784 Using Algebra to Prove
Lesson
You might have heard of the question, “Which came fi rst, the chicken
or the egg?” This question is diffi cult because: If there is a chicken,
then there must have been an egg. If there is an egg, then there must
have been a chicken.
These two if-then statements are converses. The converse of the
statement, “If p, then q” is the statement, “If q, then p.” Another way
of stating this is: the converse of an if-then statement is found by
switching the antecedent and the consequent of the statement.
QY
Converses of True Statements
Often the converse of a true statement is not true. Here
is an example.
Statement If x = 8, then x
2
= 64. (true)
Converse If x
2
= 64, then x = 8. (False; x could also be −8.)
Example 1
The given statement is true. Write its converse and explain why the converse
is not true.
a. If a quadrilateral has 4 sides of the same length and its diagonals have
the same length, then the quadrilateral is a rectangle.
b. If you live in the state of North Dakota, then you live in the United States.
Solutions
a. Converse: If a quadrilateral is
?
, then
?
. This statement is
false because
?
.
b. Converse: If
?
, then
?
The statement is false because
?
.
The Converse of an
If-Then Statement
Chapter 13
Vocabulary
converse
equivalent statements
if and only if
BIG IDEA The converse of the statement, “If A, then B.” is the
statement, “If B, then A.”
13-2
GUIDED
QY
Give the converse of the
statement: If
√
2x = 10,
then x = 50.
Evaluate when a = 2
and b = –2.
a. (a + b)
2
b. a
2
+ b
2
c. a
2
+ 2ab + b
2
Mental Math
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The Converse of an If-Then Statement 785
Sometimes the converse of a true statement is true. When a
statement and its converse are both true, then the antecedent and
consequent are equivalent statements. Equivalent statements can
be connected by the phrase if and only if. Here is an example.
Statement If x = 6, then 2
x
= 64. (true)
Converse If 2
x
= 64, then x = 6. (true)
Because the statement and its converse are both true, you can write:
x = 6 if and only if 2
x
= 64.
Example 2
The given statement is true and so is its converse. Write the converse
and then combine the statement and its converse into one if-and-only-if
statement.
a. If a quadrilateral has 4 sides of the same length and its diagonals have
the same length, then the quadrilateral is a square.
b. If you live in the largest country in South America, then you live in Brazil.
Solution
a. Converse: If a quadrilateral
?
, then
?
. You can write: A
quadrilateral has 4 sides of the same length and diagonals of the same
length if and only if
?
.
b. Converse: If you live in Brazil, then you live in the largest country in South
America. You can write:
?
if and only if
?
.
When you see the phrase “if and only if,” then you can separate the
sentence into two if-then statements.
In Hardnox High School, a student is on the honor roll if and only
if his or her grade point average is at least 3.75.
means
In Hardnox High School, if a student’s grade point average is at
least 3.75, then the student is on the honor roll.
and
If a student in Hardnox High School is on the honor roll, then his
or her grade point average is at least 3.75.
Questions
COVERING THE IDEAS
1.
State the converse of the statement: If there is smoke, then there
is fi re.
Lesson 13-2
GUIDED
Brazil is the fourth most
populous country in the
world with about 190 million
people.
Source: The World Factbook
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786 Using Algebra to Prove
2. Multiple Choice
If a statement is true, then its converse
A must be true. B may be true. C must be false.
In 3–5, a statement is given.
a. Is it true?
b. State its converse.
c. Is its converse true?
d. If either the statement or its converse is not true, correct them
so that they are both true.
3. If an integer is divisible by 3 and by 4, then it is divisible by 24.
4. If both the units and tens digits of an integer written in base 10
equal 0, then the integer is divisible by 100.
5. If 7u < 56, then u > 8.
In 6–8, write the two if-then statements that are meant by the if-and-
only-if statement.
6. You will receive full credit for this question if and only if you get
both parts correct.
7. A quadrilateral is a rectangle if and only if it has four right
angles.
8. 5x + 4y = 20 if and only if y = −1.25x + 5.
In 9 and 10, rewrite the de nition as an if-and-only-if statement.
9. Every linear function has an equation of the form f(x) = ax + b.
10. The reciprocal of a nonzero number x is the number y such that
xy = 1.
In 11–14, are statements (1) and (2) equivalent? If not, why not?
11. (1) x = 3 12. (1) y = 15
(2) 2x = 6 (2) y
4
= 50,625
13. (1) z = 8 14. (1) a + b = c
(2) z
2
+ 48 = 14z (2) c - b = a
APPLYING THE MATHEMATICS
In 15–18, a statement is given.
a. Is it true?
b. State its converse.
c. Is its converse true?
d. If either the statement or its converse is not true, correct them
so that they are both true.
15. If (x - 5)(2x + 3) = 45, then 2x
2
- 7x + 30 = 90.
Chapter 13
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The Converse of an If-Then Statement 787
16. If a polygon has 7 sides, then it is a hexagon.
17. If a parabola has an equation of the form y = (x - 5)(x + 3), then
its x-intercepts are −5 and 3.
18. If 3x + 4y = 6 and 2x - 5y = 7, then x = 6 and y = −3.
In 19–21, a statement is given.
a. Tell whether the statement is true.
b. If the statement is true, give an example. If it is not true,
modify it so that it is true.
19. A line is a vertical line if and only if its slope is undefi ned.
20. x
a
· x
b
= x
a + b
if and only if a and b are positive integers.
21. A person can be a U.S. citizen if and only if the person was born
in the United States.
REVIEW
In 22 and 23, tell whether the given if-then statement is true or false.
If it is false, provide a counterexample to the statement.
(Lessons 13-1, 9-1)
22. If a dining room table seats 8 people, then it seats 6 people.
23. If n is even, the graph of y = ax
n
crosses the x-axis twice.
24. Multiple Choice
Rectangles of width x are cut off from two
adjacent sides of a 12 in.-by-12 in. sheet of wrapping paper, as
shown at the right. What is the area, in square inches, of the
square region that remains?
(Lesson 11-6)
A 12 - x
2
B 144 + x
2
C 144 - x
2
D 144 - 24x + x
2
25. If 8 pencils and 5 erasers cost $4.69 and 3 pencils and 4 erasers
cost $2.80, fi nd the cost of 2 pencils.
(Lesson 10-5)
In 26–28, simplify the expression. (Lessons 8-6, 8-5)
26. 5(3
√
x )
2
27.
(
6 · 2
−2
)
3
28.
9 ±
√
18
_______
3
EXPLORATION
29.
Write two statements from outside of mathematics that are true
but whose converses are false.
Lesson 13-2
A valid U.S. passport is
required for U.S. citizens to
enter and leave most foreign
countries.
Source: U.S. Department of State
12 in.
x
in.
x
in.
12 in.
QY ANSWER
If x = 50, then
√
2x = 10.
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