ddljlk

10

Chapter 1

Basics of Geometry

Points, Lines, and Planes

USING UNDEFINED TERMS AND DEFINITIONS

A

uses known words to describe a new word. In geometry, some
words, such as point, line, and plane, are

Although these
words are not formally defined, it is important to have general agreement about
what each word means.

A

has no dimension. It is usually represented by a small dot.

A

extends in one dimension. It is usually represented by a straight line with
two arrowheads to indicate that the line extends without end in two directions.
In this book, lines are always straight lines.

A

extends in two dimensions. It is usually represented by a shape that
looks like a tabletop or wall. You must imagine that the plane extends without
end, even though the drawing of a plane appears to have edges.

Point A

Line ¬ or AB
¯˘
Plane M or plane ABC

A few basic concepts in geometry must also be commonly understood without
being defined. One such concept is the idea that a point lies on a line or a plane.

are points that lie on the same line.

are points that lie on the same plane.

Naming Collinear and Coplanar Points

a. Name three points that are collinear.

b. Name four points that are coplanar.

c. Name three points that are not collinear.

SOLUTION

a. Points D, E, and F lie on the same line, so they are collinear.

b. Points D, E, F, and G lie on the same plane, so they are coplanar. Also,
D, E, F, and H are coplanar, although the plane containing them is not drawn.

c. There are many correct answers. For instance, points H, E, and G do not lie
on the same line.

EXAMPLE 1

Coplanar points

Collinear points

plane

line

point

undefined terms.

definition

GOAL 1

Understand and
use the basic undefined
terms and defined terms
of geometry.

Sketch the
intersections of lines and
planes.

� To name and draw
the basic elements of
geometry, including lines
that intersect, as in the
perspective drawing in
Exs. 68–72.

Whyyou should learn it

GOAL 2

GOAL 1
Whatyou should learn
1.2

G

D
EF

H

A

B

l

A

B

C

M

A

R

E

A

L

L

I

F

E

R

E

A

L

L

I

F

E

Page 1 of 7

1.2Points, Lines, and Planes

11

Another undefined concept in geometry is the idea that a point on a line
is between two other points on the line. You can use this idea to define other
important terms in geometry.

Consider the

AB (symbolized by AB¯˘).
The

or

AB (symbolized by ABÆ)
consists of the

A and B, and all points on AB¯˘

that are between A and B.

The

AB (symbolized by ABÆ˘) consists of the

A and all points on AB¯˘ that lie on the

same side of A as point B.

Note that AB¯˘ is the same as BA¯˘, and ABÆ is the

same as BAÆ. However, ABÆ˘ and BAÆ˘ are not the same.

They have different initial points and extend in
different directions.

If C is between A and B, then CAÆ˘ and CBÆ˘ are

Like points, segments and rays are collinear if they lie on the same line. So, any
two opposite rays are collinear. Segments, rays, and lines are coplanar if they lie
on the same plane.

Drawing Lines, Segments, and Rays

Draw three noncollinear points, J, K, and L. Then draw JK¯˘, KLÆ and LJÆ˘.

SOLUTION

Draw J, K, and L.

Draw JK¯˘.

Draw KLÆ.

Draw LJÆ˘.

Drawing Opposite Rays

Draw two lines. Label points on the lines and name two pairs of opposite rays.

SOLUTION

Points M, N, and X are collinear and X is between M
and N. So, XMÆ˘ and XNÆ˘ are opposite rays.

Points P, Q, and X are collinear and X is between P
and Q. So, XPÆ˘ and XQÆ˘ are opposite rays.

EXAMPLE 3

4

3

2

1

J

K

L
J

K

L
J

K

L
J

K

L

EXAMPLE 2

opposite rays.

initial point

ray

endpoints

segment

line segment

line

X

M

P

q

N

line

A

B

segment

A

B

ray

A

B

ray

A

B

opposite rays

A

BC

Page 2 of 7

12

Chapter 1

Basics of Geometry

SKETCHING INTERSECTIONS OF LINES AND PLANES

Two or more geometric figures

if they have one or more points in
common. The

of the figures is the set of points the figures have
in common.

Sketching Intersections

Sketch the figure described.

a. a line that intersects a plane in one point

b. two planes that intersect in a line

SOLUTION

a.

b.

Draw a plane and a line.

Draw two planes.

Emphasize the point where

Emphasize the line where
they meet.

they meet.

Dashes indicate where the line is

Dashes indicate where one plane
hidden by the plane.

is hidden by the other plane.

EXAMPLE 4

intersection

intersect

GOAL 2

HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.

I

N

T

E

R

N

E

T

STUDENT HELP

Modeling Intersections

Use two index cards. Label them as shown and cut slots halfway along each card.

1. What is the intersection of ABÆ and CDÆ? of ABÆ and EFÆ?

2. Slide the cards together. What is the intersection of CDÆ and EFÆ?

3. What is the intersection of planes M and N?

4. Are CD¯˘ and EF¯˘ coplanar? Explain.

Developing
Concepts

ACTIVITY

A

B

C
DG

M

A
B

E

F

G

N

A
B

F

G

N

C

A

B

DG

M

E

Page 3 of 7

1.3Segments and Their Measures

17

Segments and Their Measures

USING SEGMENT POSTULATES

In geometry, rules that are accepted without proof are called

or
Rules that are proved are called theorems. In this lesson, you will study
two postulates about the lengths of segments.

Finding the Distance Between Two Points

Measure the length of the segment to the nearest millimeter.

SOLUTION
Use a metric ruler. Align one mark of the ruler with A. Then estimate the
coordinate of B. For example, if you align A with 3, B appears to align with 5.5.

AB = |5.5 º 3| = |2.5| = 2.5

�The distance between A and B is about 2.5 cm.
. . . . . . . . . .

It doesn’t matter how you place the ruler. For example, if the ruler in Example 1
is placed so that A is aligned with 4, then B aligns with 6.5. The difference in the
coordinates is the same.

EXAMPLE 1

axioms.

postulates

GOAL 1

Use segment
postulates.

Use the Distance
Formula to measure
distances, as applied in
Exs. 45–54.

� To solve real-life problems,
such as finding distances
along a diagonal city street
in Example 4.

Whyyou should learn it

GOAL 2

GOAL 1
Whatyou should learn
1.3

R

E

A

L

L

I

F

E

R

E

A

L

L

I

F

E

POSTULATE 1

Ruler Postulate

The points on a line can be matched
one to one with the real numbers.
The real number that corresponds to
a point is the

of the point.

The

between points A and B,
written as AB, is the absolute value of
the difference between the coordinates
of A and B.

AB is also called the

of AB
Æ.

length

distance

coordinate

POSTULATE

A

B
names of points

x1

x2
coordinates of points

A

B

A
B

AB

x1

x2

A

B

AB�|x2�x1|

Page 1 of 9

18

Chapter 1

Basics of Geometry

When three points lie on a line, you can say that one of them is

the
other two. This concept applies to collinear points only. For instance, in the
figures below, point B is between points A and C, but point E is not between
points D and F.

Point B is between points A and C.

Point E is not between points D and F.

Finding Distances on a Map

MAP READING Use the map to find the distances between the three cities
that lie on a line.

SOLUTION
Using the scale on the map, you can estimate
that the distance between Athens and Macon is

AM = 80 miles.

The distance between Macon and Albany is

MB = 90 miles.

Knowing that Athens, Macon, and Albany lie
on the same line, you can use the Segment
Addition Postulate to conclude that the
distance between Athens and Albany is

AB = AM + MB = 80 + 90 = 170 miles.
. . . . . . . . . .

The Segment Addition Postulate can be generalized
to three or more segments, as long as the segments
lie on a line. If P, Q, R, and S lie on a line as
shown, then

PS = PQ + QR + RS.

R

E

A

L

L

I

F

E

R

E

A

L

L

I

F

E

EXAMPLE 2

between

POSTULATE 2

Segment Addition Postulate

If B is between A and C, then AB + BC = AC.

If AB + BC = AC, then B is between A and C.

POSTULATE

A

CB

AC

AB

BC

0

100

Athens

Macon

Albany

170 mi
80 mi

90 mi

A

M

B

P

S

q

R

PS

Pœ

œR

RS

A

C

B

D

F

E

Page 2 of 9

1.3Segments and Their Measures

19

STUDENT HELP

Study Tip
The small numbers in
x1 and x2 are called
subscripts. You read
them as “x sub 1” and
“x sub 2.”

If A(x1, y1) and B(x2, y2) are points in
a coordinate plane, then the distance
between A and B is

AB = �(x�2�º� x�1)�2�+� (�y�2�º� y�1)�2�.

THE DISTANCE FORMULA

y

x

B(x2, y2)

A(x1, y1)

|y2�y1|

C(x2, y1)

|x2�x1|

Using
Algebra

xyxy

USING THE DISTANCE FORMULA

The

is a formula for computing the distance between two
points in a coordinate plane.

Using the Distance Formula

Find the lengths of the segments. Tell whether
any of the segments have the same length.

SOLUTION
Use the Distance Formula.

AB = �[(�º�4�)�º� (�º�1�)]�2�+� (�3� º� 1�)2�

= �(º�3�)2� +� 2�2� = �9� +� 4� = �1�3�

AC = �[3� º� (�º�1�)]�2�+� (�2� º� 1�)2�

= �4�2�+� 1�2� = �1�6� +� 1� = �1�7�

AD = �[2� º� (�º�1�)]�2�+� (�º�1� º� 1�)2�

= �3�2�+� (�º�2�)2� = �9� +� 4� = �1�3�

�So, ABÆ and ADÆ have the same length, but ACÆ has a different length.
. . . . . . . . . .

Segments that have the same length are called

For instance,
in Example 3, ABÆ and ADÆ are congruent because each has a length of �1�3�.
There is a special symbol, £, for indicating congruence.

LENGTHS ARE EQUAL.

SEGMENTS ARE CONGRUENT.

AB = AD

ABÆ £ ADÆ

“is equal to”

“is congruent to”

congruent segments.

EXAMPLE 3

Distance Formula

GOAL 2

y

x3

2

B(�4, 3)

A(�1, 1)

C(3, 2)

D(2, �1)

Page 3 of 9

20

Chapter 1

Basics of Geometry

The Distance Formula is based on the Pythagorean Theorem, which you will see
again when you work with right triangles in Chapter 9.

Finding Distances on a City Map

MAP READING On the map, the city
blocks are 340 feet apart east-west and
480 feet apart north-south.

a. Find the walking distance between A
and B.

b. What would the distance be if a diagonal
street existed between the two points?

SOLUTION

a. To walk from A to B, you would have to
walk five blocks east and three blocks north.

5 blocks • 340 �b
f
l
e
o
e
c
t
k� = 1700 feet

3 blocks • 480 �b
f
l
e
o
e
c
t
k� = 1440 feet

�So, the walking distance is 1700 + 1440,
which is a total of 3140 feet.

b. To find the diagonal distance between A and B,
use the Distance Formula.

AB = �[1�0�2�0� º� (�º�6�8�0�)]�2�+� [�9�6�0� º� (�º�4�8�0�)]�2�

= �1�7�0�0�2�+� 1�4�4�0�2�

= �4�,9�6�3�,6�0�0� ≈ 2228 feet

�So, the diagonal distance would be about 2228 feet, which is 912 feet less
than the walking distance.

R

E

A

L

L

I

F

E

R

E

A

L

L

I

F

E

EXAMPLE 4

STUDENT HELP

Study Tip
The red mark at one
corner of each triangle
indicates a right angle.

STUDENT HELP

Study Tip
If you use a calculator to
compute distances, use
the parenthesis keys to
group what needs to be
squared.

y

x

480

340

DISTANCE FORMULA

PYTHAGOREAN THEOREM
(AB)2= (x2 º x1)2+ (y2 º y1)2

c2= a2+ b2

DISTANCE FORMULA AND PYTHAGOREAN THEOREM
CONCEPT
SUMMARY

y

x

B

C

A

2228 ft

1440 ft

1700 ft

B(x2, y2)

A(x1, y1)

C(x2, y1)

|x2�x1|

|y2�y1|

b

a

c

Page 4 of 9

22

Chapter 1

Basics of Geometry

USING ALGEBRA Suppose M is between L and N. Use the Segment
Addition Postulate to solve for the variable. Then find the lengths of LM
Æ,
MN
Æ, and LN
Æ.

31. LM = 3x + 8

32. LM = 7y + 9

33. LM = �1
2�z + 2

MN = 2x º 5

MN = 3y + 4

MN = 3z + �2
3
�

LN = 23

LN = 143

LN = 5z + 2

DISTANCE FORMULA Find the distance between each pair of points.

34.

35.

36.

DISTANCE FORMULA Find the lengths of the segments. Tell whether any of
the segments have the same length.

37.

38.

39.

CONGRUENCE Use the Distance Formula to decide whether PQ
Æ £ QR
Æ.

40. P(4, º4)

41. P(º1, º6)

42. P(5, 1)

43. P(º2, 0)
Q(1, º6)

Q(º8, 5)

Q(º5, º7)

Q(10, º14)
R(º1, º3)

R(3, º2)

R(º3, 6)

R(º4, º2)

CAMBRIA INCLINE In Exercises 44 and 45, use the information about
the incline railway given below.
In the days before automobiles were available, railways called “inclines” brought
people up and down hills in many cities. In Johnstown, Pennsylvania, the
Cambria Incline was reputedly the steepest in the world when it was completed
in 1893. It rises about 514 feet vertically as it moves 734 feet horizontally.

44. On graph paper, draw a coordinate
plane and mark the axes using a scale
that allows you to plot (0, 0) and
(734, 514). Plot the points and connect
them with a segment to represent the
incline track.

45. Use the Distance Formula to estimate
the length of the track.

y

x

P(7, �6)

5

5

M(1, 7)

N(�2, �3)

L(�8, 6)

y

x

2

2

H(4, �4)

E(1, 4)

F(5, 6)

G(5, 1)

y

x4

6

A(�3, 8)

C(0, 2)

D(2, �4)

B(6, 5)

y

x
2

2

G(�2, 4)
H(5, 5)

J(4, �1)

y

x

2

2

D(�3, 6)

E(6, 8)

F(0, 2)

y

x

2

2

A(�4, 7)

B(6, 2)

C(3, �2)

xyxy

Workers constructing the
Cambria Incline

HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with Exs. 34–36.

I

N

T

E

R

N

E

T

STUDENT HELP

Page 6 of 9

26

Chapter 1

Basics of Geometry

Angles and Their Measures

USING ANGLE POSTULATES

An

consists of two different rays that
have the same initial point. The rays are the
of the angle. The initial point is the
of the angle.

The angle that has sides ABÆ˘ and ACÆ˘ is denoted
by ™BAC, ™CAB, or ™A. The point A is the
vertex of the angle.

Naming Angles

Name the angles in the figure.

SOLUTION
There are three different angles.
• ™PQS or ™SQP
• ™SQR or ™RQS
• ™PQR or ™RQP

You should not name any of these angles as ™Q because all three angles have Q
as their vertex. The name ™Q would not distinguish one angle from the others.
. . . . . . . . . .

The measure of ™A is denoted by
m™A. The measure of an angle can be
approximated with a protractor, using
units called degrees (°). For instance,
™BAC has a measure of 50°, which can
be written as

m™BAC = 50°.

Angles that have the same measure are called
For instance, ™BAC and ™DEF
each have a measure of 50°, so they are congruent.

MEASURES ARE EQUAL.

ANGLES ARE CONGRUENT.

m™BAC = m™DEF

™BAC £ ™DEF

“is equal to”

“is congruent to”

congruent angles.

EXAMPLE 1

vertex
sides

angle

GOAL 1

Use angle
postulates.

Classify angles as
acute, right, obtuse, or
straight.

� To solve real-life problems
about angles, such as the
field of vision of a horse
wearing blinkers in
Example 2.

Whyyou should learn it

GOAL 2

GOAL 1
Whatyou should learn
1.4

R

E

A

L

L

I

F

E

R

E

A

L

L

I

F

E

vertex

sides

B

A

C

0180

1800

10170

20160

30
150

40
140

130

60
120

70
110

80
100

90

100
80

110
70

120
60

13050

30

14040

17010

16020

150

123456

A

C

B

q

P

S

R

E

D

F

50�

Page 1 of 7

region seen
by both eyes

123

1.4Angles and Their Measures

27

A point is in the

of an angle if it is
between points that lie on each side of the angle.

A point is in the

of an angle if
it is not on the angle or in its interior.

POSTULATE

Calculating Angle Measures

VISION Each eye of a horse wearing
blinkers has an angle of vision that
measures 100°. The angle of vision that is
seen by both eyes measures 60°.

Find the angle of vision seen by the left
eye alone.

SOLUTION
You can use the Angle Addition Postulate.

m™2 + m™3 = 100°

Total vision for left eye is 100°.

m™3 = 100° º m™2

Subtract m™2 from each side.

m™3 = 100° º 60°

Substitute 60° for m™2.

m™3 = 40°

Subtract.

�So, the vision for the left eye alone measures 40°.

R

E

A

L

L

I

F

E

R

E

A

L

L

I

F

E

EXAMPLE 2

exterior

interior

POSTULATE 3

Protractor Postulate

Consider a point A on one side of
OB
¯˘. The rays of the form OA
Æ˘ can
be matched one to one with the
real numbers from 0 to 180.

The

of ™AOB is equal
to the absolute value of the
difference between the real
numbers for OA
Æ˘ and OB
Æ˘.

measure

POSTULATE

POSTULATE 4

Angle Addition Postulate

If P is in the interior of ™RST, then

m™RSP + m™PST = m™RST.

POSTULATE

0180

1800

10170

20160

30
150

40
140

50
130

60
120

70
110

80
100

90

100
80

110
70

120
60

13050

30

14040

17010

16020

150

123456

A

O

B

A

D

E

exterior

interior

S

R

T

P

måRST

måRSP

måPST

STUDENT HELP

Study Tip
As shown in Example 2,
it is sometimes easier
to label angles with
numbers instead of
letters.

Logical
Reasoning

Page 2 of 7

28

Chapter 1

Basics of Geometry

CLASSIFYING ANGLES

Angles are classified as

and

according to their
measures. Angles have measures greater than 0° and less than or equal to 180°.

Acute angle

Right angle

Obtuse angle

Straight angle

0° < m™A < 90°

m™A = 90°

90° < m™A < 180°

m™A = 180°

Classifying Angles in a Coordinate Plane

Plot the points L(º4, 2), M(º1, º1), N(2, 2), Q(4, º1), and P(2, º4). Then
measure and classify the following angles as acute, right, obtuse, or straight.

a. ™LMN

b. ™LMP

c. ™NMQ

d. ™LMQ

SOLUTION
Begin by plotting the points. Then use a protractor to measure each angle.

MEASURE

CLASSIFICATION

a. m™LMN = 90°

right angle

b. m™LMP = 180°

straight angle

c. m™NMQ = 45°

acute angle

d. m™LMQ = 135°

obtuse angle
. . . . . . . . .

Two angles are

if they share a common vertex and side, but
have no common interior points.

Drawing Adjacent Angles

Use a protractor to draw two adjacent acute angles ™RSP and ™PST so that
™RST is (a) acute and (b) obtuse.

SOLUTION

a.

b.

EXAMPLE 4

adjacent angles

EXAMPLE 3

straight,

obtuse,

right,

acute,

GOAL 2

A

A

A

A

y

x

L(�4, 2)

M(�1, �1)

N(2, 2)

œ(4, �1)

P(2, �4)

STUDENT HELP

Study Tip
The mark used to
indicate a right angle
resembles the corner of
a square, which has four
right angles.

0

180

90

123456

100

110

120

130

140

150

160

170

80

70

60

50

40

30

20

10

35�

25�

R

S

T

P

0

180

90

123456

100

110

120

130

140

150

160

170

80

70

60

50

40

30

20

10

65�

65�

R

S

T
P

HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.

I

N

T

E

R

N

E

T

STUDENT HELP

Page 3 of 7

1.4Angles and Their Measures

29

Match the angle with its classification.

A. acute

B. obtuse

C. right

D. straight

1.

2.

3.

4.

Use the diagram at the right to answer the questions. Explain your
answers.

5. Is ™DEF £ ™FEG?

6. Is ™DEG £ ™HEG?

7. Are ™DEF and ™FEH adjacent?

8. Are ™GED and ™DEF adjacent?

Name the vertex and sides of the angle. Then estimate its measure.

9.

10.

11.

12.

Classify the angle as acute, obtuse, right, or straight.

13. m™A = 180°

14. m™B = 90°

15. m™C = 100°

16. m™D = 45°

NAMING PARTS Name the vertex and sides of the angle.

17.

18.

19.

NAMING ANGLES Write two names for the angle.

20.

21.

22.

T

P

S

B

C

D

E

A

U

R
S

q

N

K

E

X

F

T

PRACTICE AND APPLICATIONS

T

R

S
J
H

K
M

L

N

E

D

F

D

E

H

G
F

45�

45�

B

C
A

B
C

A
B
C

A

BC

A

GUIDED PRACTICE

Vocabulary Check 

Concept Check 

Skill Check 

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 17–22
Example 2: Exs. 23–34
Example 3: Exs. 35–43
Example 4: Exs. 38, 39

Extra Practice
to help you master
skills is on pp. 803
and 804.

STUDENT HELP

Page 4 of 7