44
CHAPTER 1
■
LINE AND ANGLE RELATIONSHIPS
35. The Multiplication Property of Inequality requires that
37. Provide reasons for this proof. “If a ⫽ b and c ⫽ d, then
a ⫹ c ⫽ b ⫹ d.”
we reverse the inequality symbol when multiplying by a
negative number. Given that ⫺7 ⬍ 5, form the inequality
that results when we multiply each side by ⫺2.
PROOF
Statements
36. The Division Property of Inequality requires that we reverse
the inequality symbol when dividing by a negative number.
Given that 12 ⬎ ⫺ 4, form the inequality that results when
we divide each side by ⫺ 4.
1.
a ⫽ b
1. ?
2. a ⫹ c ⫽ b ⫹ c
3.
Reasons
c ⫽ d
2. ?
3. ?
4. a ⫹ c ⫽ b ⫹ d
4. ?
38. Write a proof for: “If a ⫽ b and c ⫽ d, then
a ⫺ c ⫽ b ⫺ d.”
(HINT: Use Exercise 37 as a guide.)
1.6
KEY CONCEPTS
Relationships: Perpendicular Lines
Vertical Line(s)
Horizontal Line(s)
Perpendicular Lines
Relations: Reflexive,
Symmetric, and
Transitive Properties
Equivalence Relation
Perpendicular Bisector
of a Line Segment
Informally, a vertical line is one that extends up and down, like a flagpole. On the other
hand, a line that extends left to right is horizontal. In Figure 1.59, ᐍ is vertical and j is
horizontal. Where lines ᐍ and j intersect, they appear to form angles of equal measure.
j
DEFINITION
Perpendicular lines are two lines that meet to form congruent adjacent angles.
Perpendicular lines do not have to be vertical and horizontal. In Figure 1.60, the slanted
lines m and p are perpendicular (m ⬜ p). As in Figure 1.60, a small square is often placed
in the opening of an angle formed by perpendicular lines.
Example 1 provides a formal proof of the relationship between perpendicular lines and
right angles. Study this proof, noting the order of the statements and reasons. The numbers
in parentheses to the left of the statements refer to the earlier statement(s) of the proof upon
which the new statement is based.
Figure 1.59
m
p
STRATEGY FOR PROOF ■ The Drawing for the Proof
General Rule: Make a drawing that accurately characterizes the “Given” information.
Illustration: For the proof of Example 1, see Figure 1.61.
THEOREM 1.6.1
Figure 1.60
If two lines are perpendicular, then they meet to form right angles.
Unless otherwise noted, 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.
1.6
GIVEN:
PROVE:
E
Relationships: Perpendicular Lines
45
EXAMPLE 1
C
A
■
! —!
AB ⬜ CD , intersecting at E (See Figure 1.61)
∠AEC is a right angle
—
PROOF
B
Statements
! —!
1. AB ⬜ CD , intersecting at E
Reasons
—
D
1. Given
(1)
2. ∠ AEC ∠ CEB
2. Perpendicular lines meet to form
congruent adjacent angles (Definition)
(2)
3. m∠ AEC ⫽ m∠ CEB
3. If two angles are congruent, their
measures are equal
4. ∠ AEB is a straight angle
and m∠ AEB ⫽ 180⬚
4. Measure of a straight angle equals 180⬚
5. m∠ AEC ⫹ m∠ CEB ⫽ m∠AEB
5. Angle-Addition Postulate
Figure 1.61
(4), (5) 6. m ∠ AEC ⫹ m∠ CEB ⫽ 180⬚
6. Substitution
(3), (6) 7. m∠ AEC ⫹ m ∠ AEC ⫽ 180⬚
or 2 ⭈ m∠ AEC ⫽ 180⬚
7. Substitution
(7)
8. m∠ AEC ⫽ 90⬚
8. Division Property of Equality
(8)
9. ∠ AEC is a right angle
9. If the measure of an angle is 90⬚,
then the angle is a right angle
EXAMPLE 2
In Figure 1.61, find the sum of m∠AEC ⫹ m∠CEB ⫹ m∠BED ⫹ m∠DEA.
SOLUTION
Because each angle of the sum measures 90⬚, the total is 4 ⭈ 90⬚ or 360⬚.
In general, “The sum of the measures of the nonoverlapping adjacent angles about a
point is 360⬚.”
RELATIONS
The relationship between perpendicular lines suggests the more general, but undefined, mathematical concept of relation. In general, a relation “connects” two elements of an associated
set of objects. Table 1.8 provides several examples of the concept of a relation R.
TABLE 1.8
EXS. 1, 2
Relation R
Objects Related
Example of Relationship
is equal to
numbers
2 ⫹ 3 ⫽ 5
is greater than
numbers
7 ⬎ 5
is perpendicular to
lines
ᐍ⬜m
is complementary to
angles
∠ 1 is comp. to ∠2
is congruent to
line segments
AB CD
is a brother of
people
Matt is a brother of Phil
Unless otherwise noted, 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.
46
CHAPTER 1
■
LINE AND ANGLE RELATIONSHIPS
There are three special properties that may exist for a given relation R. Where a, b, and
c are objects associated with relation R, the properties consider one object (reflexive), two objects in either order (symmetric), or three objects (transitive). For the properties to exist, it is
necessary that the statements be true for all objects selected from the associated set. These
properties are generalized, and specific examples are given below:
Reminder
Numbers that measure may be
equal (AB = CD or m∠ 1 = m∠ 2),
whereas geometric figures may be
congruent (AB CD or ∠ 1 ∠2).
Reflexive property: aRa (5 ⫽ 5; equality of numbers has a reflexive property)
Symmetric property: If aRb, then bRa. (If ᐍ ⬜ m, then m ⬜ ᐍ; perpendicularity of lines has
a symmetric property)
Transitive property: If aRb and bRc, then aRc. (If ∠1 ∠ 2 and ∠2 ∠ 3, then
∠ 1 ∠ 3; congruence of angles has a transitive property)
EXAMPLE 2
© Karel Broz̆ /Shutterstock.com
Geometry in Nature
Does the relation “is less than” for numbers have a reflexive property? a symmetric
property? a transitive property?
SOLUTION Because “2 ⬍ 2” is false, there is no reflexive property.
“If 2 ⬍ 5, then 5 ⬍ 2” is also false; there is no symmetric property.
“If 2 ⬍ 5 and 5 ⬍ 9, then 2 ⬍ 9” is true; there is a transitive property.
NOTE: The same results are obtained for choices other than 2, 5, and 9.
An icicle formed from freezing
water assumes a vertical path.
Congruence of angles (or of line segments) is closely tied to equality of angle measures (or line segment measures) by the definition of congruence.
PROPERTIES FOR THE CONGRUENCE OF ANGLES
Reflexive:
Symmetric:
Transitive:
EXS. 3–9
Technology Exploration
Use computer software if available.
—!
—!
1. Construct AC and BD to intersect
at point O. [See Figure 1.62(a).]
2. Measure ∠ 1, ∠2, ∠ 3, and ∠4.
3. Show that m∠1 = m ∠3 and
∠ 1 ∠ 1; an angle is congruent to itself.
If ∠ 1 ∠ 2, then ∠ 2 ∠1.
If ∠ 1 ∠2 and ∠ 2 ∠ 3, then ∠1 ∠3.
Any relation (such as congruence of angles) that has reflexive, symmetric, and transitive properties is known as an equivalence relation. In later chapters, we will see that congruence of triangles and similarity of triangles also have reflexive, symmetric, and
transitive properties; therefore, these relations are
also equivalence relations.
A
B
3
Returning to the formulation of a proof, the fiO
2
4
nal example in this section is based on the fact that
D
C
1
vertical angles are congruent when two lines intersect. See Figure 1.62(a). Because there are two pairs
Figure 1.62(a)
of congruent angles, the Prove could be stated
Prove: ∠1 ∠3 and ∠2 ∠ 4
Such a conclusion is a conjunction and would be proved if both congruences were
established. For simplicity, the Prove of Example 3 is stated
Prove: ∠2 ∠4
Study this proof of Theorem 1.6.2, noting the order of the statements and reasons.
m ∠ 2 = m ∠4.
THEOREM 1.6.2
If two lines intersect, then the vertical angles formed are congruent.
Unless otherwise noted, all content on this page is © Cengage Learning.g.
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.
1.6
A
3
O
2
D
4
1
B
EXAMPLE 3
C
GIVEN:
PROVE:
Figure 1.62(b)
■
Relationships: Perpendicular Lines
47
!
—!
AC intersects BD at O [See Figure 1.62(b).]
∠2 ∠4
—
PROOF
Statements
!
—!
1. AC intersects BD at O
—
A
B
X
(a)
Reasons
1. Given
2. ∠ s AOC and DOB are straight ∠ s, with
m∠ AOC ⫽ 180 and m∠ DOB ⫽ 180
2. The measure of a straight angle is 180⬚
3. m∠ DOB ⫽ m∠ AOC
3. Substitution
4. m∠ 1 ⫹ m∠ 4 ⫽ m ∠ DOB and
m∠ 1 ⫹ m∠ 2 ⫽ m ∠ AOC
4. Angle-Addition Postulate
5. m∠ 1 ⫹ m∠ 4 ⫽ m∠ 1 ⫹ m∠ 2
5. Substitution
6. m∠ 4 ⫽ m∠ 2
6. Subtraction Property of Equality
7. ∠ 4 ∠ 2
7. If two angles are equal in measure,
the angles are congruent
8. ∠ 2 ∠ 4
8. Symmetric Property of Congruence
of Angles
In the preceding proof, there is no need to reorder the congruent angles from statement 7 to statement 8 because congruence of angles is symmetric; in the later work,
statement 7 will be written to match the Prove statement even if the previous line does not
have the same order. The same type of thinking applies to proving lines perpendicular or
parallel: The order is simply not important!
CONSTRUCTIONS LEADING TO PERPENDICULAR LINES
A
C
X
(b)
D
B
Construction 2 in Section 1.2 determined not only the midpoint of AB but also that of the
perpendicular bisector of AB. In many instances, we need the line perpendicular to another line at a point other than the midpoint of a segment.
CONSTRUCTION 5 To construct the line perpendicular to a given line at a
specified point on the given line.
—!
GIVEN: AB with point X in Figure 1.63(a)
—!
—!
—!
CONSTRUCT: A line EX , so that EX ⬜ AB
E
A
C
X
D
CONSTRUCTION: Figure 1.63(b): Using
X as the center, mark off arcs of equal
—!
radii on each side of X to intersect AB at C and D.
Figure 1.63(c): Now, using C and D as centers, mark off arcs of equal radii
with a length greater than
XD so that these arcs intersect either above
—!
(as shown) or below AB .
—!
Calling
the
point of intersection E, draw EX , which is the desired line; that is,
!
!
—
—
EX ⬜ AB .
B
(c)
Figure 1.63
The theorem that Construction 5 is based on is a consequence of the Protractor
Postulate, and we state it without proof.
THEOREM 1.6.3
In a plane, there is exactly one line perpendicular to a given line at any point on the line.
Unless otherwise noted, 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.
48
CHAPTER 1
■
LINE AND ANGLE RELATIONSHIPS
Construction 2, which was used to locate the midpoint of a line segment in Section 1.2,
is also the method
for constructing the perpendicular bisector of a line segment. In
—!
Figure 1.64, XY is the perpendicular bisector of RS. The following theorem can be proved
by methods developed later in this book.
X
M
S
THEOREM 1.6.4
R
The perpendicular bisector of a line segment is unique.
Y
Figure 1.64
EXS. 10–14
Exercises
1.6
In Exercises 1 and 2, supply reasons.
∠1 ∠3
∠ MOP ∠ NOQ
1. Given:
Prove:
2. Given:
M
N
1
2
O
Reasons
1. ∠1 ∠ 3
1. ?
2. m∠1 ⫽ m ∠3
2. ?
3. m∠1 ⫹ m ∠2 ⫽ m∠ MOP
and
m∠2 ⫹ m ∠ 3 ⫽ m∠ NOQ
3. ?
4. m∠1 ⫹ m ∠ 2 ⫽
m∠2 ⫹ m ∠ 3
4. ?
5. m∠MOP ⫽ m∠ NOQ
5. ?
6. ∠ MOP ∠ NOQ
6. ?
C
1
2
3
O
PROOF
3
Q
Reasons
—
Statements
1
Prove:
Statements
!
—!
1. AB intersects CD at O
PROOF
A
P
!
—!
AB intersects CD at O so that ∠ 1 is a right ∠
(Use the figure following Exercise 1.)
∠ 2 and ∠ 3 are complementary
—
2
1. ?
2. ∠ AOB is a straight ∠ , so
m∠ AOB ⫽ 180
2. ?
3. m∠ 1 ⫹ m∠ COB ⫽
m ∠ AOB
3. ?
4. m∠ 1 ⫹ m∠ COB ⫽ 180
4. ?
5. ∠ 1 is a right angle
5. ?
6. m∠ 1 ⫽ 90
6. ?
7. 90 ⫹ m∠ COB ⫽ 180
7. ?
8. m ∠COB ⫽ 90
8. ?
9. m ∠2 ⫹ m∠ 3 ⫽
m∠ COB
10. m∠ 2 ⫹ m∠ 3 ⫽ 90
11. ∠2 and ∠ 3 are
complementary
9. ?
10. ?
11. ?
B
In Exercises 3 and 4, supply statements.
3
D
Exercise 2
Exercise 3
∠ 1 ∠ 2 and ∠2 ∠ 3
Prove:
∠1 ∠3
(Use the figure following Exercise 1.)
3. Given:
PROOF
Statements
Reasons
1. ?
1. Given
2. ?
2. Transitive Property
of Congruence
Unless otherwise noted, 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.
1.6
m∠ AOB ⫽ m∠ 1
m ∠BOC
⫽ m∠ 1
!
OB bisects ∠ AOC
4. Given:
Prove:
■
Relationships: Perpendicular Lines
49
In Exercises 11 and 12, provide the missing statements and
reasons.
11. Given:
A
Prove:
∠ s 1 and 3 are complementary
∠ s 2 and 3 are complementary
∠1 ∠2
B
C
1
1
O
2
PROOF
Statements
1. ?
1. Given
2. ?
2. Substitution
3. ?
3. Angles with equal
measures are congruent
4. ?
4. If a ray divides an angle
into two congruent
angles, then the ray
bisects the angle
In Exercises 5 to 9, use a compass and a straightedge to
complete the constructions.
5. Given:
Point N on line s
Line m through N so that m ⬜ s
Construct:
3
Reasons
4
PROOF
Statements
Reasons
1. ∠ s 1 and 3 are
complementary;
∠ s 2 and 3 are
complementary
1. ?
2. m∠ 1 ⫹ m∠ 3 ⫽ 90;
m ∠2 ⫹ m∠ 3 ⫽ 90
2. The sum of the measures
of complementary
∠ s is 90
(2) 3. m∠ 1 ⫹ m∠ 3 ⫽
m∠ 2 ⫹ m ∠3
3. ?
4. ?
4. Subtraction Property
of Equality
(4) 5. ?
5. If two ∠ s are ⫽ in
measure, they are s
N
!
OA
Right angle BOA
6. Given:
Construct:
12. Given:
!
(HINT: Use a straightedge to extend OA to the left.)
O
Prove:
1
A
2
Line ᐍ containing point A
A 45° angle with vertex at A
7. Given:
Construct:
∠ 1 ∠2; ∠ 3 ∠ 4
∠ s 2 and 3 are complementary
∠ s 1 and 4 are complementary
3
4
A
PROOF
8. Given:
Construct:
AB
The perpendicular bisector of AB
A
9. Given:
Construct:
Statements
1. ∠ 1 ∠ 2 and
∠3 ∠4
2. ? and ?
B
Triangle ABC
The perpendicular bisectors of sides AB, AC,
and BC
C
(3)
A
B
10. Draw a conclusion based on the results of Exercise 9.
3. ∠ s 2 and 3 are
complementary
4. ?
(2), (4) 5. m ∠1 ⫹ m∠ 4
⫽ 90
6. ?
Reasons
1. ?
2. If two ∠ s are , then
their measures are equal
3. ?
4. The sum of the measures
of complementary
∠ s is 90
5. ?
6. If the sum of the measures of two angles is 90,
then the angles are
complementary
Unless otherwise noted, 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.
50
CHAPTER 1
■
LINE AND ANGLE RELATIONSHIPS
13. Does the relation “is perpendicular to” have a reflexive
26. The Angle-Addition Postulate
property (consider line ᐍ)? a symmetric property (consider
lines ᐍ and m)? a transitive property (consider lines ᐍ, m,
and n)?
14. Does the relation “is greater than” have a reflexive property
(consider real number a)? a symmetric property (consider
real numbers a and b)? a transitive property (consider real
numbers a, b, and c)?
L
M
1
2
3
H
4
N
K
27. If there were no understood restriction to lines in a plane in
Theorem 1.6.4, the theorem would be false. Explain why
the following statement is false: “In space, the perpendicular
bisector of a line segment is unique.”
15. Does the relation “is complementary to” for angles have a
reflexive property (consider one angle)? a symmetric
property (consider two angles)? a transitive property
(consider three angles)?
G
can be generalized as follows:
“The measure of an angle equals
the sum of the measures of its
parts.” State a general conclusion
about m∠ GHK based on the
figure shown.
*28. In the proof below, provide the missing reasons.
Given:
16. Does the relation “is less than” for numbers have a reflexive
property (consider one number)? a symmetric property
(consider two numbers)? a transitive property (consider
three numbers)?
Prove:
∠ 1 and ∠ 2 are complementary
∠ 1 is acute
∠ 2 is also acute
PROOF
17. Does the relation “is a brother of” have a reflexive property
Statements
(consider one male)? a symmetric property (consider two
males)? a transitive property (consider three males)?
18. Does the relation “is in love with” have a reflexive property
(consider one person)? a symmetric property (consider two
people)? a transitive property (consider three people)?
19. This textbook has used numerous symbols and abbrevia-
tions. In this exercise, indicate what word is represented or
abbreviated by each of the following:
a) ⬜
b) ∠s
c) supp.
d) rt.
e) m∠ 1
20. This textbook has used numerous symbols and abbrevia-
tions. In this exercise, indicate what word is represented or
abbreviated by each of the following:
a) post.
b) ´
c) ⭋
d) <
e) pt.
21. This textbook has used numerous symbols and abbrevia-
tions. In this exercise, indicate what word is represented or
abbreviated by each of the following:
!
a) adj.
b) comp.
c) AB
d) e) vert.
22. If there were no understood restriction to lines in a plane in
Theorem 1.6.3, the theorem would be false. Explain why the
following statement is false: “In space, there is exactly one
line perpendicular to a given line at any point on the line.”
(1)
Reasons
1. ∠ 1 and ∠ 2 are complementary
1. ?
2. m∠ 1 ⫹ m∠ 2 ⫽ 90
2. ?
3. ∠ 1 is acute
3. ?
(3)
4. Where m∠ 1 ⫽ x,
0 ⬍ x ⬍ 90
4. ?
(2)
5. x ⫹ m∠ 2 ⫽ 90
5. ?
(5)
6. m∠ 2 ⫽ 90 ⫺ x
6. ?
(4)
7. ⫺x ⬍ 0 ⬍ 90 ⫺ x
7. ?
(7)
8. 90 ⫺ x ⬍ 90 ⬍
180 ⫺ x
8. ?
(7), (8) 9. 0 ⬍ 90 ⫺ x ⬍ 90
9. ?
(6), (9) 10. 0 ⬍ m∠ 2 ⬍ 90
(10)
10. ?
11. ∠ 2 is acute
11. ?
29. Without writing a proof, explain the conclusion for
the following problem.
Given:
Prove:
!
AB
m∠ 1 ⫹ m ∠2 ⫹ m∠ 3 ⫹ m ∠ 4 ⫽ 180
—
23. Prove the Extended Segment Addition Property by using the
Drawing, the Given, and the Prove that follow.
Given:
M-N-P-Q on MQ
Prove:
MN ⫹ NP ⫹ PQ ⫽ MQ
M
N
P
2
1
A
Q
24. The Segment-Addition Postulate can be generalized as fol-
lows: “The length of a line segment equals the sum of the
lengths of its parts.” State a general conclusion about AE
based on the following figure.
A
B
C
D
E
4
O
B
Exercises 29, 30
30. Without writing a proof, explain the conclusion for the fol-
lowing problem.
—!
Given:
AB, ∠ s 2 and 3 are complementary.
Prove:
∠ s 1 and 4 are complementary.
(HINT: Use the result from Exercise 29.)
25. Prove the Extended Angle
Addition Property by using
the Drawing, the Given, and
the Prove that follow.
!
!
Given:
∠ TSW with SU and SV
Prove:
m ∠ TSW ⫽ m ∠TSU ⫹
m∠ USV ⫹ m∠VSW
3
T
U
S
V
W
Unless otherwise noted, 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.