1.5
41. Refer to the circle with center O.
■
Introduction to Geometric Proof
37
44. Refer to the circle with center P.
a) Use a protractor to find m∠ B.
b) Use a protractor to find m∠ D.
c) Compare results in parts (a) and (b).
a) Use a protractor to find m∠ 1.
b) Use a protractor to find m∠ 2.
c) Compare results in parts (a) and (b).
R
B
P
O
A
C
1
S
V
2
T
D
45. On the hanging sign, the three angles ( ∠ ABD, ∠ABC, and
42. If m∠ TSV ⫽ 38⬚, m ∠ USW ⫽ 40⬚, and m ∠TSW ⫽ 61⬚,
find m ∠USV.
∠ DBC) at vertex B have! the sum of measures 360⬚. If
m∠ DBC ⫽ 90⬚ and BA bisects the indicated reflex angle,
find m∠ ABC.
T
A
U
D
B
S
V
C
W
Exercises 42, 43
43. If m∠ TSU ⫽ x ⫹ 2z, m∠ USV ⫽ x ⫺ z, and
m∠ VSW ⫽ 2x ⫺ z, find x if m∠TSW ⫽ 60.
Also, find z if m∠ USW ⫽ 3x ⫺ 6.
1.5
KEY CONCEPTS
Reminder
Additional properties and techniques
of algebra are found in Appendix A.
46. With 0 ⬍ x ⬍ 90, an acute angle has measure x. Find
the difference between the measure of its supplement and
the measure of its complement.
Introduction to Geometric Proof
Algebraic Properties
Proof
Given Problem and
Prove Statement
To believe certain geometric principles, it is necessary to have proof. This section introduces some guidelines for establishing the proof of these geometric properties. Several
examples are offered to help you develop your own proofs. In the beginning, the form of
proof will be a two-column proof, with statements in the left column and reasons in the
right column. But where do the statements and reasons come from?
To deal with this question, you must ask “What” is known (Given) and “Why” the
conclusion (Prove) should follow from this information. In correctly piecing together a
proof, you will usually scratch out several conclusions, discarding some and reordering the
rest. Each conclusion must be justified by citing the Given (hypothesis), a previously
stated definition or postulate, or a theorem previously proved.
Selected properties from algebra are often used as reasons to justify statements. For
instance, we use the Addition Property of Equality to justify adding the same number to
each side of an equation. Reasons found in a proof often include the properties found in
Tables 1.5 and 1.6 on page 38.
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38
CHAPTER 1
■
LINE AND ANGLE RELATIONSHIPS
TABLE 1.5
Properties of Equality (a, b, and c are real numbers)
Addition Property of Equality:
If a ⫽ b, then a ⫹ c ⫽ b ⫹ c.
Subtraction Property of Equality:
If a ⫽ b, then a ⫺ c ⫽ b ⫺ c.
Multiplication Property of Equality:
If a ⫽ b, then a ⭈ c ⫽ b ⭈ c.
Division Property of Equality:
If a ⫽ b and c 苷 0, then
a b
= .
c c
As we discover in Example 1, some properties can be used interchangeably.
EXAMPLE 1
Which property of equality justifies each conclusion?
a) If 2x ⫺ 3 ⫽ 7, then 2x ⫽ 10.
b) If 2x ⫽ 10, then x ⫽ 5.
SOLUTION
a) Addition Property of Equality; added 3 to each side of the equation.
1
b) Multiplication Property of Equality; multiplied each side of the equation by 2 .
OR Division Property of Equality; divided each side of the equation by 2.
TABLE 1.6
Further Algebraic Properties of Equality (a, b, and c are real numbers)
Reflexive Property:
a ⫽ a.
Symmetric Property:
If a ⫽ b, then b ⫽ a.
Distributive Property:
a(b ⫹ c) ⫽ a ⭈ b ⫹ a ⭈ c.
Substitution Property:
If a ⫽ b, then a replaces b in any equation.
Transitive Property:
If a ⫽ b and b ⫽ c, then a ⫽ c.
Before considering geometric proof, we study algebraic proof in Examples 2 and 3.
Each statement in the proof is supported by the reason why we can make that statement
(claim). The first claim in the proof is the Given statement; and the sequence of steps must
conclude with a final statement representing the claim to be proved (called the Prove
statement).
In Example 2, we construct the algebraic proof of the claim, “If 2x ⫺ 3 ⫽ 7, then
x ⫽ 5.” Where P represents the statement “2x ⫺ 3 ⫽ 7,” and R represents “x ⫽ 5,” the
theorem has the form “If P, then R.” We also use letter Q to name the intermediate conclusion “2x ⫽ 10.” Using the letters P, Q, and R, we show the logical development for the
proof at the left. This logical format will not be provided in future proofs.
EXAMPLE 2
GIVEN:
PROVE:
2x ⫺ 3 ⫽ 7
x ⫽ 5
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1.5
■
Introduction to Geometric Proof
39
PROOF
Logical Format
EXS. 1–4
Statements
Reasons
P
1. 2x ⫺ 3 ⫽ 7
1. Given
If P, then Q
2. 2x ⫺ 3 ⫹ 3 ⫽ 7 ⫹ 3
2. Addition Property of Equality
Q
3. 2x ⫽ 10
3. Substitution
If Q, then R
4.
R
5. x ⫽ 5
10
2x
⫽
2
2
4. Division Property of Equality
5. Substitution
Study Example 3. Then cover the reasons and provide the reason for each statement.
In turn, with statements covered, find the statement corresponding to each reason.
EXAMPLE 3
2(x ⫺ 3) ⫹ 4 ⫽ 10
x ⫽ 6
GIVEN:
PROVE:
PROOF
Statements
Reasons
1. 2(x ⫺ 3) ⫹ 4 ⫽ 10
1. Given
2. 2x ⫺ 6 ⫹ 4 ⫽ 10
2. Distributive Property
3. 2x ⫺ 2 ⫽ 10
3. Substitution
4. 2x ⫽ 12
4. Addition Property of Equality
5. x ⫽ 6
5. Division Property of Equality
Alternatively, Step 5 could use the reason Multiplication Property of Equality
(multiply by 12 ).
NOTE 2: The fifth step is the final step because the Prove statement (x ⫽ 6) has been
made and justified.
NOTE 1:
EXS. 5–7
Discover
In the diagram, the wooden trim
pieces are mitered (cut at an angle)
to be equal and to form a right angle
when placed together. Use the
properties of algebra to explain why
the measures of ⬔1 and ⬔2 are both
45°. What you have done is an
informal “proof.”
1
The Discover activity at the left suggests that formal geometric proofs also exist. The
typical format for a problem requiring geometric proof is
GIVEN: ________
DRAWING
PROVE: ________
Consider this problem:
2
GIVEN:
A-P-B on AB (Figure 1.54)
A
PROVE:
AP ⫽ AB ⫺ PB
Figure 1.54
P
B
First consider the Drawing (Figure 1.54), and relate it to any additional information
described by the Given. Then consider the Prove statement. Do you understand the claim,
and does it seem reasonable? If it seems reasonable, intermediate claims can be ordered
and supported to form the contents of the proof. Because a proof must begin with the
Given and conclude with the Prove, the proof of the preceding problem has this form:
ANSWER
m⬔1 ⫹ m⬔2 ⫽ 90°. Because m⬔1 ⫽
m⬔2, we see that m⬔1 ⫹ m⬔1 ⫽ 90°.
Thus, 2 ⭈ m⬔1 ⫽ 90°, and, dividing by 2, we
see that m⬔1 ⫽ 45°. Then m⬔2 ⫽ 45° also.
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40
CHAPTER 1
■
LINE AND ANGLE RELATIONSHIPS
PROOF
Statements
Reasons
1. A-P-B on AB
1. Given
2. ?
2. ?
.
.
.
.
.
.
?. AP = AB - PB
?. ?
To construct the preceding proof, you must deduce from the Drawing and the
Given that
AP ⫹ PB ⫽ AB
In turn, you may conclude (through subtraction) that AP ⫽ AB ⫺ PB. The complete
proof problem will have the appearance of Example 4, which follows the first of several
“Strategy for Proof” features used in this textbook.
STRATEGY FOR PROOF ■ The First Line of Proof
General Rule: The first statement of the proof includes the “Given” information; also,
the first reason is Given.
Illustration: See the first line in the proof of Example 4.
EXAMPLE 4
A
P
B
Figure 1.55
GIVEN:
PROVE:
A-P-B on AB (Figure 1.55)
AP ⫽ AB ⫺ PB
PROOF
Statements
EXS. 8–10
Reasons
1. A-P-B on AB
1. Given
2. AP ⫹ PB ⫽ AB
2. Segment-Addition Postulate
3. AP ⫽ AB ⫺ PB
3. Subtraction Property of Equality
Some properties of inequality (see Table 1.7) are useful in geometric proof.
TABLE 1.7
Properties of Inequality (a, b, and c are real numbers)
Addition Property of Inequality:
If a ⬎ b, then a ⫹ c ⬎ b ⫹ c.
If a ⬍ b, then a ⫹ c ⬍ b ⫹ c.
Subtraction Property of Inequality:
If a ⬎ b, then a ⫺ c ⬎ b ⫺ c.
If a ⬍ b, then a ⫺ c ⬍ b ⫺ c.
SAMPLE PROOFS
Consider Figure 1.56 and this problem:
GIVEN:
MN ⬎ PQ
PROVE:
MP ⬎ NQ
M
N
P
Q
Figure 1.56
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1.5
■
Introduction to Geometric Proof
41
To understand the situation, first study the Drawing (Figure 1.56) and the related Given.
Then read the Prove with reference to the Drawing. What may be confusing here is that the
Given involves MN and PQ, whereas the Prove involves MP and NQ. However, this is
easily remedied through the addition of NP to each side of the inequality MN ⬎ PQ; see
Step 2 in the proof of Example 5.
EXAMPLE 5
M
N
P
GIVEN: MN ⬎ PQ (Figure 1.57)
PROVE: MP ⬎ NQ
Q
Figure 1.57
PROOF
Statements
Reasons
1. MN ⬎ PQ
1. Given
2. MN ⫹ NP ⬎ NP ⫹ PQ
2. Addition Property of Inequality
3. MN ⫹ NP ⫽ MP and
NP ⫹ PQ ⫽ NQ
3. Segment-Addition Postulate
4. MP ⬎ NQ
4. Substitution
NOTE: The final reason may come as a surprise. However, the Substitution Axiom of
Equality allows you to replace a quantity with its equal in any statement—including
an inequality! See Appendix A.3 for more information.
STRATEGY FOR PROOF ■ The Last Statement of the Proof
General Rule: The final statement of the proof is the “Prove” statement.
Illustration: See the last statement in the proof of Example 6.
EXAMPLE 6
T
R
U
V
Study this proof, noting the order of the statements and reasons.
!
GIVEN:
ST! bisects ∠RSU
SV bisects ∠USW (Figure 1.58)
PROVE:
m∠RST ⫹ m∠VSW ⫽ m ∠TSV
PROOF
S
W
Figure 1.58
Statements
!
1. ST bisects ∠RSU
2. m∠ RST ⫽ m∠ TSU
!
3. SV bisects ∠ USW
EXS. 11, 12
Reasons
1. Given
2. If an angle is bisected, then the measures
of the resulting angles are equal.
3. Same as reason 1
4. m∠ VSW ⫽ m∠ USV
4. Same as reason 2
5. m∠ RST ⫹ m∠ VSW ⫽
m ∠TSU ⫹ m∠ USV
5. Addition Property of Equality (use the
equations from statements 2 and 4)
6. m∠ TSU ⫹ m∠ USV ⫽ m∠ TSV
6. Angle-Addition Postulate
7. m∠ RST ⫹ m∠ VSW ⫽ m ∠ TSV
7. Substitution
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42
CHAPTER 1
■
Exercises
LINE AND ANGLE RELATIONSHIPS
1.5
In Exercises 1 to 6, which property justifies the conclusion of
the statement?
1. If 2x ⫽ 12, then x ⫽ 6.
In Exercises 23 and 24, fill in the missing reasons for the algebraic proof.
23. Given:
Prove:
2. If x ⫹ x ⫽ 12, then 2x ⫽ 12.
3(x ⫺ 5) ⫽ 21
x ⫽ 12
PROOF
3. If x ⫹ 5 ⫽ 12, then x ⫽ 7.
Statements
4. If x ⫺ 5 ⫽ 12, then x ⫽ 17.
5. If
x
5
⫽ 3, then x ⫽ 15.
6. If 3x ⫺ 2 ⫽ 13, then 3x ⫽ 15.
In Exercises 7 to 10, state the property or definition that
justifies the conclusion (the “then” clause).
then ∠s 3 and 4 are supplementary.
!
9. Given ∠RSV and ST as shown,
then m∠ RST ⫹ m ∠TSV ⫽
m ∠RSV.
10. Given that
! m∠ RST ⫽ m∠ TSV,
3.
3x ⫽ 36
3. ?
4.
x ⫽ 12
4. ?
PROOF
Statements
S
Reasons
1. 2x ⫹ 9 ⫽ 3
V
1. ?
2.
2x ⫽ ⫺ 6
2. ?
3.
x ⫽ ⫺3
3. ?
Exercises 9, 10
In Exercises 25 and 26, fill in the missing statements for the
algebraic proof.
25. Given:
Prove:
2(x ⫹ 3) ⫺ 7 ⫽ 11
x ⫽ 6
PROOF
B
Statements
Exercises 11, 12
11. Given:
A-M-B; Segment-Addition Postulate
12. Given:
M is the midpoint of AB; definition of midpoint
13. Given:
m ∠1 ⫽ m∠ 2; definition
of angle bisector
!
EG bisects ∠ DEF;
definition of angle bisector
14. Given:
2x ⫹ 9 ⫽ 3
x ⫽ ⫺3
T
In Exercises 11 to 22, use the Given information to draw a
conclusion based on the stated property or definition.
M
2. ?
Prove:
then ST bisects ∠ RSV.
A
1. ?
2. 3x ⫺ 15 ⫽ 21
R
supplementary, then
m∠ 1 ⫹ m∠ 2 ⫽ 180⬚.
8. Given that m ∠ 3 ⫹ m∠ 4 ⫽ 180⬚,
1. 3(x ⫺ 5) ⫽ 21
24. Given:
7. Given that ∠ s 1 and 2 are
Reasons
Reasons
1. ?
1. Given
2. ?
2. Distributive Property
3. ?
3. Substitution (Addition)
4. ?
4. Addition Property
of Equality
5. ?
5. Division Property
of Equality
D
G
1
15. Given:
∠s 1 and 2 are
2
F
complementary; definition E
of complementary angles Exercises 13–16
26. Given:
Prove:
x
5
⫹ 3 ⫽ 9
x ⫽ 30
16. Given:
m ∠1 ⫹ m∠ 2 ⫽ 90⬚; definition of
complementary angles
17. Given:
2x ⫺ 3 ⫽ 7; Addition Property of Equality
1. ?
1. Given
18. Given:
3x ⫽ 21; Division Property of Equality
2. ?
19. Given:
7x ⫹ 5 ⫺ 3 ⫽ 30; Substitution Property
of Equality
2. Subtraction Property
of Equality
3. ?
20. Given:
1
2
⫽ 0.5 and 0.5 ⫽ 50%; Transitive Property
of Equality
3. Multiplication Property
of Equality
21. Given:
3(2x ⫺ 1) ⫽ 27; Distributive Property
22. Given:
x
5
PROOF
Statements
Reasons
⫽ ⫺ 4; Multiplication Property of Equality
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1.5
In Exercises 27 to 30, fill in the missing reasons for each
geometric proof.
!
D-E-F on DF
DE ⫽ DF ⫺ EF
—
27. Given:
Prove:
D
E
1. ?
2. DE ⫹ EF ⫽ DF
2. ?
3. DE ⫽ DF ⫺ EF
3. ?
28. Given:
Reasons
E is the midpoint of DF
DE ⫽ 12(DF)
Prove:
Statements
2. DE ⫽ EF
2. ?
!
∠ ABC and BD (See figure for Exercise 29.)
m∠ ABD ⫽ m ∠ ABC ⫺ m ∠DBC
30. Given:
Prove:
F
PROOF
Statements
!
1. ∠ ABC and BD
Reasons
1. ?
2. m∠ ABD ⫹ m∠DBC
⫽ m∠ ABC
2. ?
3. m∠ ABD ⫽ m∠ ABC
⫺ m ∠DBC
3. ?
31. Given:
Reasons
1. ?
M-N-P-Q on MQ M
MN ⫹ NP ⫹ PQ ⫽ MQ
Prove:
N
P
Statements
Reasons
3. DE ⫹ EF ⫽ DF
3. ?
4. DE ⫹ DE ⫽ DF
4. ?
2. MN ⫹ NQ ⫽ MQ
2. ?
5. 2(DE) ⫽ DF
5. ?
3. NP ⫹ PQ ⫽ NQ
3. ?
1
2 (DF)
6. ?
4. ?
4. Substitution Property
of Equality
!
BD bisects ∠ ABC
m ∠ABD ⫽ 12(m ∠ ABC)
29. Given:
Prove:
Q
PROOF
1. ?
6. DE ⫽
43
In Exercises 31 and 32, fill in the missing statements
and reasons.
PROOF
1. E is the midpoint of DF
Introduction to Geometric Proof
Exercises 27, 28
PROOF
Statements
—!
1. D-E-F on DF
■
1. ?
!
!
∠ TSW with SU and SV
32. Given:
Prove:
m∠ TSW ⫽ m∠ TSU ⫹ m∠ USV ⫹ m∠VSW
A
T
D
U
B
S
C
V
Exercises 29, 30
W
PROOF
Statements
PROOF
Reasons
!
1. BD bisects ∠ ABC
1. ?
2. m ∠ABD ⫽ m∠ DBC
2. ?
3. m ∠ ABD ⫹ m∠ DBC
⫽ m∠ ABC
Statements
Reasons
1. ?
1. ?
2. ?
3. ?
2. m∠ TSW ⫽ m∠ TSU
⫹ m∠ USW
4. m∠ ABD ⫹ m∠ ABD
⫽ m ∠ABC
3. m∠ USW ⫽ m∠ USV
⫹ m ∠VSW
3. ?
4. ?
5. 2(m ∠ABD) ⫽ m ∠ABC
4. ?
5. ?
4. Substitution Property
of Equality
6. m∠ ABD ⫽ 12(m ∠ ABC)
6. ?
33. When the Distributive Property is written in its symmetric
form, it reads a ⭈ b ⫹ a ⭈ c ⫽ a(b ⫹ c). Use this form to
rewrite 5x ⫹ 5y.
34. Another form of the Distributive Property (see Exercise 33)
reads b ⭈ a ⫹ c ⭈ a ⫽ (b ⫹ c)a. Use this form to rewrite
5x ⫹ 7x. Then simplify.
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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.
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