Download Ch 1

Exploring Geometry:
Points, lines, and
Angles in the Plane
A
ir traffic controllers keep airplanes moving in a safe and
orderly way. Each air traffic controller watches one or
more "sectors" or sections of airspace. They
direct airplanes in their sector when to take off, land, or change
flight paths. Therefore, they must keep track of the speed, altitude,
and direction the airplanes are traveling.
An air traffic controller's computer screen uses points and lines to
show airplanes and their routes. Even though the airplanes are all
flying at different altitudes, their routes are shown on a flat screen,
or plane. Points and lines are two building blocks in your study of
plane geometry.
In Chapter 1, you'll learn about points, lines, and angles in the
plane.
Goals for Learning
 To recognize a point, a line, and a plane
 To identify line segments and rays
 To use postulates to determine how to use a ruler with
geometric figures
 To construct angles-copying and bisecting
 To measure and classify angles
 To identify complementary and supplementary angles
 To use algebra to solve problems in geometry
1
Geometry
The study of points,
lines, angles, surfaces,
and solids
Point
·A
A location in space
represented by a dot
Line
H
AB
A set of many points
that extend in opposite
directions without
ending
In geometry, you study the size, shape, and position of objects.
One way to make that study easier is to concentrate on the outline
of an object. Look closely at the points and lines that make up
geometric figures.
A point is the simplest of geometric figures and it has no formal
definition. In geometry, a point is represented by a dot on a piece of
paper. Capital letters are used to name points. A point has a location
but no size or shape. You might think of a point as a star in the night
sky. Here are some examples of points.
A
0
•
•
c
•
Plane 0 A twoB
dimensional fiat surface
•
These points are named A, B, C, and D.
Please note that no
formal definition can
be given for point,
line, and plane.
There are two other concepts in geometry that have no formal
definition-a line and a plane. A straight two-headed arrow
represents a line. A line is endless in two directions. A line can be
named by naming any two points on it, or it can be named with a
lowercase letter. You might think of a line as a straight, thin wire or
thread, but a geometric line has no thickness. Here is an example of
a line.
~~--------._--------~ m
A
B
H H
This line is named AB or BA or m.
~
o:
2
The four-sided shape at the left represents a
plane. A plane is a flat surface. It extends in
all directions with no end and has no
thickness. A plane is named using capital
letters. You might think of a page from this
book or a tabletop as part of a plane.
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
Ii'
t
\J This is
plane W.
I
Name this line in seven different ways.
Collinear
Points on the same line
A
Line segment AB
AB is the set of points A,
B
B, and all the points
between A and B
c
.---.
Endpoints A B
A and B are the
endpoints of AB where
AB is the set of points A,
B, and all the points
between A and B
~~~~~~
This line can be named AB, BA, AC, CA, BC, CB, or C.
Points on the same line are called collinear points. In the
example above, A, B, and C are collinear points on line -e.
Which three of these points are collinear?
G
D
A
•
•
E
•
·
c
H
•
F
B
Points 0, E, and F appear to be on the same line.
Recall that a set
is a collection of
particular things, like
the set of points on
a line segment.
If you draw a line between any two
points, such as A and B, you have drawn
a line segment. A line segment is part of
a line.
A------B
This segment ~
named AB or BA.
AB is a line segment, the set of points A, B, and all points
between A and B. A and Bare called the endpoints of the
segment. The symbol AB is read "line segment AB."
Draw and name all the line
segments between points P,
0, and R.
P.
•
Remember that you need two
endpoints for each~gmen~
You can draw PO, OR, and RP.
Q
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1 3
~
Ray
AB
A set of points that is
part of a line; a ray has
one endpoint and
extends in one direction
with no end
Another part of a line is called a ray. A ray is a line with one
endpoint. You might think of it as a ray of light leaving the sun
and traveling into space forever. A straight arrow is used to
represent a ray. The symbol jj is read "ray AB."
•
•
A
B
••
~
This ray is named AB. A is its endpoint.
A ray extends from its endpoint in one direction with no end.
This chart brings together all of the geometric figures and their
definitions. Come back to this chart if you have any questions.
about what a diagram or symbol means .
..,._
~fffim1
..n. _"
~.lE.llL
~'HJ.' •
Point A
'A
A
H
Line AB or line BA
•••••A
H
ABor BA
~
RayAB
AB
•••
B
•
•••
A
B
B
A
Line segment AB
-
or line segment BA
-
ABor BA
AB
Length of AB or
Exercise A Which three points are collinear? There may be
A
B
distance from A to B AB
more than one correct answer.
OW
Plane
e
B
1.
~ W
2. ~
'B
•
-$-
F·
•
·e
•
E
0
•
•
G
3. ~
'
0
E·
F·
•
'H
e·
4
o
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
F
E
•
•
G
•
Exercise B Tell whether each figure is a line, a line segment, or
a ray.
4.
5.
6.
I
~
7.~
8.
..•.
'"
••
I
9 .. ~.
Exercise C Name the line as many ways as you can.
_m
10
.
H
G
Exercise D Answer each question.
11. How many endpoints does a line segment have?
12. How many endpoints does a line have?
13. How many endpoints does a ray have?
14. Are two points always collinear? Explain why or why not.
15. What geometric term(s) can you use to describe the lights in
this picture?
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1
16. What geometric term(s) can you use to describe the lights on a
straight string of holiday lights?
17. What geometric term can you use to describe the light
from a flashlight?
Exercise E Answer the following questions.
18. How many line segments can you
draw using 2 points as endpoints?
19. How many line segments can you
draw using 3 points as endpoints?
20. Complete the following table to show
how many segments can be drawn
using the given number of points as
endpoints. Hint: Draw a triangle (3
sides), square (4 sides), 5- sided figure,
6-sided figure, and 8- sided figure.
Connect the corners (endpoints).
Count the segments.
2
Hint: Let n =
6
6
You experimented by counting segments. Can you
discover a pattern or formula to give you the answer?
the number of sides or points.
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
Postulate
A statement about
geometric figures
accepted as true
without proof
Measurement is an important part of geometry. You use
measurement every day. The measuring tool you use most often is
a ruler. You can think of a ruler as a line with numbers on it
Modern geometry has set some rules on how to use a ruler for
geometric figures. The rules are called postulates. A postulate is a
statement about geometric figures accepted as true without proof.
Here are three postulates involving rulers.
Ruler Postulate:
The points on a line can be placed in a one-to-One
correspondence with real numbers so that
1. for every point on the line, there is exactly one real number.
2. for every real number, there is exactly one point on the line.
1. the distance between any two points is the absolute value of
the difference of the corresponding real numbers.
Line
B
A
Line as Ruler
Remember that
the absolute value of
a number is
equivalent to its
distance from zero
on the number line.
Inl is the symbol for
the absolute value
of n. For example,
1-31 = 3.
A
011(
I
B
t
••
t
- to -1
2
0
A corresponds
O. B corresponds
to 3. 3
4
5
2
The distance between A and B is 3.
Caution: When writing a distance, use AB without a
bar to represent a number. AB with a bar names a line
segment, not its length.
Ruler Postulate
AB =
13 - 01 or 10 - 31 = 3
Notice that A was placed at 0 and Bat 3. This makes it easy to
find the distance between A and B--you can read it directly from
the number line. The following postulate allows you to do this
for any two points.
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1 7
Ruler Placement postulate:
Given two points, A and B on a line, the number line can be
chosen so that A is at zero and B is at a positive number.
The set of real
numbers contains
rational and
irrational numbers.
A rational number
can be written as a
fraction whose
numerator and
denominator are whole
numbers. Any number
that is not rational is
called irrational. For
example, 1t and V2
are irrational numbers.
Given AB.Find AB, the length of AB.
e-
-B
A
Ruler postulate
A
B
I
~t
-1
I
I
\
I
\
t
\
I
\
0
1
2
3
4
5
\
-2
I
\
~
AB = \-2 - 2\ = 4 or \2 - (-2)\ = 4
Ruler Placement postulate
A
B
I
~t
0
I
\
\
\
\
1
2
I
t
3
®
I
\
~
\
\
\
5
6
7
The distance between A and B = \4 - 0\ or \0 - 4\.
Using the Ruler Placement
postulate, you can read the distance
AB = 4
between A and B directly from the number line "ruler:'
Segment Addition postulate:
If B is between A and C on a line, then AB + BC = AC.
Prove that B is between A and C.
\
o
I
\
1
2
Bt
\
34
5
\
Ct ~
6
7
Segment Addition postulate
AB = 4, BC = 3, AC = 7
AB + BC = 4 + 3 = 7
B is between A and C because 4 + 3 = 7.
Also, 4 is between 0 and 7 on the number line.
8 Chapter 1 Exploring Geometry: points, Lines, and Angles in the Plane
~Ign'v
Prove that 0 is between C and E.
C
•• I
--4
t
-3
E
D
-2
t
-1
t
2
0
.-
3
Segment Addition Postulate
CO = 2, DE = 2, CE = 4
CD + DE = 2 + 2 = 4
D is between C and E because 2 + 2 = 4.
Also, 0 is between - 2 and 2 on the number line.
Exercise A Use the Ruler Postulate to name the real number
corresponding to each letter.
1
.
o
-3 -2 -1
2
3
4
5
6
. Exercise B Use the Ruler Postulate to name the letter
corresponding to each real number.
A
B C D E
2.
F
G
I \ •0t iii
I Ii 4t
l_\12
•• t
-3
1t
-2
.-
4
Exercise C Use the Ruler Placement Postulate to find the
distance between the points. (Hint: Think 0 at the point at
the left.)
A
3.
t
9
4.
10 11
12
13 14
D
t
-6 --'5 --4 -3 -2 -1
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1
9
Exercise D Use a ruler to measure these segments in inches.
Round to the nearest inch.
5.
6.
Exercise E Use a ruler to measure these segments in
centimeters. Round to the nearest centimeter.
7. --
9.
8.
Exercise f Use the Segment Addition Postulate to prove that B is
between A and C.
10
.
B
c
2
6
A
o
11
.
A
B
c
o
4
1
2
Exercise G Answer the questions.
12. Which postulate places numbers on a line?
12. Which postulate can you use to measure distances
between points?
14. How did you use the Ruler Postulate in problems 5-9?
15. Do you think you can use
the Ruler Postulate if the
points A, B, and C are not
collinear? Why or why not?
•
•
it~~;
Accurate
measurements of distances and angles,
such as flight paths and re-entry
angles, are important to space
exploration. The Mars Climate Orbiter,
which traveled 416,000,000 miles to
Mars, may have failed because it came
about 12t miles too close to the
planet's surface. That's an error of
about 0.000003%!
10
C
A
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
B•
Angle L A figure
made up of two sides, or
rays, with a cornman
endpoint
Vertex
The point cornman to
both sides of an angle
Geometric figures exist in a variety of shapes and sizes. As you
look around, you will see one of the most important geometric
figures, angles.
An angle is a geometric figure made up of two rays with a
common endpoint called the vertex.
Examples of angles
<
L
The symbol for an angle is L. Angles can be named in
several ways.
AL
An angle can be named by its vertex.
Angle A or LA
r
vertex
L
Z
c
An angle can be named by a letter
or number "inside" the angle. Angle
3 or L3
Angle xor Lx
An angle can be named by the rays
that form it. The vertex is always in
the middle of the name.
LABCorLCBA
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1
11
~ Copy LA using a compass and straighted//
A
Step 1 Draw a ray. Label the endpoint P.
.
P --------+
Step 2 Place the needle of the compass at A, the vertex of
LA. Draw any size arc that crosses each ray of
LA. Once you draw the arc, do
not change the opening of the compass.
AL....------+--
...•.
Label the points where the arc crosses the
rays Band C.
Step 3 Place the compass needle at point P. Keeping the
compass opening the same, draw an arc that
crosses the ray. Label that point Q.
Step 4 Place the compass needle at point B. Fix the
compass opening so that the pencil touches
point C. Keeping the same opening, place the
needle at point Q. Draw an arc that crosses the
arc that passes through Q.
Label the point where the arc crosses R.
~QR
P
e-,
----01----+-----
~
Step 5 Draw PRo LRPQ is a copy of LA.
If you are asked to copy an angle, you will be expected to do all five
Compass
A tool used to draw
circles and parts of
circles called arcs
steps on the given angle and its copy. The construction will look
like this.
••
..
Step 5
C
Arc
R
Step 4
Part of a circle
Construction
Process of making a line,
angle, or figure according
to specific requirements
12
Given
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
Cop
y
p
Q~
Step
1
Step 3
To bisect an angle means to divide an angle into two equal parts. ~ Bisect LA
using a compass and straighted/ A-·
Step 1 Place the compass needle at A. Draw an arc of
any size that crosses each ray of the angle.
Label the points where the arc crosses the
rays Band C.
Step 2 Place the compass needle at B. Draw an arc.
Once you draw the arc, do not change the
opening of the compass. Place the needle
at C and draw an arc that crosses the arc
you just drew. Label the point where the
arcs cross D.
A
42
B
Step 3 Draw XD. XD bisects LA.
~
Angle A has been bisected. AD is the
angle bisector. Two angles are formed.
LCAD and LDAB are equal.
Bisect
To divide into two equal
parts
~
A
B
If you are asked to bisect an angle, you will be expected to do all
three of the steps on the given angle. The construction will look like
this.
Angle bisector Ray
that divides an angle into
two equal parts
~Step3
Step 2
Step 1
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1
13
Exercise
Name each angle in
2.L··
3. LR
Q
R----+--5
Exercise B On a separate sheet of paper, draw an angle similar to
each of the angles shown. Be sure to use a ruler or straightedge and be
neat and accurate. Then copy each angle you have drawn.
5.
4
.
6
.
Exercise C On a sheet of paper, draw an angle similar to each
of the angles shown. Remember to be accurate. Then bisect each angle you
have drawn. You will need to use a compass and straightedge.
8
.
7.
9.
Exercise D Answer the following question.
~
~
10. Can BA and Be form an angle? Why or why not?
14
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
Degree
A unit of angle measure
Protractor
The ancient Babylonians created several systems of
measurement. Historians believe that they were the first to divide
a circle into 360 equal parts, called degrees. No matter who
invented them, we all use degrees to measure angles.
A tool used to draw or
measure angles
Note that the numbers increase in a
counterclockwise direction
-the direction opposite to the
direction the hands of a clock move.
1800
The symbol for a degree is 0. An angle measurement of 10 degrees
is written 10°. To measure an angle, you can use a protractor. A
protractor is a tool shaped like half a circle with degree markings
from 0° to 180°.
Find the measure of La.
L
'
a
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1
15
(continued)
Step 1 Place the center point (labeled 0 below) on the base of the protractor at the vertex of
the angle. Turn the protractor so that one ray of the angle passes through the 0°
mark on the protractor.
Step 2 Follow the second ray outward from point 0 on the protractor. Read the degree
mark where the ray crosses the protractor.
The measure of angle a is 40°. You can write this as mLa = 40°.
You can classify an angle using its measure.
Acute angle
An angle whose measure
is greater than 0° and
less than 90°
An acute angle is an angle whose measure is greater than 0° and less
than 90°.
Examples of acute angles
Rightangle
An angle whose measure
is 90°
~
The symbol D is
sometimes used to
show a right angle.
16
A right angle is an angle whose measure = 90°.
Examples of right angles
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
Obtuse angle
An angle whose measure
is greater than 900 but
less than 180 0
An obtuse angle is an angle whose measure is greater than 90°
and less than 180°.
Examples of obtuse angles
Straight angle
An angle whose measure
is 1800
Perpendicular
lines
.1
Lines that form right
angles
A straight angle is an angle whose measure = 180°.
Examples of straight angles
Dc
Write a description
of three objects. Tell
whether the objects
have acute, right, or
obtuse angles. Point
out any
perpendiculars.
B
A
o
Look again at the right angles. Notice the rays that make
up the angles.
Lines that form right angles are said to be perpendicular lines.
Examples of perpendicular lines, rays, and segments
~ j: .:
+,
a.1 b
e
m
m.1.£
The symbol for "is perpendicular to" is .1. Here, a
is perpendicular to b,
is perpendicular to Bt, PQ is
~
~
-perpendicular
to
QR,
5 is perpendicular to r, and m is
BA.1 Be
PQ.1 QR
5.1 r
perpendicular to .£. As part of your geometry study, you
will need to know how to construct perpendicular lines.
BA
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1 17
~ Ci.j~hljm'j[·j~DConstruct a line perpendicular to a given line e
~
and passing through a given point P.
Step 1 Place the needle of the compass at P. Draw any size
p
e
arc that crosses on both sides of point P.
Label one place where the arc crosses A. Label
the other place B.
p
Step 2 Place the compass needle at A. Open the compass
so that it reaches past point P. Keep the needle on
A and draw an arc above P.
Step 3
Step 4
Then, keeping the same compass opening, place the
compass needle at point B. Draw a second arc above
P. Be sure the arcs cross one another. Label the
point where the arcs cross C.
Draw tc. PC l_ e.
~
•..e ~ -A+-( --1'___)t=B_ •.
Exercise A Classify each angle. Write acute, right, obtuse, or
straight.
1. illLa = 15°
2. illLABC = 100°
3. mz z = 180°
4. illLd = 90°
5.
6
.
7.
8. .... .• 1(------.._----; •.•.
18
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
o
Exercise B Draw an example of each type of angle.
Then use a protractor to measure your angle.
9. acute
10. right
11. obtuse
Exercise C For each exercise, name a pair of perpendicular lines,
rays, or segments.
12
.
14.
x
y
m
Exercise D Answer the following question.
15. Explain with a sentence or a construction how you can
make four right angles using only two lines.
Talking over geometry
problems and definitions with
friends can help you to learn
the fundamental ideas of
geometry. This is also a
terrific way to study.
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1 19
z
Adjacent angles
Angles with a common
vertex and one common
side
Complementary
angles
Two angles whose
measures add to 90°
You have seen that angles can be classified by their measures.
Now you will see that pairs of angles can be classified by their
positions and their measures.
In the drawing on the right, L.a and Zb are
classified as adjacent angles.
They have a common vertex, C.
They also have a common side, e.
a
C ._._-=.b __ __.
Adjacent angles have a common vertex and one common side. The
0 symbol in the drawing tells you that mL.C = 90°.
So, mL.a + mL.b = 90°.
Two angles whose measures add to 90° are called complementary
angles. In this case, angles a and b are also called complementary
angles. Note: Two angles do not have to be adjacent in order to be
complementary. For example, the following pairs of angles are
complementary angles.
Suppose you have a straight angle such as the angle below.
o
If you draw any ray from 0, you will form two adjacent angles,
cand d.
o
The drawing tells you that mL.O = 180°.
20
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
Supplementary
angles
Two angles whose
measures add to 1800
Intersect
Two angles whose measures add to 180° are supplementary angles.
So, mLe + mLd = 180°. In this case, angles e and dare also called
supplementary angles. Note: Two angles do not have to be adjacent
in order to be supplementary. For example, the following pairs of
angles are supplementary angles.
To meet at a point;
to cross or overlap
each other
Vertical angles
Angles that have a
common vertex and
whose sides are formed
by the same lines
When two lines intersect, four angles are formed.
a
d
Find three other
words that mean the
same thing as
intersect. Use these
words to write three
sentences about
angles.
Vertical angles are angles that have a common vertex and whose
sides are formed by the same lines. In the drawing above, La and L
d are vertical angles and L band L e are vertical angles. Whenever
two lines intersect, two pairs of equal vertical angles are formed. If
mLa = 45°, then mLd = 45°. If mLb = 135°, then mLe = 135°.
That is, mLa = mLd, and mLb = mLe.
Note that if the two lines are
perpendicular, then four right
angles are formed.
In this drawing, more than two
lines intersect to create three pairs
of vertical angles: L nand Lq,
La and Lr, and Ls and Lp.
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1 21
Exercise A Classify each pair of angles. Write supplementary or
complementary. Identify any pairs that are adjacent.
1. \
.
~nd
2.
3.
Given that the two angles are
supplementary, find the measure
of the unknown angle x.
Use your calculator to find the answer.
Press 18G El'GS, then press ~ or §l
The answer is 75°.
Exercise B Use your calculator to find the measure of an angle
that is supplementary to the given angle measure.
10. 125°
22
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
Exercise C Name the pairs of vertical angles in each figure.
14.
13.
m/
0(
yP
tu
vw
•
16
.
a
15.
c
f
d
! ji i j! ! !
Writing About
Mathematics
:j:!
Think of a realworld example of
vertical angles.
Describe the
example, telling
about the angles.
Exercise D Find the measure of each angle.
17. Lf
18. Lb
b
19. Ld
e
c = 90°
d
f
9 = 45°
20. Le
Urban planners try to layout cities so that the streets are
parallel and perpendicular, making a grid. This is not always possible.
Study a map of your city. Find streets that are not perpendicular and
measure the angles they form. Are there any streets that form
complementary, supplementary, or vertical angles? Are there any
adjacent angles? Which type of angle did you find most often? Explain
why you think this is so.
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1 23
You can use the definitions of complementary, supplementary, and
vertical angles along with algebra to solve problems in geometry.
The measure of one of two complementary angles is
20°. What is the measure of the second angle?
To solve for x in an
algebraic equation,
you must get x by
itself. To do this, you
might use addition,
subtraction,
multiplication, or
division. For
example, to solve
x + 18 = 40, you
subtract 18 from
both sides of the
equation. To solve
5x = 95, you divide
both sides of the
equation by 5.
Step 1
Call the unknown angle measure x.
Step 2
20° + x = 90° by definition of complementary angles
Step 3
x
x
90° - 20°
70°
The measure of the
second angle is 70°.
Use algebra and the definitions of angles to solve this problem.
The measure of one of two supplementary angles is
35°. What is the measure of the second angle?
-.- -.-.-.-.-
-.-.-.-
-.--;:-
~".: W;iti';g:Abo~t
':
~'Mathe"iatics
.
Two perpendicular
~i~",~')_
~~"
~
_ __
="
lines always form
four right angles.
Write an explanation of why this
is so.
Step 1 Call the unknown angle measure y.
Step 2 35° + Y = 180° by definition of supplementary angles
:,c.
Step 3
Y
180° - 35°
Y
145°
The measure of the second angle is 145°.
and
24
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
D Two lines cross each other, creating two pairs of
----
supplementary angles. The measure of one
supplementary angle is twice the measure of the
second supplementary angle. What are the measures
of each of the four angles?
Step 1 Let w be the measure of the smaller angle.
Let 2w be the measure of the larger angle.
Step 2 2w + w = 180° by definition of supplementary angles
Step 3
3w
w
180°
60°
The measure of one of the four angles is 60°.
w = 60°
x
z
y
Now, find the measures of the other three angles.
Step 4
x + 60°
x
x
Step 5
180° by definition of supplementary
angles
180° - 60°
1 20° so mLx = 1 20°
180° by definition of supplementary
angles
180° - 60°
y
120° so mLy = 120°
y
OR mLy = 120° because vertical angles are equal
Step 6 z + 1 20°
180° by definition of supplementary
angles
z
180° - 120°
z
60° so mLz = 60°
OR mLz = 60° because vertical angles are equal
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1 25
Exercise A Solve for the missing angle .
1.
2.
xy
89° Z
3.
Estimate: Estimate mL2, if mLl = 58° and Ll and L2 are
complementary angles.
Round the measure of Ll .
= 58° = 60°
Subtract.
90° - 60° = 30°
The solution, mL2, should be close to 30°.
Solution: mLl
Exercise B Find the measure of each numbered angle, given
mL2 = 42°.
4. mLl
5. mL3
6. mL4
A
7. mLS
G
8. mL6
26
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
F
Exercise C Use the figure from Exercise B to do the following. 9.
Add mL6 and mL3.
10. Give another name for L6.
11. Name a pair of vertical angles.
12. Name a pair of complementary angles.
Exercise D Answer the following questions.
13. Draw a set of vertical angles.
Compare the measures of the vertical
angles. Write a statement that
describes your observation about the
measures of vertical angles.
14. The measure of one complementary
angle is five times as large as the
measure of the second angle. What is
the measure of each angle?
15. The measure of one supplementary
angle is four times as large as the
measure of the second angle. What is
the measure of each angle?
16. One complementary angle is half as
large as the second angle. What is the
measure of each angle?
17. One supplementary angle is two-
thirds as large as the second angle.
What is the measure of each angle?
18. The measure of one complementary
angle is four times as large as the
measure of the second angle. What is
the measure of each angle?
19. The measure of one supplementary
angle is five times as large as the
measure of the second angle. What is
the measure of each angle?
20. One complementary angle is twothirds as large as the second angle.
What is the measure of each angle?
Many companies have logos that are based upon the
letters of their name. Use straws to create a logo design based on the
first letters of your first, middle, and last name. Use at least one of each
of the following angles: acute, obtuse, right, straight, vertical,
supplementary, adjacent, and complementary. Glue your design to
construction paper. Label one of each type of angle formed in your
design.
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1 27
The expression 103 is shorthand for 10 • 10 • 10.
Exponent
D
x2
103 .-- 3 is called the exponent.
The number of times
a base is multiplied
by itself
x
~ lOis called the base.
x
In x- x = x2, X is the base and 2 is the exponent.
Read "x to the second power" or "x squared."
In y' y' y = y ,y 3is the ase band 3 is the
Base
The number being
multiplied by itself
exponent. Read "y to the third power"
or "y cubed."
L§J:
Y
:: y3
/--------~
y
In a' a' a' a = a4, a is the base and 4 is the exponent.
Read" a to the fourth power."
Understanding exponents is important when working with area or
volume. These topics will be covered later in this textbook.
We can multiply or divide expressions with the same base by
adding or subtracting the exponents.
To multiply expressions with the same base, add the
exponents.
5 -7- y2 =
_t_ = y. y. y. y. Y= y3 y2 y. Y
Y
So, y5 -7- y2 = y5 - 2 = y3
To divide expressions with the same base, subtract the
exponents.
Special Cases: Any number (except 0) divided by itself equals l.
2 2
2
xo 2x l---x - -x -
3
- a3 a-
1
-
3 3
l---a - -- ao
= ;; =
yn - n = yO
xm
For any m and any x*- 0, 1 = ------m = x'" - m = x", so 1 = xO. x
Also, for any y, y1 = y.
Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
(rJ!t1~hd¥ DFor what value of n is the statement true? 0 • x5
3
•0
Solution: 0 • x • 0 = 0
5
3
4
3
+ 4•x
5
4
= On • x
5
J,
5
= 0 • x The
7
statement is true for n = 7.
Rule: xm. x" = x'" + n
For x =!= 0, x'" -i- x" = x'" - n
Special Case: For x =!= 0, x'" -:-- x'" = x'" - m = xO = 1
Exercise A Simplify multiplication.
4. w4• W
Exercise B Simplify division.
9.
13.
y3 -:-- y2
10. alO -:-- as
14. X13 -i- x10
11. Xl -i- X
15. y10 -:-- y2
12.
4
w
-i-
4
w
16.
17. n21 -:-- n16
Z15 -:-- Z5
18. e9 -i- e9
19
19. m19 -:-m
20. a4 -:-- a3
bB -:-- b
Exercise C Find the value of n that makes each statement true.
4
21.
y3. b2 • y2 = b2 • yn
26.
22.
as. b • a = an • b
27.
= yl
y
3
a • X2 • a4 • X5 = an • Xl
28.
b4• e3
3
B
3
• b4 • e6 =
bB • en
29. p5 • r6 • p2 • r = r
30.
.
rl
x3• y3 • X • y4 = x" • yl
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1
29
Chapter 1
REV lEW - continued
Tell whether each figure is a point, a line, a line segment, or
a ray. Then use symbols to name each figure.
Example: Q
------------. W
Solution: line segment, QW
9.
·
X
10.
R
s
11. A
":>«:
12. X ------------. y
Use a ruler to measure these segments in inches. Round to the
nearest inch.
Example:
Solution: 2 in.
14
.
15.
Use a ruler to measure these segments in centimeters. Round to the
nearest centimeter.
Example:
Solution: 2 em
16.
17.
18.
L
Use a straightedge and compass to copy and bisect each angle.
Example:
190
_j
Solution
L k
32 Chapter 1 Exploring Geometry: Points, Lines, and Angles in the Plane
20o____/
Classify each angle. Write acute, right, obtuse, or straight.
Example: mLC = 90°
23'L
Solution: right
21. mLx = 105°
22. mLXYZ = 90°
Find the measure of an angle complementary to the given angle.
Example: mLC = 35°
24. mLy = 15°
Solution: 90° - 35° = 55°
25. mLx = 30°
26. mLw = 45°
Find the measure of an angle supplementary to the given angle.
Example: mLC = 135°
Solution: 180° - 135° = 45°
27. mLm = 115° 28. mLn = 60°
29. mLo = 40°
L
Draw an example of each type of angle, then use a protractor to
measure your angle.
Example: right
30. obtuse
Solution:
31. straight
Solve to find the measure of the unknown angles.
Example: The measure of an angle
is two times the measure of its
complement. What is the
measure of the smaller angle?
Solution: x + 2x = 90
3x = 90
x = 30
So, the smaller angle measures 30°.
32. Supplementary angles, one angle = 65°.
33. Complementary angles, one angle = 12°.
34. One supplementary angle is half as large as the second
supplementary angle. What is the measure of the larger angle?
35. One complementary angle is nine times as large as the second
complementary angle. What is the measure of the smaller angle?
To prepare for a geometry test, study in short sessions for several days
rather than one long session the night before the test.
Exploring Geometry: Points, Lines, and Angles in the Plane Chapter 1 33