ÷ 4 = 29 in. = 2 ft. 5 in.

6. d. M = number of persons in trade and finance.
Since M = 25% of the total, 4M = total number
of city workers. Number of persons in manufac4M
total number of workers
turing = ᎏᎏᎏ = ᎏ3ᎏ.
3

382


– THE GED MATHEMATICS EXAM –

9.1.

A
B

E

y

D

5
4
3

C

B (3,5)
A (1,3)

2
1

7. c. Since pairs of alternate interior angles of parallel
lines have equal measures, m∠BCD = m∠ABC.
Thus, m∠BCD = 112°.

x
−5 −4 −3 −2 −1
−1
−2
−3
−4
−5

m∠ECD = ᎏ1ᎏm∠BCD = ᎏ1ᎏ(112) = 56°
2
2
8. e. In two weeks, Mr. Delandro worked a total of
(42 + 37) hours and earned $12 for each hour.
Therefore, the total number of dollars he earned
was 12(42 + 37).

1 2 3 4 5

1

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9

9

9

9

•

The coordinates of point A are (1,3). The
coordinates of point B are (3,5). Use the slope
formula:
y2 − y1
ᎏᎏ
x2 − x1

Substitute and solve:

5−3
ᎏᎏ
3−1

383

= ᎏ2ᎏ, or ᎏ1ᎏ = 1
2
1


– THE GED MATHEMATICS EXAM –

10. 58.

Pretest Assessment

58
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8

8

8

8

9

9

9

9

How did you do on the math pretest? If you answered
seven or more questions correctly, you have earned the
equivalent of a passing score on the GED Mathematics
Test. But remember that this pretest covers only a fraction of the material you might face on the GED exam. It
is not designed to give you an accurate measure of how
you would do on the actual test. Rather, it is designed to
help you determine where to focus your study efforts.
For success on the GED, review all of the chapters in this
section thoroughly. Focus on the sections that correspond to the pretest questions you answered incorrectly.

•

5

9

3(2x − y) + (3 + x)2, x = 4 and y = 5.
3(2 × 4 − 5) + (3 + 4)2 = 3(8 − 5) + (7)2 =
3(3) + 49 = 9 + 49 = 58.

384


C H A P T E R

40

About the GED

Mathematics
Exam
IN THIS chapter, you will learn all about the GED Mathematics
Exam, including the number and type of questions, the topics and
skills that will be tested, guidelines for the use of calculators, and
recent changes in the test.

What to Expect on the GED Mathematics Exam
The GED Mathematics Exam measures your understanding of the mathematical knowledge needed in everyday
life. The questions are based on information presented in words, diagrams, charts, graphs, and pictures. In addition to testing your math skills, you will also be asked to demonstrate your problem-solving skills. Examples of
some of the skills needed for the mathematical portion of the GED are:
■
■
■
■
■
■
■

understanding the question
organizing data and identifying important information
selecting problem-solving strategies
knowing when to use appropriate mathematical operations
setting up problems and estimating
computing the exact, correct answer
reflecting on the problem to ensure the answer you choose is reasonable

This section will give you lots of practice in the basic math skills that you use every day as well as crucial
problem-solving strategies.

385


– ABOUT THE GED MATHEMATICS EXAM –

The GED Mathematics Test is given in two separate
sections. The first section permits the use of a calculator;
the second does not. The time limit for the GED is 90
minutes, meaning that you have 45 minutes to complete
each section. The sections are timed separately but
weighted equally. This means that you must complete
both sections in one testing session to receive a passing
grade. If only one section is completed, the entire test
must be retaken.
The test contains 40 multiple-choice questions and
ten gridded-response questions for a total of 50 questions overall. Multiple-choice questions give you several
answers to choose from and gridded-response questions
ask you to come up with the answer yourself. Each
multiple-choice question has five answer choices, a
through e. Gridded response questions use a standard
grid or a coordinate plane grid. (The guidelines for
entering a gridded-response question will be covered
later in this section.)

Formula Page

A page with a list of common formulas is provided with
all test forms. You are allowed to use this page when you
are taking the test. It is necessary for you to become
familiar with the formula page and to understand when

and how to use each formula. An example of the formula
page is on page 388 of this book.
Gridded-Response and Set-Up
Questions

There are ten non-multiple-choice questions in the math
portion of the GED. These questions require you to find
an answer and to fill in circles on a grid or on a coordinate axis.
S TANDARD G RID - IN Q UESTIONS

When you are given a question with a grid like the one
below, keep these guidelines in mind:
■

Test Topics

The math section of the GED tests you on the following
subjects:
■
■
■
■
■

measurement and geometry
algebra, functions, and patterns
number operations and number sense
data analysis, statistics, and probability

■

■

Each of these subjects is detailed in this section along
with tips and strategies for solving them. In addition, 100
practice problems and their solutions are given at the end
of the subject lessons.
Using Calculators

The GED Mathematics Test is given in two separate
booklets, Part I and Part II. The use of calculators is permitted on Part I only. You will not be allowed to use your
own. The testing facility will provide a calculator for you.
The calculator that will be used is the Casio fx-260. It is
important for you to become familiar with this calculator as well as how to use it. Use a calculator only when it
will save you time or improve your accuracy.

First, write your answer in the blank boxes at the
top of the grid. This will help keep you organized
as you “grid in” the bubbles and ensure that you
fill them out correctly.
You can start in any column, but leave enough
columns for your whole answer.
You do not have to use all of the columns. If your
answer only takes up two or three columns, leave
the others blank.
You can write your answer by using either fractions or decimals. For example, if your answer
is ᎏ1ᎏ, you can enter it either as a fraction or as a
4
decimal, .25.

The slash “/” is used to signify the fraction bar of the
fraction. The numerator should be bubbled to the left of
the fraction bar and the denominator should be bubbled
in to the right. See the example on the next page.

386


– ABOUT THE GED MATHEMATICS EXAM –

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These questions measure your ability to recognize the

correct procedure for solving a problem. They ask you to
choose an expression that represents how to “set up” the
problem rather than asking you to choose the correct
solution. About 25 percent of the questions on the GED
Mathematics Test are set-up questions.

/

1

■

S ET-U P Q UESTIONS

9

9

9

9

9

9

9

9

9

Example: Samantha makes $24,000 per year at a new
job. Which expression below shows how much
she earns per month?
a. $24,000 + 12
b. $24,000 12
c. $24,000 ì 12
d. $24,000 ữ 12
e. 12 ÷ $24,000

2

When your answer is a mixed number, it must be
represented on the standard grid in the form of
an improper fraction. For example, for the
answer 1ᎏ1ᎏ, grid in ᎏ5ᎏ.
4
4
When you are asked to plot a point on a coordinate grid like the one below, simply fill in the
bubble where the point should appear.
6
5

Answer: d. You know that there are 12 months in a
year. To find Samantha’s monthly income, you
would divide the total ($24,000) by the number
of months (12). Option e is incorrect because it
means 12 is divided by $24,000.
Graphics

Many questions on the GED Mathematics Test use
diagrams, pie charts, graphs, tables, and other visual
stimuli as references. Sometimes, more than one of these
questions will be grouped under a single graphic. Do not
let this confuse you. Learn to recognize question sets by
reading both the questions and the directions carefully.

4

What’s New for the GED?

3
2
1
−6 −5 −4 −3 −2 −1

0
−1
−2
−3
−4
−5
−6

1

2

3

4

5

6

The structure of the GED Mathematics Test, revised in
2002, ensures that no more than two questions should
include “not enough information is given” as a correct
answer choice. Given this fact, it is important for you to
pay attention to how many times you select this answer
choice. If you find yourself selecting the “not enough
information is given” for the third time, be sure to check
the other questions for which you have selected this
choice because one of them must be incorrect.
The current GED has an increased focus on “math in
everyday life.” This is emphasized by allowing the use of
a calculator on Part I as well as by an increased emphasis on data analysis and statistics. As a result, griddedresponse questions and item sets are more common. The
number of item sets varies.

387


Formulas
Area of a:
square

Area = side2

rectangle

Area = length ϫ width

parallelogram

Area = base ϫ height

triangle

1
Area = ᎏᎏ ϫ base ϫ height
2

trapezoid

1
Area = ᎏᎏ ϫ (base1 + base2) ϫ height
2

circle

Area = π ϫ radius2; π is approximately equal to 3.14

Perimeter of a:
square

Perimeter = 4 ϫ side

rectangle

Perimeter = 2 ϫ length + 2 ϫ width

triangle

Perimeter = side1 + side2 + side3

Circumference of a circle

Circumference = π ϫ diameter; π is approximately equal to 3.14

Volume of a:
cube

Volume = edge3

rectangular solid

Volume = length ϫ width ϫ height

square pyramid

1
Volume = ᎏᎏ ϫ (base edge)2 ϫ height
3

cylinder

π ϫ radius2 ϫ height π is approximately equal to 3.14

cone

1
Volume = ᎏᎏ ϫ π ϫ radius2 ϫ height; π is approximately equal to 3.14
3

Coordinate Geometry

distance between points = ͙(x2 – xෆ2 – y1)2 (x1,y1) and (x2,y2) are two points
ෆ1)2 + (yෆෆ;
in a plane
y2 y1
slope of a line = ᎏ–ᎏ; (x1,y1) and (x2,y2) are two points on the line
x2 – x1

Pythagorean Relationship a2 + b2 = c2; a and b are legs and c is the hypotenuse of a right triangle
Measures of
Central Tendency

x1 + x2 + . . . + xn
mean = ᎏᎏ, where the x's are the values for which a mean is desired,
n
and n is the total number of values for x.
median = the middle value of an odd number of ordered scores, and halfway
between the two middle values of an even number of ordered scores.

Simple Interest

interest = principal ϫ rate ϫ time

Distance

distance = rate ϫ time

Total Cost

total cost = (number of units) ϫ (price per unit)

Adapted from official GED materials.

388


C H A P T E R

41

Measurement
and Geometry
THE GED Mathematics Test emphasizes real-life applications of
math concepts, and this is especially true of questions about measurement and geometry. This chapter will review the basics of measurement systems used in the United States and other countries,
performing mathematical operations with units of measurement, and
the process of converting between different units. It will also review
geometry concepts you’ll need to know for the exam, such as properties of angles, lines, polygons, triangles, and circles, as well as the
formulas for area, volume, and perimeter.

T

measurement enables you to form a connection between mathematics and the real world.
To measure any object, assign a unit of measure. For instance, when a fish is caught, it is often weighed

in ounces and its length measured in inches. This lesson will help you become more familiar with the
types, conversions, and units of measurement.
Also required for the GED Mathematics Test is knowledge of fundamental, practical geometry. Geometry is the
study of shapes and the relationships among them. A comprehensive review of geometry vocabulary and concepts, after this measurement lesson, will strengthen your grasp on geometry.
HE USE OF

389


– MEASUREMENT AND GEOMETRY –

Types of Measurements

5 feet = how many inches?
5 feet × 12 inches (the number of inches in a single
foot) = 60 inches
Therefore, there are 60 inches in 5 feet.

The types of measurements used most frequently in the
United States are listed below:

Try another:

Units of Length

Change 3.5 tons to pounds.
3.5 tons = how many pounds?
3.5 tons × 2,000 pounds (the number of pounds in
a single ton) = 6,500 pounds
Therefore, there are 6,500 pounds in 3.5 tons.

12 inches (in.) = 1 foot (ft.)
3 feet = 36 inches = 1 yard (yd.)
5,280 feet = 1,760 yards = 1 mile (mi.)
Units of Volume

8 ounces* (oz.) = 1 cup (c.)
2 cups = 16 ounces = 1 pint (pt.)
2 pints = 4 cups = 32 ounces = 1 quart (qt.)
4 quarts = 8 pints = 16 cups = 128 ounces = 1 gallon
(gal.)

■

Units of Weight

16 ounces* (oz.) = 1 pound (lb.)
2,000 pounds = 1 ton (T.)

To change a smaller unit to a larger unit, simply
divide the specific number of smaller units by the
number of smaller units in only one of the larger
units.
For example, to find the number of pints in 64
ounces, simply divide 64, the smaller unit, by 16,
the number of ounces in one pint.
specific number of the smaller unit
ᎏᎏᎏᎏᎏ
the number of smaller units in one larger unit

Units of Time

64 ounces
ᎏᎏ = 4 pints
16 ounces

60 seconds (sec.) = 1 minute (min.)
60 minutes = 1 hour (hr.)
24 hours = 1 day
7 days = 1 week
52 weeks = 1 year (yr.)
12 months = 1 year
365 days = 1 year

Therefore, 64 ounces are equal to four pints.
Here is one more:
Change 24 ounces to pounds.
32 ounces
ᎏᎏ = 2 pounds
16 ounces

*Notice that ounces are used to measure both the volume and
weight.

Therefore, 32 ounces are equal to two pounds.

Converting Units
When performing mathematical operations, it is necessary to convert units of measure to simplify a problem.
Units of measure are converted by using either multiplication or division:
■

To change a larger unit to a smaller unit, simply
multiply the specific number of larger units by
the number of smaller units that makes up one of
the larger units.

Basic Operations with
Measurement
It will be necessary for you to review how to add, subtract, multiply, and divide with measurement. The
mathematical rules needed for each of these operations
with measurement follow.
Addition with Measurements

To add measurements, follow these two steps:

For example, to find the number of inches in 5
feet, simply multiply 5, the number of larger units,
by 12, the number of inches in one foot:

390

1. Add like units.
2. Simplify the answer.


– MEASUREMENT AND GEOMETRY –

Example: Add 4 pounds 5 ounces to 20 ounces.
4 lb. 5 oz. Be sure to add ounces to ounces.
+ 20 oz.

4 lb. 25 oz. Because 25 ounces is more than 16
ounces (1 pound), simplify by
dividing by 16. Then add the 1
pound to the 4 pounds.

Multiplication with Measurements

1. Multiply like units.
2. Simplify the answer.
Example: Multiply 5 feet 7 inches by 3.
5 ft. 7 in. Multiply 7 inches by 3, then multiply 5
×3
feet by 3. Keep the units separate.
15 ft. 21 in. Since 12 inches = 1 foot, simplify 21
inches.
15 ft. 21 in. = 15 ft. + 1 ft. + 9 inches =
16 feet 9 inches

4 lb. + 25 oz.
1 lb.
4 lb. + 16ͤ25
ෆ
−16
9 oz.

Example: Multiply 9 feet by 4 yards.
First, change yards to feet by multiplying the
number of feet in a yard (3) by the number of
yards in this problem (4).

4 pounds 25 ounces =
4 pounds + 1 pound 9 ounces =
5 pounds 9 ounces

3 feet in a yard × 4 yards = 12 feet
Then, multiply 9 feet by 12 feet =
108 square feet.
(Note: feet × feet = square feet)

Subtraction with Measurements

1. Subtract like units.
2. Regroup units when necessary.
3. Write the answer in simplest form.

Division with Measurements

1. Divide into the larger units first.
2. Convert the remainder to the smaller unit.
3. Add the converted remainder to the existing
smaller unit if any.
4. Then, divide into smaller units.
5. Write the answer in simplest form.

For example, to subtract 6 pounds 2 ounces
from 9 pounds 10 ounces,
9 lb. 10 oz. Subtract ounces from ounces.
− 6 lb. 2 oz. Then, subtract pounds from pounds.
3 lb. 8 oz.
Sometimes, it is necessary to regroup units when

subtracting.
Example: Subtract 3 yards 2 feet from 5 yards 1
foot.
4

4

5 yd. 1 ft.
΋
΋
− 3 yd. 2 ft.
1 yd. 2 ft.
From 5 yards, regroup 1 yard to 3 feet. Add 3
feet to 1 foot. Then subtract feet from feet and
yards from yards.

391

Example:
Divide 5 quarts 4 ounces by 4.
1 qt. R1
First, divide 5 ounces
1. 4ͤ5 ෆ
ෆ
by 4, for a result of 1
−4
quart and a reminder
1
of one.
2. R1 = 32 oz.

Convert the remainder
to the smaller unit
(ounces).
3. 32 oz. + 4 oz. = 36 oz. Add the converted
remainder to the
existing smaller unit.
4.
9 oz.
Now divide the smaller
4ͤ36
ෆ
units by 4.
5. 1 qt. 9 oz.


– MEASUREMENT AND GEOMETRY –

Metric Measurements

The chart shown here illustrates some common relationships used in the metric system:

The metric system is an international system of measurement also called the decimal system. Converting units
in the metric system is much easier than converting
units in the English system of measurement. However,
making conversions between the two systems is much
more difficult. Luckily, the GED test will provide you
with the appropriate conversion factor when needed.
The basic units of the metric system are the meter,
gram, and liter. Here is a general idea of how the two systems compare:
M ETRIC S YSTEM

E NGLISH S YSTEM

1 meter

A meter is a little more than a
yard; it is equal to about 39 inches.

1 gram

A gram is a very small unit of
weight; there are about 30 grams
in one ounce.

1 liter

Length

Weight

Volume

1 km = 1,000 m

1 kg = 1,000 g

1 kL = 1,000 L

1 m = .001 km

1 g = .001 kg

1 L = .001 kL

1 m = 100 cm

1 g = 100 cg

1 L = 100 cL

1 cm = .01 m

1 cg = .01 g

1 cL = .01 L

1 m = 1,000 mm

1 g = 1,000 mg

1 L = 1,000 mL

1mm = .001 m

1 mg = .001 g

1 mL = .001 L

Conversions within the Metric
System

A liter is a little more than a quart.

Prefixes are attached to the basic metric units listed
above to indicate the amount of each unit.
For example, the prefix deci means one-tenth (ᎏ1ᎏ);
10
therefore, one decigram is one-tenth of a gram, and one
decimeter is one-tenth of a meter. The following six prefixes can be used with every metric unit:
Kilo
(k)
1,000

Hecto
(h)
100

Deka
(dk)
10

Deci
(d)

Centi
(c)
1
ᎏᎏ
100

1
ᎏᎏ
1,000

Making Easy Conversions within
the Metric System

When you multiply by a power of ten, you move the decimal point to the right. When you divide by a power of
ten, you move the decimal point to the left.
To change from a large unit to a smaller unit, move
the decimal point to the right.
kilo

Milli
(m)

1
ᎏᎏ
10

An easy way to do conversions with the metric system is
to move the decimal point to either the right or the left
because the conversion factor is always ten or a power of
ten. As you learned previously, when you change from a
large unit to a smaller unit, you multiply, and when you
change from a small unit to a larger unit, you divide.

hecto deka

UNIT

deci

centi milli

To change from a small unit to a larger unit, move the
decimal point to the left.

Examples:
■ 1 hectometer = 1 hm = 100 meters
1
■ 1 millimeter = 1 mm = ᎏᎏ meter =
1,000
.001 meter
■ 1 dekagram = 1 dkg = 10 grams
1
■ 1 centiliter = 1 cL* = ᎏᎏ liter = .01 liter
100
■ 1 kilogram = 1 kg = 1,000 grams
1
■ 1 deciliter = 1 dL* = ᎏᎏ liter = .1 liter
10

Example:
Change 520 grams to kilograms.
Step 1: Be aware that changing meters to kilometers is going from small units to larger units, and
thus, you will move the decimal point three places
to the left.
Step 2: Beginning at the UNIT (for grams), you
need to move three prefixes to the left.

‫ی یی‬
k h dk unit d c m

*Notice that liter is abbreviated with a capital letter—“L.”

392


– MEASUREMENT AND GEOMETRY –

Step 3: Move the decimal point from the
end of 520 to the left three places.
Place the decimal point before the 5.

Geometr y
520.
.520

Your answer is 520 grams = .520 kilograms.
Example:
You are packing your bicycle for a trip from
New York City to Detroit. The rack on the back
of your bike can hold 20 kilograms. If you
exceed that limit, you must buy stabilizers for
the rack that cost $2.80 each. Each stabilizer can
hold an additional kilogram. If you want to pack
23,000 grams of supplies, how much money will
you have to spend on the stabilizers?

As previously defined, geometry is the study of shapes

and the relationships among them. Basic concepts in
geometry will be detailed and applied in this section. The
study of geometry always begins with a look at basic
vocabulary and concepts. Therefore, here is a list of definitions of important terms:
area—the space inside a two-dimensional figure
bisect—cut in two equal parts
circumference—the distance around a circle
diameter—a line segment that goes directly through
the center of a circle—the longest line you can
draw in a circle
equidistant—exactly in the middle of
hypotenuse—the longest leg of a right triangle,
always opposite the right angle
line—an infinite collection of points in a straight
path
point—a location in space
parallel—lines in the same plane that will never
intersect
perimeter—the distance around a figure
perpendicular—two lines that intersect to form 90degree angles
quadrilateral—any four-sided closed figure
radius—a line from the center of a circle to a point
on the circle (half of the diameter)
volume—the space inside a three-dimensional
figure

Step 1: First, change 23,000 grams to kilograms.
‫ی ی ی‬
kg hg dkg g dg cg mg
Step 2: Move the decimal point three places to the

left.
23,000 g = 23.000 kg = 23 kg
Step 3: Subtract to find the amount over the limit.
23 kg − 20 kg = 3 kg
Step 4: Because each stabilizer holds one kilogram
and your supplies exceed the weight limit of the
rack by three kilograms, you must purchase three
stabilizers from the bike store.
Step 5: Each stabilizer costs $2.80, so multiply
$2.80 by 3: $2.80 × 3 = $8.40.

393


– MEASUREMENT AND GEOMETRY –

Angles

■

An angle is formed by an endpoint, or vertex, and two
rays.

An acute angle is an angle that measures less than
90 degrees.

ra
y

Acute

Angle

■

ray

A right angle is an angle that measures exactly 90
degrees. A right angle is represented by a square
at the vertex.

Endpoint (or Vertex)
Naming Angles

There are three ways to name an angle.
Right
Angle
B

1

■

D

An obtuse angle is an angle that measures more
than 90 degrees, but less than 180 degrees.

2
A

C

Obtuse Angle

1. An angle can be named by the vertex when no
other angles share the same vertex: ∠A.
2. An angle can be represented by a number written
across from the vertex: ∠1.
3. When more than one angle has the same vertex,
three letters are used, with the vertex always
being the middle letter: –1 can be written as
∠BAD or as ∠DAB; –2 can be written as ∠DAC
or as ∠CAD.

■

A straight angle is an angle that measures 180
degrees. Thus, its sides form a straight line.
Straight Angle
180°

Classifying Angles

Angles can be classified into the following categories:
acute, right, obtuse, and straight.

394


– MEASUREMENT AND GEOMETRY –

Angles of Intersecting Lines

C OMPLEMENTARY A NGLES

Two angles are complementary if the sum of their measures is equal to 90 degrees.

1
2

When two lines intersect, two sets of nonadjacent angles
called vertical angles are formed. Vertical angles have
equal measures and are supplementary to adjacent
angles.

Complementary
Angles

2
1

3

∠1 + ∠2 = 90°

4

S UPPLEMENTARY A NGLES

Two angles are supplementary if the sum of their measures is equal to 180 degrees.

■
■
■

Supplementary
Angles

Bisecting Angles and Line
Segments

1

2

m∠1 = m∠3 and m∠2 = m∠4
m∠1 + m∠2 = 180 and m∠2 + m∠3 = 180
m∠3 + m∠4 = 180 and m∠1 + m∠4 = 180

Both angles and lines are said to be bisected when
divided into two parts with equal measures.

∠1 + ∠2 = 180°
A DJACENT A NGLES

Example

Adjacent angles have the same vertex, share a side, and do
not overlap.

S

C

A

∠1 and ∠2 are adjacent.

B

Adjacent
Angles

2

S

1

Line segment AB is bisected at point C.

The sum of the measures of all adjacent angles around
the same vertex is equal to 360 degrees.

2

1

35°

∠1 + ∠2 + ∠3 + ∠4 = 360°

C

3

35°

4

A

According to the figure, ∠A is bisected by ray AC.

395


– MEASUREMENT AND GEOMETRY –

Solution:
Because both sets of lines are parallel, you know
that x can be added to x + 10 to equal 180. The
equation is thus, x + x + 10 = 180.

Angles Formed by Parallel Lines

When two parallel lines are intersected by a third line,
vertical angles are formed.

d

a

Example
Solve for x:
2x + 10 = 180
−10 −10
2x
ᎏᎏ
2

170
= ᎏ2ᎏ
x
= 85
Therefore, m∠x = 85° and the obtuse angle is
equal to 180 − 85 = 95°.

c

e

f

Angles of a Triangle

h

The measures of the three angles in a triangle always
equal 180 degrees.

B

g

In the above figure:

b

+

+

c

a

C

Examples
m∠b + m∠d = 180°
m∠c + m∠e = 180°
m∠f + m∠h = 180°
m∠g + m∠a = 180°

c
= 180°

E XTERIOR A NGLES

B

b

Example
In the figure below, if m || n and a || b, what is
the value of x?
b

=

b

d

= 180° and

+

a

c

A

d

n
(x + 10)°

+

C

m

a

x°

c

a

d

■

∠b, ∠c, ∠f, and ∠g are all acute and equal.
∠a, ∠d, ∠e, and ∠h are all obtuse and equal.
Also, any acute angle added to any obtuse angle
will be supplementary.

a

■

b

■

A

■

Of these vertical angles, four will be equal and
acute, four will be equal and obtuse, or all four
will be right angles.
Any combination of an acute and an obtuse angle
will be supplementary.

b

■

An exterior angle can be formed by extending a side from
any of the three vertices of a triangle. Here are some rules
for working with exterior angles:
■

396

An exterior angle and interior angle that share the
same vertex are supplementary.


– MEASUREMENT AND GEOMETRY –
■

■

An exterior angle is equal to the sum of the

nonadjacent interior angles.
The sum of the exterior angles of a triangle
equals 360 degrees.

70°

Acute

60°

50°

Triangles
Right

Classifying Triangles

It is possible to classify triangles into three categories
based on the number of equal sides:
Scalene

Isosceles

Equilateral

(no equal sides)

(two equal sides)

(all sides equal)

Obtuse
150°

Scalene

Angle-Side Relationships

Knowing the angle-side relationships in isosceles, equilateral, and right triangles will be useful in taking the
GED exam.
■

Isoceles

In isosceles triangles, equal angles are opposite
equal sides.
C

Isosceles
m∠a = m∠b

b

B
■

Equilateral

a

A

In equilateral triangles, all sides are equal and all
angles are equal.

60°
5

It is also possible to classify triangles into three categories based on the measure of the greatest angle:
Acute

Right
greatest angle

greatest angle

is acute

is 90°

60°

60°
5

Obtuse

greatest angle

5

is obtuse

397

Equilateral


– MEASUREMENT AND GEOMETRY –
■

In a right triangle, the side opposite the right
angle is called the hypotenuse. This will be the
longest side of the right triangle.

45-45-90 Right Triangles

45°

e

s
nu
te

po

Hy

45°

Right

A right triangle with two angles each measuring 45
degrees is called an isosceles right triangle. In an isosceles
right triangle:
Pythagorean Theorem
■

The Pythagorean theorem is an important tool for working with right triangles. It states: a2 + b2 = c2, where a and
b represent the legs and c represents the hypotenuse.
This theorem allows you to find the length of any side
as along as you know the measure of the other two.

■

The length of the hypotenuse is ͙2 multiplied by
ෆ
the length of one of the legs of the triangle.
͙ෆ
ᎏ
The length of each leg is ᎏ2 multiplied by the
2
length of the hypotenuse.

10

x
c

2

y
1

a2

b2

x=y=

c2

+ =
12 + 22 = c 2
1 + 4 = c2
5 = c2
͙5 = c
ෆ

͙2
ෆ
ᎏ
2

͙2
ෆ

10
ෆ
× ᎏ1ᎏ = 10 ᎏ = 5͙2

2

30-60-90 Triangles

In a right triangle with the other angles measuring 30
and 60 degrees:
■

■

The leg opposite the 30-degree angle is half the
length of the hypotenuse. (And, therefore, the
hypotenuse is two times the length of the leg
opposite the 30-degree angle.)
The leg opposite the 60-degree angle is ͙3 times
ෆ
the length of the other leg.
60°
s

2s

30°
s√¯¯¯
3

398


– MEASUREMENT AND GEOMETRY –

Example

D

A
60°
x

7

B
30°

y
ෆ
x = 2 × 7 = 14 and y = 7͙3

C
Step 2: Determine whether this is enough
information to prove the triangles are
congruent.
Yes, two angles and the side between them are
equal. Using the ASA rule, you can determine
that triangle ABD is congruent to triangle CBD.

Comparing Triangles

Triangles are said to be congruent (indicated by the symbol Х) when they have exactly the same size and shape.
Two triangles are congruent if their corresponding parts

(their angles and sides) are congruent. Sometimes, it is
easy to tell if two triangles are congruent by looking.
However, in geometry, you must be able to prove that the
triangles are congruent.
If two triangles are congruent, one of the three criteria listed below must be satisfied.

Polygons and Parallelograms
A polygon is a closed figure with three or more sides.
B

Side-Side-Side (SSS)

The side measures for both
triangles are the same.
Side-Angle-Side (SAS) The sides and the angle
between them are the same.
Angle-Side-Angle (ASA) Two angles and the side
between them are the same.
Example: Are triangles ᭝ABC and ᭝BCD
congruent?
Given: ∠ABD is congruent to ∠CBD and ∠ADB
is congruent to ∠CDB. Both triangles share
side BD.

C

A

F

■

■

B
■

■

C
Step 1: Mark the given congruencies on the
drawing.

E

Terms Related to Polygons

D

A

D

Vertices are corner points, also called endpoints,
of a polygon. The vertices in the above polygon
are: A, B, C, D, E, and F.
A diagonal of a polygon is a line segment between
two nonadjacent vertices. The two diagonals in
the polygon above are line segments BF and AE.
A regular polygon has sides and angles that are all

equal.
An equiangular polygon has angles that are all
equal.

Angles of a Quadrilateral

A quadrilateral is a four-sided polygon. Since a quadrilateral can be divided by a diagonal into two triangles,
399