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  • An Introduction to Geometry

Prealgebra

Alan S. Tussy, R. David Gustafson, Diane R. Koenig

Chapter 9

An Introduction to Geometry - all with Video Answers

Educators

Section 1

Basic Geometric Figures; Angles

Problem 1

Fill in the blanks.
Three undefined words in geometry are ____ ____ and ____.

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Problem 2

Fill in the blanks.
A line ____ has two endpoints.

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Problem 3

Fill in the blanks.
A ____ divides a line segment into two parts of equal length.

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Problem 4

Fill in the blanks.
A____ is the part of a line that begins at some point and continues forever in one direction.

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Problem 5

Fill in the blanks.
$\mathrm{An}$_____ is formed by two rays with a common endpoint.

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Problem 6

An angle is measured in ______ .

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Problem 7

Fill in the blanks.
A____ is used to measure angles.

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Problem 8

Fill in the blanks.
The measure of an ____ angle is less than $90^{\circ}$.

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Problem 9

Fill in the blanks.
The measure of a ____ angle is $90^{\circ}$.

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Problem 10

The measure of an _____ angle is greater than $90^{\circ}$ but less than $180^{\circ}$.

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Problem 11

Fill in the blanks.
The measure of a straight angle is _____.

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Problem 12

Fill in the blanks.
When two segments have the same length, we say that they are _____.

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Problem 13

Fill in the blanks.
____angles have the same vertex, are side-by side, and their interiors do not overlap.

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Problem 14

Fill in the blanks.
When two lines intersect, pairs of nonadjacent angles are called _____ angles.

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Problem 15

Fill in the blanks.
When two angles have the same measure, we say that they are ____.

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Problem 16

Fill in the blanks.
The word sum indicates the operation of ____.

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00:45

Problem 17

Fill in the blanks.
The sum of two complementary angles is ____.

Sheryl Ezze
Sheryl Ezze
Numerade Educator

Problem 18

Fill in the blanks.
The sum of two ____ angles is $180^{\circ}$.

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Problem 19

a. Given two points (say, $M$ and $N$ ), how many different lines pass through these two points?
b. Fill in the blank: In general, two different points determine exactly one _____.

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Problem 20

Fill in the blanks.
Refer to the figure.
a. Name $\overrightarrow{N M}$ in another way.
b. Do $\overrightarrow{M N}$ and $\overrightarrow{N M}$ name the same ray?
(FIGURE NOT COPY)

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Problem 21

Consider the acute angle shown below.
a. What two rays are the sides of the angle?
b. What point is the vertex of the angle?
c. Name the angle in four ways.
(FIGURE NOT COPY)

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Problem 22

Estimate the measure of each angle. Do not use a protractor.
(FIGURE NOT COPY)

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Problem 23

Draw an example of each type of angle.
a. an acute angle
b. an obtuse angle
c. a right angle
d. a straight angle

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Problem 24

Fill in the blanks with the correct symbol.
a. If $\mathrm{m}(\overline{A B})=\mathrm{m}(\overline{C D}),$ then $\overline{A B} \quad \overline{C D}$
b. If $\angle A B C \approx \angle D E F,$ then $\mathrm{m}(\angle A B C) \quad \mathrm{m}(\angle D E F)$

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Problem 25

a. Draw a pair of adjacent angles. Label them $\angle A B C$ and $\angle C B D$
b. Draw two intersecting lines. Label them lines $l_{1}$ and $l_{2} .$ Label one pair of vertical angles that are formed as $\angle 1$ and $\angle 2$
c. Draw two adjacent complementary angles.
d. Draw two adjacent supplementary angles.

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Problem 26

Fill in the blank:
If $\angle M N O \cong \angle B F G,$ then $\mathrm{m}(\angle M N O) \quad \mathrm{m}(\angle B F G)$

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01:13

Problem 27

Fill in the blank:
The vertical angle property: Vertical angles are ____.

Ashley High
Ashley High
Numerade Educator

Problem 28

Refer to the figure below. Fill in the blanks.
a. $\angle X Y Z$ and $\angle \longrightarrow$ are vertical angles.
b. $\angle X Y Z$ and $\angle Z Y W$ are $\quad$ angles
c. $\angle Z Y W$ and $\angle X Y V$ are
(FIGURE NOT COPY)

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Problem 29

Refer to the figure below and tell whether each statement is true.
a. $\angle A G F$ and $\angle B G C$ are vertical angles.
b. $\angle F G E$ and $\angle B G A$ are adjacent angles.
c. $\mathrm{m}(\angle A G B)=\mathrm{m}(\angle B G C)$
d. $\angle A G C \approx \angle D G F$
(FIGURE NOT COPY)

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Problem 30

Refer to the figure below and tell whether the angles are congruent.
a. $\angle 1$ and $\angle 2$
b. $\quad \angle F G B$ and $\angle C G E$
c. $\angle A G F$ and $\angle F G E$
d. $\angle C G D$ and $\angle C G B$
(FIGURE NOT COPY)

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Problem 31

Refer to the figure above and tell whether each statement is true.
$\angle 1$ and $\angle C G D$ are adjacent angles.

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Problem 32

Refer to the figure above and tell whether each statement is true.
$\angle F G A$ and $\angle A G C$ are supplementary.

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Problem 33

Refer to the figure above and tell whether each statement is true.
$\angle A G B$ and $\angle B G C$ are complementary.

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Problem 34

Refer to the figure above and tell whether each statement is true.
$\angle A G F$ and $\angle 2$ are complementary.

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Problem 35

The symbol $\overrightarrow{A B}$ is read as "____$AB$."

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Problem 36

Fill in the blanks.
The symbol $\overline{A B}$ is read as $"$_____AB."

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Problem 37

Fill in the blanks.
The symbol $\overrightarrow{A B}$ is read as" ____ $AB$."

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Problem 38

Fill in the blanks.
We read $\mathrm{m}(\overrightarrow{A B})$ as "the ____ of segment $A B$."

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Problem 39

Fill in the blanks.
We read $\angle A B C$ as "_____$ABC$."

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Problem 40

Fill in the blanks.
We read $\mathrm{m}(\angle A B C)$ as "the____ of angle $ABC."$

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Problem 41

Fill in the blanks.
The symbol for ____is a small raised circle, $^{0}.$

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Problem 42

Fill in the blanks.
The symbol 7 indicates a ____ angle.

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Problem 43

Fill in the blanks.
The symbol \cong is read as "is ____ to."

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Problem 44

Fill in the blanks.
The symbol $l_{1}$ can be used to name a line. It is read as "line $l$ ____ one."

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Problem 45

Draw each geometric figure and label it completely. See Objective 1
a. Point $T$
b. $\overrightarrow{J K}$
c. Plane $A B C$

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Problem 46

Draw each geometric figure and label it completely. See Objectives 2 and 3
a. $\overline{R S}$
b. $\overrightarrow{P Q}$
c. $\angle X Y Z$
d. $\angle L$

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Problem 47

Refer to the figure and find the length of each segment. See Objective 2 .
(FIGURE NOT COPY)
a. $\overline{A B}$
b. $\overline{C E}$
c. $\overline{D C}$
$\overline{E A}$

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Problem 48

Refer to the figure and find the length of each segment. See Objective 2.
a. Find the midpoint of $\overline{A D}$.
b. Find the midpoint of $\overline{B E}$.
c. Find the midpoint of $\overline{E A}$.

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Problem 49

Use the protractor to find each angle measure listed below.
$$\mathrm{m}(\angle G D E)$$
(FIGURE NOT COPY)

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Problem 50

Use the protractor to find each angle measure listed below.
$$\mathrm{m}(\angle A D E)$$
(FIGURE NOT COPY)

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Problem 51

Use the protractor to find each angle measure listed below.
$$m(\angle E D S)$$
(FIGURE NOT COPY)

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Problem 52

Use the protractor to find each angle measure listed below.
$$\mathrm{m}(\angle E D R)$$
(FIGURE NOT COPY)

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Problem 53

Use the protractor to find each angle measure listed below.
$$\mathrm{m}(\angle C D R)$$
(FIGURE NOT COPY)

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Problem 54

Use the protractor to find each angle measure listed below.
$$\mathrm{m}(\angle C D A)$$
(FIGURE NOT COPY)

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Problem 55

Use the protractor to find each angle measure listed below.
$$\mathrm{m}(\angle C D G)$$
(FIGURE NOT COPY)

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Problem 56

Use the protractor to find each angle measure listed below.
$$\mathrm{m}(\angle C D S)$$
(FIGURE NOT COPY)

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Problem 57

Classify the following angles in the figure as an acute angle, a right angle, an obtuse angle, or a straight angle. See Example $1 .$
$$\angle M N O$$
(FIGURE NOT COPY)

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Problem 58

Classify the following angles in the figure as an acute angle, a right angle, an obtuse angle, or a straight angle.
$$\angle O P N$$
(FIGURE NOT COPY)

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Problem 59

Classify the following angles in the figure as an acute angle, a right angle, an obtuse angle, or a straight angle.
$$<N Q P$$
(FIGURE NOT COPY)

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Problem 60

Classify the following angles in the figure as an acute angle, a right angle, an obtuse angle, or a straight angle.
$$\angle P O S$$
(FIGURE NOT COPY)

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Problem 61

Classify the following angles in the figure as an acute angle, a right angle, an obtuse angle, or a straight angle.
$$\angle M P Q$$
(FIGURE NOT COPY)

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Problem 62

Classify the following angles in the figure as an acute angle, a right angle, an obtuse angle, or a straight angle.
$$\angle P N O$$
(FIGURE NOT COPY)

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Problem 63

Classify the following angles in the figure as an acute angle, a right angle, an obtuse angle, or a straight angle.
$$\angle Q P O$$
(FIGURE NOT COPY)

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Problem 64

Classify the following angles in the figure as an acute angle, a right angle, an obtuse angle, or a straight angle.
$$\angle M N Q$$
(FIGURE NOT COPY)

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Problem 65

Find
(FIGURE NOT COPY)

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Problem 66

Find $x$.
(FIGURE NOT COPY)

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Problem 67

Find
(FIGURE NOT COPY)

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Problem 68

Find $x$.
(FIGURE NOT COPY)

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Problem 69

Refer to the figure below. Find the measure of each angle.
$\angle 1$
(figure not copy)

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Problem 70

Refer to the figure below. Find the measure of each angle.
$\angle M Y X$
(figure not copy)

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Problem 71

Refer to the figure below. Find the measure of each angle.
$\angle N Y Z$
(figure not copy)

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Problem 72

Refer to the figure below. Find the measure of each angle.
$\angle 2$
(figure not copy)

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Problem 73

First find $x$. Then find $m(\angle A B D)$ and $m(\angle D B E)$.
(figure not copy)

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Problem 74

First find $x$. Then find $m(\angle A B D)$ and $m(\angle D B E)$.
(figure not copy)

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Problem 75

First find $x$. Then find $m(\angle Z Y Q)$ and $m(\angle P Y Q)$.
(figure not copy)

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Problem 76

First find $x$. Then find $m(\angle Z Y Q)$ and $m(\angle P Y Q)$.
(figure not copy)

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Problem 77

Let $x$ represent the unknown angle measure. Write an equation and solve it to find $x$.
Find the complement of a $30^{\circ}$ angle.

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Problem 78

Let $x$ represent the unknown angle measure. Write an equation and solve it to find $x$.
Find the supplement of a $30^{\circ}$ angle.

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Problem 79

Let $x$ represent the unknown angle measure. Write an equation and solve it to find $x$.
Find the supplement of a $105^{\circ}$ angle.

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Problem 80

Let $x$ represent the unknown angle measure. Write an equation and solve it to find $x$.
Find the complement of a $75^{\circ}$ angle.

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Problem 81

Refer to the figure in the next column and tell whether each statement is true. If a statement is false, explain why.
a. $\overrightarrow{G F}$ has point $G$ as its endpoint.
b. $\overline{A G}$ has no endpoints.
c. $\overrightarrow{C D}$ has three endpoints.
d. Point $D$ is the vertex of $\angle D G B$.
e. $m(\angle A G C)=m(\angle B G D)$
f. $\angle A G F \cong \angle B G E$
(figure not copy)

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Problem 82

Refer to the figure for Problem 81 and tell whether each angle is an acute angle, a right angle, an obtuse angle, or a straight angle.
a. $\angle A G C$
b. $\angle E G A$
c. $\angle F G D$
d. $\angle B G A$

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Problem 83

Use a protractor to measure each angle.
(figure not copy)

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Problem 84

Use a protractor to measure each angle.
(figure not copy)

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Problem 85

Use a protractor to measure each angle.
(figure not copy)

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Problem 86

Use a protractor to measure each angle.
(figure not copy)

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Problem 87

Refer to the figure below, in which $\mathrm{m}(\angle 1)=50^{\circ} .$ Find the measure of each angle or sum of angles.
a. $\angle 3$
b. $\angle 4$
c. $m(\angle 1)+m(\angle 2)+m(\angle 3)$
d. $m(\angle 2)+m(\angle 4)$
(figure not copy)

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Problem 88

Refer to the figure below, in which $m(\angle 1)+m(\angle 3)+$ $\mathrm{m}(\angle 4)=180^{\circ}, \angle 3 \cong \angle 4,$ and $\angle 4 \cong \angle 5 .$ Find the
measure of each angle.
a. $\angle 1$
b. $\angle 2$
c. $\angle 3$
d. $\angle 6$
(figure not copy)

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Problem 89

Refer to the figure below where $\angle 1 \cong \angle A C D$ $\angle 1 \cong \angle 2,$ and $\angle B A C \cong \angle 2$
a. What is the complement of $\angle B A C ?$
b. What is the supplement of $\angle B A C ?$
(figure not copy)

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Problem 90

Refer to the figure below where $\angle E B S \cong \angle B E S$.
a. What is the measure of $\angle A E F ?$
b. What is the supplement of $\angle A E T ?$
(figure not copy)

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Problem 91

Find the supplement of the complement of a $51^{\circ}$ angle.

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Problem 92

Find the complement of the supplement of a $173^{\circ}$ angle.

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Problem 93

Find the complement of the complement of a
$1^{\circ}$ angle.

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Problem 94

Find the supplement of the supplement of a $6^{\circ}$ angle.

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Problem 95

MUSICAL INSTRUMENTS Suppose that you are a beginning band teacher describing the correct posture needed to play various instruments. Using the diagrams shown below, approximate the angle measure (in degrees) at which each instrument should be held in relation to the student's body.
a. flute
b. clarinet
c. trumpet
(picture not copy)

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Problem 96

PLANETS The figures below show the direction of rotation of several planets in our solar system. They also show the angle of tilt of each planet.
a. Which planets have an angle of tilt that is an acute angle?
b. Which planets have an angle of tilt that is an bbtuse angle?
(image can't copy)

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Problem 97

a. AVIATION How many degrees from the horizontal position are the wings of the airplane?
(image can't copy)
b. GARDENING What angle does the handle of the lawn mower make with the ground?
(image can't copy)

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Problem 98

SYNTHESIZER Find $x$ and $y$
(image can't copy)

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Problem 99

PHRASES Explain what you think each of these phrases means. How is geometry involved?
a. The president did a complete 180 -degree flip on the subject of a tax cut.
b. The rollerblader did a " $360 "$ as she jumped off the ramp.

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Problem 100

In the statements below, the $^{\circ}$ symbol is used in two different ways. Explain the difference.
$\mathrm{m}(\angle A)=85^{\circ}$ and $85^{\circ} \mathrm{F}$

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Problem 101

Can two angles that are complementary be equal? Explain.

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Problem 102

Explain why the angles highlighted below are not vertical angles.
(image can't copy)

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Problem 103

Add: $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}$

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Problem 104

Subtract: $\frac{3}{4}-\frac{1}{8}-\frac{1}{2}$

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Problem 105

Multiply: $\frac{5}{8} \cdot \frac{2}{15} \cdot \frac{6}{5}$

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Problem 106

Divide: $\frac{12}{17} \div \frac{4}{34}$

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