Download Section 1.3

■
1.3
39.
*48. In the drawing, m ∠1 ⫽ x and
∠ABC is a straight angle. Using your protractor, you
can show that m∠ 1 ⫹ m ∠2 ⫽ 180⬚. Find m∠ 1 if
m ∠2 ⫽ 56⬚.
R
m ∠2 ⫽ y. If m∠ RSV ⫽ 67⬚ and
x ⫺ y ⫽ 17⬚, find x and y.
(HINT: m∠ 1 ⫹ m ∠ 2 ⫽ m∠ RSV.)
S
1
C
Exercises 39, 40
40. Find m∠ 1 if m ∠ 1 ⫽ 2x and m ∠2 ⫽ x.
(HINT: See Exercise 39.)
In Exercises 41 to 44, m⬔1 ⫹ m⬔2 ⫽ m⬔ABC.
41. Find m∠ ABC if m∠1 ⫽ 32⬚ and
m∠ 2 ⫽ 39⬚.
49. Find the bearing of airplane B relative to the control tower.
50. Find the bearing of airplane C relative to the control tower.
A
42. Find m∠ 1 if m∠ ABC ⫽ 68⬚ and
D
m∠ 1 ⫽ m∠ 2.
1
43. Find x if m ∠ 1 ⫽ x, m∠ 2 ⫽ 2x ⫹ 3,
N
2
B
and m∠ ABC ⫽ 72⬚.
C
B
Exercises 41–44
22
mi
44. Find an expression for m∠ ABC if m ∠ 1 ⫽ x and
300
m ∠2 ⫽ y.
control
tower
45. A compass was used to mark off three congruent segments,
AB, BC, and CD. Thus, AD has been trisected at points B
and C. If AD ⫽ 32.7, how long is AB?
W
E
50
24
mi
325
2
A
B
V
For Exercises 49 and 50, use the following information.
Relative to its point of departure or some other point of reference, the angle that is used to locate the position of a ship or
airplane is called its bearing. The bearing may also be used to
describe the direction in which the airplane or ship is moving.
By using an angle between 0 ⬚ and 90 ⬚, a bearing is measured
from the North-South line toward the East or West. In the diagram, airplane A (which is 250 miles from Chicago’s O’Hare
airport’s control tower) has a bearing of S 53⬚ W.
2
B
T
1
2
D
A
19
Early Definitions and Postulates
C
D
53
E
mi
C
A
46. Use your compass and straightedge to bisect EF.
E
F
*47. In the figure, m∠ 1 ⫽ x and
S
m ∠2 ⫽ y. If x ⫺ y ⫽ 24⬚,
find x and y.
(HINT:
m ∠1 ⫹ m ∠2 ⫽ 180⬚.)
1.3
KEY CONCEPTS
Exercises 49, 50
D
1
A
2
B
C
Early Definitions and Postulates
Mathematical System
Axiom or Postulate
Theorem
Ruler Postulate
Distance
Segment-Addition
Postulate
Congruent Segments
Midpoint of a Line
Segment
Ray
Opposite Rays
Intersection of Two
Geometric Figures
Parallel Lines
Plane
Coplanar Points
Space
A MATHEMATICAL SYSTEM
Like algebra, the branch of mathematics called geometry is a mathematical system. The formal study of a mathematical system begins with undefined terms. Building on this foundation, we can then define additional terms. Once the terminology is sufficiently developed,
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20
CHAPTER 1
■
LINE AND ANGLE RELATIONSHIPS
certain properties (characteristics) of the system become apparent. These properties are
known as axioms or postulates of the system; more generally, such statements are called
assumptions in that they are assumed to be true. Once we have developed a vocabulary and
accepted certain postulates, many principles follow logically as we apply deductive methods.
These statements can be proved and are called theorems. The following box summarizes the
components of a mathematical system (sometimes called a logical system or deductive system).
FOUR PARTS OF A MATHEMATICAL SYSTEM
1.
2.
3.
4.
Undefined terms
Defined terms
Axioms or postulates
Theorems
f
vocabulary
f
principles
Discover
CHARACTERISTICS OF A GOOD DEFINITION
Although we cannot actually define
line and plane, we can compare
them in the following analogy.
Please complete: A _?
__ is to straight
as a _?
__ is to flat.
Terms such as point, line, and plane are classified as undefined because they do not fit into
any set or category that has been previously determined. Terms that are defined, however,
should be described precisely. But what is a good definition? A good definition is like a
mathematical equation written using words. A good definition must possess four characteristics, which we illustrate with a term that we will redefine at a later time.
ANSWERS
line; plane
DEFINITION
An isosceles triangle is a triangle that has two congruent sides.
In the definition, notice that: (1) The term being defined—isosceles triangle—is named.
(2) The term being defined is placed into a larger category (a type of triangle). (3) The
distinguishing quality (that two sides of the triangle are congruent) is included.
(4) The reversibility of the definition is illustrated by these statements:
“If a triangle is isosceles, then it has two congruent sides.”
“If a triangle has two congruent sides, then it is an isosceles triangle.”
CHARACTERISTICS OF A GOOD DEFINITION
1.
2.
3.
4.
C
It names the term being defined.
It places the term into a set or category.
It distinguishes the defined term from other terms without providing unnecessary facts.
It is reversible.
The reversibility of a definition is achieved by using the phrase “if and only if.” For
instance, we could define congruent angles by saying “Two angles are congruent if
and only if these angles have equal measures.” The “if and only if” statement has the
following dual meaning:
E
“If two angles are congruent, then they have equal measures.”
“If two angles have equal measures, then they are congruent.”
Figure 1.30
B
A
Figure 1.31
When represented by a Venn Diagram, the definition above would relate set C = {congruent angles} to set E = {angles with equal measures} as shown in Figure 1.30. The sets
C and E are identical and are known as equivalent sets.
Once undefined terms have been described, they become the building blocks for other
terminology. In this textbook, primary terms are defined within boxes, whereas related
terms are often boldfaced and defined within statements. Consider the following definition
(see Figure 1.31).
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Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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1.3
■
21
Early Definitions and Postulates
DEFINITION
EXS. 1–4
Geometry in the Real World
B
6
5
C
D
4
E
12
A line segment is the part of a line that consists of two points, known as endpoints, and
all points between them.
EXAMPLE 1
State the four characteristics of a good definition of the term “line segment.”
5
6
1. The term being defined, line segment, is clearly present in the definition.
2. A line segment is defined as part of a line (a category).
F
3. The definition distinguishes the line segment as a specific part of a line.
10
4. The definition is reversible.
i) A line segment is the part of a line between and including two points.
ii) The part of a line between and including two points is a line segment.
A
On the road map, driving distances
between towns are shown. In traveling from town A to town D, which
path traverses the least distance?
Solution A to E, E to C, C to D:
10 ⫹ 4 ⫹ 5 ⫽ 19
INITIAL POSTULATES
Recall that a postulate is a statement that is assumed to be true.
POSTULATE 1
Through two distinct points, there is exactly one line.
C
D
Figure 1.32
Postulate 1 is sometimes stated in the form “Two points determine
a line.” See Figure
—!
1.32, in which points C and D determine exactly one line, namely CD. Of course, Postulate
1 also implies that there is a unique line segment determined by two distinct points used as
endpoints. Recall Figure 1.31, in which points A and B determine AB.
NOTE: In geometry, the reference numbers used with postulates (as in Postulate 1) need
not be memorized.
EXAMPLE 2
In Figure 1.33, how many distinct lines can be drawn through
a) point A?
b) both points A and B at the same time?
c) all points A, B, and C at the same time?
SOLUTION
a) An infinite (countless) number
b) Exactly one
c) No line contains all three points.
A
B
C
Figure 1.33
Recall from Section 1.2 that the symbol for line segment AB, named by its endpoints,
is AB. Omission of the bar from AB, as in AB, means that we are considering the length of
the segment. These symbols are summarized in Table 1.3.
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22
CHAPTER 1
■
LINE AND ANGLE RELATIONSHIPS
TABLE 1.3
Symbol
Words for Symbol
!
AB
Line AB
AB
Line segment AB
AB
Length of segment AB
—
Geometry in the Real World
Geometric Figure
A
B
A
B
A number
A ruler is used to measure the length of a line segment such as AB. This length may be
represented by AB or BA (the order of A and B is not important). However, AB must be a
positive number.
POSTULATE 2 ■ Ruler Postulate
The measure of any line segment is a unique positive number.
In construction, a string joins two
stakes. The line determined is
described in Postulate 1 on the
previous page.
We wish to call attention to the term unique and to the general notion of uniqueness.
The Ruler Postulate implies the following:
1. There exists a number measure for each line segment.
2. Only one measure is permissible.
Characteristics 1 and 2 are both necessary for uniqueness! Other phrases that may replace
the term unique include
One and only one
Exactly one
One and no more than one
A more accurate claim than the commonly heard statement “The shortest distance between
two points is a straight line” is found in the following definition.
DEFINITION
The distance between two points A and B is the length of the line segment AB that joins
the two points.
A
X
B
Figure 1.34
As we saw in Section 1.2, there is a relationship between the lengths of the line segments
determined in Figure 1.34. This relationship is stated in the third postulate. The title and
meaning of the postulate are equally important! The title “Segment-Addition Postulate”
will be cited frequently in later sections.
POSTULATE 3 ■ Segment-Addition Postulate
If X is a point of AB and A-X-B, then AX ⫹ XB ⫽ AB.
Technology Exploration
EXAMPLE 3
Use software if available.
1. Draw line segment XY.
2. Choose point P on XY.
In Figure 1.34, find AB if
a) AX ⫽ 7.32 and XB ⫽ 6.19.
b) AX ⫽ 2x ⫹ 3 and XB ⫽ 3x ⫺ 7.
3. Measure XP, PY, and XY.
4. Show that XP ⫹ PY ⫽ XY.
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1.3
■
Early Definitions and Postulates
23
SOLUTION
a) AB ⫽ 7.32 ⫹ 6.19, so AB ⫽ 13.51.
b) AB ⫽ (2x ⫹ 3) ⫹ (3x ⫺ 7), so AB ⫽ 5x ⫺ 4.
DEFINITION
Congruent () line segments are two line segments that have the same length.
A
B
C
D
E
In general, geometric figures that can be made to coincide (fit perfectly one on top of the
other) are said to be congruent. The symbol is a combination of the symbol ~, which
means that the figures have the same shape, and =, which means that the corresponding parts
of the figures have the same measure. In Figure 1.35, AB CD, but AB EF (meaning that
AB and EF are not congruent). Does it appear that CD EF?
F
Figure 1.35
EXAMPLE 4
In the U.S. system of measures, 1 foot ⫽ 12 inches. If AB ⫽ 2.5 feet and CD ⫽ 2 feet
6 inches, are AB and CD congruent?
SOLUTION Yes, AB CD because 2.5 feet ⫽ 2 feet ⫹ 0.5 feet or 2 feet ⫹
0.5(12 inches), or 2 feet 6 inches.
DEFINITION
The midpoint of a line segment is the point that separates the line segment into two
congruent parts.
C
A
M
B
In Figure 1.36, if A, M, and B are collinear and AMMB, then M is the midpoint of
AB. Equivalently, M is the midpoint of AB if AM ⫽ MB. Also, if AM MB, then CD is
described as a bisector of AB.
If M is the midpoint of AB in Figure 1.36, we can draw any of these conclusions:
AM ⫽ MB
AM ⫽ 12(AB)
MB ⫽ 12(AB)
AB ⫽ 2 (AM)
AB ⫽ 2 (MB)
D
EXAMPLE 5
Figure 1.36
GIVEN:
M is the midpoint of EF (not shown). EM ⫽ 3x ⫹ 9 and
MF ⫽ x ⫹ 17
FIND:
x, EM, and MF
SOLUTION Because M is the midpoint of EF, EM ⫽ MF. Then
Discover
3x ⫹ 9
2x ⫹ 9
2x
x
Assume that M is the midpoint of AB
in Figure 1.36. Can you also conclude
that M is the midpoint of CD?
ANSWER
⫽
⫽
⫽
⫽
x ⫹ 17
17
8
4
By substitution, EM ⫽ 3(4) ⫹ 9 ⫽ 12 ⫹ 9 ⫽ 21 and MF ⫽ 4 ⫹ 17 ⫽ 21.
Thus, x ⫽ 4 while EM ⫽ MF ⫽ 21.
No
In geometry, the word union is used to describe the joining or combining of two
figures or sets of points.
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24
CHAPTER 1
■
LINE AND ANGLE RELATIONSHIPS
DEFINITION
!
—!
Ray AB, denoted by AB, is the union of AB and all points X on AB such that B is between
A and X.
!
!
!
!
—!
In Figure 1.37, AB , AB, and BA are shown in that order; note that AB and BA are not
the same ray.
Line AB
(AB has no endpoints)
A
B
A
B
A
B
Ray AB
(AB has endpoint A)
Ray BA
(BA has endpoint B)
Neil St.
Figure 1.37
Opposite rays are two rays with a! common
! endpoint; also, the union of opposite rays
is a straight line. In Figure 1.39(a), BA and BC are opposite rays.
The intersection of two geometric figures is the set of points that the two figures have
in common. In everyday life, the intersection of Bradley Avenue and Neil Street is the part
of the roadway that the two roads have in common (Figure 1.38).
Bradley Ave.
Figure 1.38
POSTULATE 4
A
B
C
If two lines intersect, they intersect at a point.
(a)
When two lines share two (or more) points,
the lines coincide; in this
situation, we say there
—!
—!
—!
is only one line. In Figure 1.39(a), AB and BC are the same as AC. In Figure 1.39(b), lines
ᐍ and m intersect at point P.
m
P
DEFINITION
(b)
Parallel lines are lines that lie in the same plane but do not intersect.
Figure 1.39
In Figure 1.40, suppose that ᐍ and n are parallel; in symbols, ᐍ n and ᐍ n ⫽ ⭋.
However, ᐍ and m are not parallel because they intersect at point A; so ᐍ m and
ᐍ m ⫽ A.
EXS. 5–12
EXAMPLE 6
In Figure 1.40, ᐍ n. What is the intersection of
n
a) lines n and m?
b) lines ᐍ and n?
A
B
m
SOLUTION
a) Point B
b) Parallel lines do not intersect.
Figure 1.40
Another undefined term in geometry is plane. A plane is two-dimensional; that is, it
has infinite length and infinite width but no thickness. Except for its limited size, a flat surface such as the top of a table could be used as an example of a plane. An uppercase letUnless otherwise noted, all content on this page is © Cengage Learning.
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1.3
■
Early Definitions and Postulates
25
ter can be used to name a plane. Because a plane (like a line) is infinite, we can show only
a portion of the plane or planes, as in Figure 1.41.
R
A
V
T
Planes R and S
B
C
S
D
Planes T and V
Figure 1.41
E
A plane is two-dimensional, consists of an infinite number of points, and contains an
infinite number of lines. Two distinct points may determine (or “fix”) a line; likewise, exactly three noncollinear points determine a plane. Just as collinear points lie on the same
line, coplanar points lie in the same plane. In Figure 1.42, points B, C, D, and E are coplanar, whereas A, B, C, and D are noncoplanar.
In this book, points shown in figures are generally assumed to be coplanar unless
otherwise stated. For instance, points A, B, C, D, and E are coplanar in Figure 1.43(a), as
are points F, G, H, J, and K in Figure 1.43(b).
Figure 1.42
A
C
K
G
D
J
F
B
H
E
(a)
Geometry in the Real World
Figure 1.43
© Dja65/Shutterstock.com
The tripod illustrates Postulate 5 in
that the three points at the base
enable the unit to sit level.
(b)
POSTULATE 5
Through three noncollinear points, there is exactly one plane.
On the basis of Postulate 5, we can see why a three-legged table sits evenly but a fourlegged table would “wobble” if the legs were of unequal length.
Space is the set of all possible points. It is three-dimensional, having qualities of
length, width, and depth. When two planes intersect in space, their intersection is a line. An
opened greeting card suggests this relationship, as does Figure 1.44(a). This notion gives
rise to our next postulate.
R
S
R
S
M
N
(a)
(b)
(c)
Figure 1.44
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26
CHAPTER 1
■
LINE AND ANGLE RELATIONSHIPS
Discover
POSTULATE 6
During a baseball game, the catcher
and the third baseman follow the
path of a foul pop fly toward the
grandstand. Does it appear that there
is a play on the baseball?
If two distinct planes intersect, then their intersection is a line.
The intersection of two planes is infinite because it is a line. [See Figure 1.44(a) on
page 25.] If two planes do not intersect, then they are parallel. The parallel vertical planes
R and S in Figure 1.44(b) may remind you of the opposite walls of your classroom. The parallel horizontal planes M and N in Figure 1.44(c) suggest the relationship between the ceiling and the floor.
Imagine a plane and two points of that plane, say
points A and B. Now think of the line
—!
containing the two points and the relationship of AB to the plane. Perhaps your conclusion
can be summed up as follows.
POSTULATE 7
VISITORS
Given two distinct points in a plane, the line containing these points also lies in the plane.
ANSWER
No; the baseball will land in the stands.
Because the uniqueness of the midpoint of a line segment can be justified, we call the
following statement a theorem. The “proof” of the theorem is found in Section 2.2.
EXS. 13–16
THEOREM 1.3.1
The midpoint of a line segment is unique.
A
M
If M is the midpoint of AB in Figure 1.45, then no other point can separate AB into two
congruent parts. The proof of this theorem is based on the Ruler Postulate. M is the point
that is located 12(AB) units from A (and from B).
The numbering system used to identify Theorem 1.3.1 need not be memorized. However, this theorem number may be used in a later reference. The numbering system works
as follows:
B
Figure 1.45
1
CHAPTER
where
found
EXS. 17–20
Exercises
B
1
ORDER
found in
section
A summary of the theorems presented in this textbook appears at the end of the book.
1.3
In Exercises 1 and 2, complete the statement.
A
3
SECTION
where
found
C
Exercises 1, 2
1. AB ⫹ BC ⫽ __?_
2. If AB ⫽ BC, then B is the __?_ of AC.
In Exercises 3 and 4, use the fact that 1 foot ⫽ 12 inches.
3. Convert 6.25 feet to a measure in inches.
4. Convert 52 inches to a measure in feet and inches.
In Exercises 5 and 6, use the fact that 1 meter 3.28 feet
(measure is approximate).
5. Convert
1
2
meter to feet.
6. Convert 16.4 feet to meters.
7. In the figure, the 15-mile road from
A to C is under construction. A
detour from A to B of 5 miles and
then from B to C of 13 miles must
be taken. How much farther is the
“detour” from A to C than the road
from A to C?
B
C
A
Exercises 7, 8
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■
1.3
8. A cross-country runner jogs at a rate of 15 feet per second.
If she runs 300 feet from A to B, 450 feet from B to C, and
then 600 feet from C back to A, how long will it take her to
return to point A? See figure for Exercise 7.
In Exercises 9 to 28, use the drawings as needed to answer the
following questions.
B
A
b) noncollinear.
10. How many lines can be drawn through
a)
b)
c)
d)
C
point A?
points A and B?
points A, B, and C?
points A, B, and D?
D
Exercises 9, 10
!
—!
11. Give the meanings of CD , CD, CD, and CD.
—
a) CD and DC .
b) CD and DC.
21. Make a sketch of
a) two intersecting lines that are perpendicular.
b) two intersecting lines that are not perpendicular.
c) two parallel lines.
a) two intersecting planes.
b) two parallel planes.
c) two parallel planes intersected by a third plane that is
not parallel to the first or the second plane.
23. Suppose that (a) planes M and N intersect, (b) point A lies in
both planes M and N, and (c) point B lies in !both planes M
—
and N. What can you conclude regarding AB ?
(b) AB ⬎ AC. Which point can you conclude cannot lie
between the other two?
c) CD! and DC.!
d) CD and DC .
25. Suppose that points A, R, and V are collinear. If AR ⫽ 7
13. Name two lines that appear to be
a) parallel.
—
in plane X. What can you conclude regarding CD ?
24. Suppose that (a) points A, B, and C are collinear and
12. Explain! the difference,
if any, between
!
—
20. Suppose that (a) point C lies in plane X and (b) point
D lies
!
22. Make a sketch of
9. Name three points that appear to be
a) collinear.
27
Early Definitions and Postulates
and RV ⫽ 5, then which point cannot possibly lie between
the other two?
b) nonparallel.
26. Points A, B, C, and D are
coplanar; B, C, and D are
collinear; point E is not
in plane M. How many
planes contain
a) points A, B, and C?
b) points B, C, and D?
c) points A, B, C, and D?
d) points A, B, C, and E?
m
p
14. Classify as true or false:
A
M
B
C
d) AB ⫹ BC ⫹ CD ⫽ AD
e) AB ⫽ BC
D
Exercises 14–17
15. Given:
Find:
16. Given:
Find:
17. Given:
Find:
C
D
M
a) is the midpoint of AE.
b) is the endpoint of a segment of length 4, if the other
endpoint is point G.
c) has a distance from B equal to 3(AC).
A
B
C
D
E
F
G
H
–3
–2
–1
0
1
2
3
4
Exercises 27, 28
M is the midpoint of AB
AM ⫽ 2x ⫹ 1 and MB ⫽ 3x ⫺ 2
x and AM
28. Consider the figure for Exercise 27. Given that B is the
midpoint of AC and C is the midpoint of BD, what can you
conclude about the lengths of
a) AB and CD?
c) AC and CD?
b) AC and BD?
M is the midpoint of AB
AM ⫽ 2(x ⫹ 1) and MB ⫽ 3(x ⫺ 2)
x and AB
In Exercises 29 to 32, use only a compass and a straightedge
to complete each construction.
AM ⫽ 2x ⫹ 1, MB ⫽ 3x ⫹ 2, and
AB ⫽ 6x ⫺ 4
x and AB
29. Given:
18. Can a segment bisect a line? a segment? Can a line bisect a
Construct:
AB and CD (AB ⬎ CD)
MN on line ᐉ so that MN ⫽ AB ⫹ CD
segment? a line?
A
19. In the figure, name
a) two opposite rays.
b) two rays that are not
opposite.
A
B
27. Using the number line provided, name the point that
t
a) AB ⫹ BC ⫽ AD
b) AD ⫺ CD ⫽ AB
c) AD ⫺ CD ⫽ AC
E
C
B
A
B
D
C
O
D
Exercises 29, 30
30. Given:
Construct:
AB and CD (AB ⬎ CD)
EF on line ᐉ so that EF ⫽ AB ⫺ CD
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28
■
CHAPTER 1
31. Given:
LINE AND ANGLE RELATIONSHIPS
AB as shown in the figure
PQ on line n so that PQ ⫽ 3(AB)
Construct:
A
B
n
Exercises 31, 32
32. Given:
AB as shown in the figure
TV on line n so that TV ⫽ 12(AB)
Construct:
38.
!
—!
AB and EF are said to be skew lines because they
neither intersect nor are parallel. How many planes are
determined by
a) parallel lines AB and DC?
b) intersecting lines AB and BC?
c) skew lines AB and EF?
d) lines AB, BC, and DC?
e) points A, B, and F?
f) points A, C, and H?
g) points A, C, F, and H?
—
33. Can you use the construction for the midpoint of a segment
to divide a line segment into
a) three congruent parts?
b) four congruent parts?
c) six congruent parts?
d) eight congruent parts?
G
F
A
B
H
E
34. Generalize your findings in Exercise 33.
35. Consider points A, B, C, and D, no three of which are
collinear. Using two points at a time (such as A and B), how
many lines are determined by these points?
36. Consider noncoplanar points A, B, C, and D. Using three
points at a time (such as A, B, and C), how many planes are
determined by these points?
37. Line ᐍ is parallel to plane P (that is, it will not intersect P
even if extended). Line m intersects line ᐍ. What can you
conclude about m and P?
P
D
C
Exercises 38–40
39. In the “box” shown for Exercise 38, use intuition to answer
each question.
a) Are AB and DC parallel?
b) Are AB and FE skew line segments?
c) Are AB and FE perpendicular?
40. In the “box” shown for Exercise 38, use intuition to answer
each question.
a) Are AG and BC skew line segments?
b) Are AG and BC congruent line segments?
c) Are GF and DC parallel?
*41. Let AB ⫽ a and BC ⫽ b. Point M is the midpoint of BC.
m
If AN ⫽ 23(AB), find the length of NM in terms of a and b.
A
1.4
KEY CONCEPTS
N
B
M
C
Angles and Their Relationships
Angle: Sides of Angle,
Vertex of Angle
Protractor Postulate
Acute, Right, Obtuse,
Straight, and Reflex
Angles
Angle-Addition
Postulate
Adjacent Angles
Congruent Angles
Bisector of an Angle
Complementary Angles
Supplementary Angles
Vertical Angles
This section introduces you to the language of angles. Recall from Sections 1.1 and 1.3 that
the word union means that two sets or figures are joined.
A
DEFINITION
An angle is the union of two rays that share a common endpoint.
1
B
C
Figure 1.46
!
!
The preceding definition is illustrated in Figure 1.46, in which BA and BC have the
common endpoint B.
All content on this page is © Cengage Learning.
Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.