Download 9.1 Points, Lines, Planes, and Angles

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CHAPTER 9
9.1
Geometry
Points, Lines, Planes, and Angles
The Geometry of Euclid
Let no one unversed in geometry enter here.
—Motto over the door of Plato’s Academy
Euclid’s Elements as translated
by Billingsley appeared in 1570
and was the first English language
translation of the text—the most
influential geometry text ever
written.
Unfortunately, no copy of
Elements exists that dates back to
the time of Euclid (circa 300 B.C.),
and most current translations are
based upon a revision of the work
prepared by Theon of Alexandria.
Although Elements was only
one of several works of Euclid, it
is, by far, the most important. It
ranks second only to the Bible as
the most published book in
history.
To the ancient Greeks, mathematics meant geometry above all—a rigid kind of
geometry from a modern-day point of view. The Greeks studied the properties of figures identical in shape and size (congruent figures) as well as figures identical in
shape but not necessarily in size (similar figures). They absorbed ideas about area
and volume from the Egyptians and Babylonians and established general formulas.
The Greeks were the first to insist that statements in geometry be given rigorous
proof.
The Greek view of geometry (and other mathematical ideas) was summarized
in Elements, written by Euclid about 300 B.C. The influence of this book has been
extraordinary; it has been studied virtually unchanged to this day as a geometry textbook and as the model of deductive logic.
The most basic ideas of geometry are point, line, and plane. In fact, it is not really
possible to define them with other words. Euclid defined a point as “that which has
no part,” but this definition is so vague as to be meaningless. Do you think you could
decide what a point is from this definition? But from your experience in saying “this
point in time” or in sharpening a pencil, you have an idea of what he was getting at.
Even though we don’t try to define point, we do agree that, intuitively, a point has
no magnitude and no size.
Euclid defined a line as “that which has breadthless length.” Again, this definition is vague. Based on our experience, however, we know what Euclid meant. The
drawings that we use for lines have properties of no thickness and no width, and they
extend indefinitely in two directions.
What do you visualize when you read Euclid’s definition of a plane: “a surface
which lies evenly with the straight lines on itself”? Do you think of a flat surface,
such as a tabletop or a page in a book? That is what Euclid intended.
The geometry of Euclid is a model of deductive reasoning. In this chapter, we
will present geometry from an inductive viewpoint, using objects and situations
found in the world around us as models for study.
Points, Lines, and Planes There are certain universally accepted conventions and symbols used to represent points, lines, planes, and angles. A capital letter
usually represents a point. A line may be named by two capital letters representing
points that lie on the line, or by a single (usually lowercase) letter, such as . Subscripts are sometimes used to distinguish one line from another when a lowercase
letter is used. For example, 1 and 2 would represent two distinct lines. A plane may
be named by three capital letters representing points that lie in the plane, or by a letter of the Greek alphabet, such as (alpha), (beta), or (gamma).
Figure 1 depicts a plane that may be represented either as or as plane ADE.
Contained in the plane is the line DE (or, equivalently, line ED), which is also labeled in the figure.
Selecting any point on a line divides the line into three parts: the point itself, and
two half-lines, one on each side of the point. For example, in Figure 2, point A divides the line into three parts, A itself and two half-lines. Point A belongs to neither
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9.1
Points, Lines, Planes, and Angles
493
half-line. As the figure suggests, each half-line extends indefinitely in the direction
opposite to the other half-line.
A Ray AB B
Half-line
A
Line segment AB
A
α
FIGURE 1
Given any three points that are
not in a straight line, a plane can
be passed through the points. That
is why camera tripods have three
legs—no matter how irregular the
surface, the tips of the three legs
determine a plane. On the other
hand, a camera support with four
legs would wobble unless all four
legs were carefully extended just
the right amount.
B
A
E
D
A Ray BA B
Half-line
FIGURE 2
FIGURE 3
Including an initial point with a half-line gives a ray. A ray is named with two
letters, one for the initial point of the ray, and one for another point contained in the
half-line. For example, in Figure 3 ray AB has initial point A and extends in the direction of B. On the other hand, ray BA has B as its initial point and extends in the
direction of A.
A line segment includes both endpoints and is named by its endpoints. Figure 3
shows line segment AB, which may also be designated as line segment BA.
The following chart shows these figures along with the symbols used to represent them.
Name
Figure
Symbol
i
Line AB or line BA
Half-line AB
Half-line BA
Ray AB
Ray BA
Segment AB or segment BA
A
B
A
B
A
B
A
B
A
B
A
B
i
AB or BA
l
AB
l
BA
•l
AB
•l
BA
•–• •–•
AB or BA
For a line, the symbol above the two letters shows two arrowheads, indicating
that the line extends indefinitely in both directions. For half-lines and rays, only one
arrowhead is used since these extend in only one direction. An open circle is used
for a half-line to show that the endpoint is not included, while a solid circle is used
for a ray to indicate the inclusion of the endpoint. Since a segment includes both endpoints and does not extend in either direction, solid circles are used to indicate endpoints of line segments.
The geometric definitions of “parallel” and “intersecting” apply to two or more
lines or planes. (See Figure 4 on the next page.) Parallel lines lie in the same plane
and never meet, no matter how far they are extended. However, intersecting lines
do meet. If two distinct lines intersect, they intersect in one and only one point.
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We use the symbol to denote parallelism. If 1 and 2 are parallel lines, as in
Figure 4, then this may be indicated as 1!2.
Parallel planes also never meet, no matter how far they are extended. Two distinct intersecting planes form a straight line, the one and only line they have in common. Skew lines do not lie in the same plane, and they never meet, no matter how
far they are extended.
1
2
Intersecting lines
Parallel lines
Intersecting planes
Parallel planes
Skew lines
FIGURE 4
Vertex
A
Side
X
B
Side
C
D
E
F
J
K
L
FIGURE 5
Angles
An angle is the union of two rays that have a common endpoint, as shown
in Figure 5. It is important to remember that the angle is formed by points on the rays
themselves, and no other points. In Figure 5, point X is not a point on the angle. (It is
said to be in the interior of the angle.) Notice that “angle” is the first basic term in
this section that is actually defined, using the undefined terms ray and endpoint.
The rays forming an angle are called its sides. The common endpoint of the rays
is the vertex of the angle. There are two standard ways of naming angles. If no confusion will result, an angle can be named with the letter marking its vertex. Using
this method, the angles in Figure 5 can be named, respectively, angle B, angle E, and
angle K. Angles also can be named with three letters: the first letter names a point
on one side of the angle; the middle letter names the vertex; the third names a point
on the other side of the angle. In this system, the angles in the figure can be named
angle ABC, angle DEF, and angle JKL.
The symbol for representing an angle is . Rather than writing “angle ABC,”
we may write “ABC.”
An angle can be associated with an amount of rotation. For example, in Figure 6(a),
•l
•l
•l
we let BA first coincide with BC—as though they were the same ray. We then rotate BA
(the endpoint remains fixed) in a counterclockwise direction to form ABC.
A
A
A
B
C
B
C
B
C
(a)
150°
360°
90°
X
C
10°
45°
N
(b)
FIGURE 6
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M
9.1
Why 360? The use of the
number 360 goes back to the
Babylonian culture. There are
several theories regarding why 360
was chosen for the number of
degrees in a complete rotation
around a circle. One says that 360
was chosen because it is close to
the number of days in a year, and
is conveniently divisible by 2, 3,
4, 5, 6, 8, 9, 10, 12, and other
numbers.
Angles are key to the study of
geodesy, the measurement of
distances on the earth’s surface.
Points, Lines, Planes, and Angles
495
Angles are measured by the amount of rotation, using a system that dates back
to the Babylonians some two centuries before Christ. Babylonian astronomers chose
the number 360 to represent the amount of rotation of a ray back onto itself. Using
360 as the amount of rotation of a ray back onto itself, one degree, written 1°, is defined to be 1360 of a complete rotation. Figure 6(b) shows angles of various degree
measures.
Angles are classified and named with reference to their degree measures. An
angle whose measure is between 0° and 90° is called an acute angle. Angles M and N
in Figure 6(b) are acute. An angle that measures 90° is called a right angle. Angle C
in the figure is a right angle. The squared symbol
at the vertex denotes a right
angle. Angles that measure more than 90° but less than 180° are said to be obtuse
angles (angle X, for example). An angle that measures 180° is a straight angle. Its
sides form a straight line.
Our work in this section will be devoted primarily to angles whose measures are
less than or equal to 180°. Angles whose measures are greater than 180° are studied
in more detail in trigonometry courses.
A tool called a protractor can be used to measure angles. Figure 7 shows a protractor measuring an angle. To use a protractor, position the hole (or dot) of the protractor on the vertex of the angle. The 0-degree measure on the protractor should be
placed on one side of the angle, while the other side should extend to the degree
measure of the angle. The figure indicates an angle whose measure is 135°.
0
0
12
110
100
90
80
70
0
15
30
0
14
40
20
10
180 170
160
60
50
13
Protractor
FIGURE 7
Q
R
When two lines intersect to form right angles they are called perpendicular
lines. Our sense of vertical and horizontal depends on perpendicularity.
In Figure 8, the sides of NMP have been extended to form another angle,
RMQ. The pair NMP and RMQ are called vertical angles. Another pair of
vertical angles have been formed at the same time. They are NMQ and PMR.
An important property of vertical angles follows.
M
N
Property of Vertical Angles
P
Vertical angles have equal measures.
FIGURE 8
For example, NMP and RMQ in Figure 8 have equal measures. What other pair
of angles in the figure have equal measures?
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CHAPTER 9
Geometry
EXAMPLE
(4x + 19)°
(6x – 5)°
1
Refer to the appropriate figure to solve each problem.
(a) Find the measure of each marked angle in Figure 9.
Since the marked angles are vertical angles, they have the same measure. Set
4x 19 equal to 6x 5 and solve.
4x 19 6x 5
FIGURE 9
4x 4x 19 4x 6x 5
Add 4x.
19 2x 5
19 5 2x 5 5
Add 5.
24 2x
12 x
(3x – 30)°
(4x)°
FIGURE 10
Divide by 2.
Since x 12, one angle has measure 412 19 67 degrees. The other has
the same measure, since 612 5 67 as well. Each angle measures 67°.
(b) Find the measure of each marked angle in Figure 10.
The measures of the marked angles must add to 180° since together they form
a straight angle. The equation to solve is
3x 30 4x 180 .
7x 30 180
7x 30 30 180 30
Combine like terms.
Add 30.
7x 210
x 30
Divide by 7.
To find the measures of the angles, replace x with 30 in the two expressions.
3x 30 330 30 90 30 60
4x 430 120
The two angle measures are 60° and 120°.
If the sum of the measures of two acute angles is 90°, the angles are said to be
complementary, and each is called the complement of the other. For example, angles
measuring 40° and 50° are complementary angles, because 40° 50° 90°. If two
angles have a sum of 180°, they are supplementary. The supplement of an angle
whose measure is 40° is an angle whose measure is 140°, because 40° 140° 180°.
If x represents the degree measure of an angle, 90 x represents the measure of its
complement, and 180 x represents the measure of its supplement.
A
(2x + 20)°
(12x)°
B
C
FIGURE 11
EXAMPLE 2
Find the measures of the angles in Figure 11, given that
ABC is a right angle.
The sum of the measures of the two acute angles is 90° (that is, they are complementary), since they form a right angle. We add their measures to obtain a sum of
90 and solve the resulting equation.
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Points, Lines, Planes, and Angles
497
2x 20 12x 90
14x 20 90
14x 70
Subtract 20.
x5
Divide by 14.
The value of x is 5. Therefore, the measures of the angles are
2x 20 25 20 30 degrees
12x 125 60 degrees .
and
EXAMPLE 3
The supplement of an angle measures 10° more than three
times its complement. Find the measure of the angle.
Let
Then
and
x the degree measure of the angle.
180 x the degree measure of its supplement,
90 x the degree measure of its complement.
10 more than





180 x
10 three times
its complement.





measures





Moiré Patterns A set of
parallel lines with equidistant
spacing intersects an identical set,
but at a small angle. The result is a
moiré pattern, named after the
fabric moiré (“watered”) silk. You
often see similar effects looking
through window screens with
bulges. Moiré patterns are related
to periodic functions, which
describe regular recurring
phenomena (wave patterns such as
heartbeats or business cycles).
Moirés thus apply to the study of
electromagnetic, sound, and water
waves, to crystal structure, and to
other wave phenomena.
Supplement





Now use the words of the problem to write the equation.
390 x
180 x 10 270 3x
Distributive property
180 x 280 3x
2x 100
Add 3x ; subtract 180.
x 50
Divide by 2.
The angle measures 50°. Since its supplement (130°) is 10° more than three times
its complement (40°), that is, 130 10 340 is true, the answer checks.
Parallel lines are lines that lie in the same plane and do not intersect. Figure 12
shows parallel lines m and n. When a line q intersects two parallel lines, q is called
a transversal. In Figure 12, the transversal intersecting the parallel lines forms eight
angles, indicated by numbers. Angles 1 through 8 in the figure possess some special
properties regarding their degree measures, as shown in the following chart.
q
1
3
5
7
2
4
6
8
m
n
FIGURE 12
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CHAPTER 9
Geometry
Name
Figure
Rule
Alternate interior angles
Angle measures
are equal.
q
m
4
5
n
(also 3 and 6)
Alternate exterior angles
Angle measures
are equal.
q
1
m
n
(also 2 and 7)
8
Interior angles on same
side of transversal
Angle measures
add to 180°.
q
m
4
6
n
(also 3 and 5)
Corresponding angles
Angle measures
are equal.
q
2
6
m
n
(also 1 and 5, 3 and 7, 4 and 8)
The converses of the above also are true. That is, if alternate interior angles are
equal, then the lines are parallel, with similar results valid for alternate exterior angles, interior angles on the same side of a transversal, and corresponding angles.
(3x + 2)°
m
n
(5x – 40)°
FIGURE 13
EXAMPLE 4
Find the measure of each marked angle in Figure 13, given
that lines m and n are parallel.
The marked angles are alternate exterior angles, which are equal. This gives
3x 2 5x 40
42 2x
Subtract 3x and add 40.
21 x .
Divide by 2.
One angle has a measure of 3x 2 3 21 2 65 degrees, and the other has a
measure of 5x 40 5 21 40 65 degrees.
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