Geometry 201 unit 2.2

UNIT 2.2 CONDITIONAL
STATEMENTS

Warm Up
Determine if each statement is true or
false.
1. The measure of an obtuse angle is less than
90°.
2. All perfect-square numbers are positive.
3. Every prime number is odd.
4. Any three points are coplanar.
F
T
F
T

Identify, write, and analyze the truth
value of conditional statements.
Write the inverse, converse, and
contrapositive of a conditional
statement.
Objectives

conditional statement
hypothesis
conclusion
truth value
negation
converse
inverse
contrapostive
logically equivalent statements
Vocabulary

By phrasing a conjecture as an if-then statement,
you can quickly identify its hypothesis and
conclusion.

Identify the hypothesis and conclusion of each
conditional.
Example 1: Identifying the Parts of a Conditional
Statement
A. If today is Thanksgiving Day, then today is
Thursday.
B. A number is a rational number if it is an
integer.
Hypothesis: Today is Thanksgiving Day.
Conclusion: Today is Thursday.
Hypothesis: A number is an integer.
Conclusion: The number is a rational number.

Check It Out! Example 1
"A number is divisible by 3 if it is divisible by 6."
Identify the hypothesis and conclusion of the
statement.
Hypothesis: A number is divisible by 6.
Conclusion: A number is divisible by 3.

“If p, then q” can also be written as “if p, q,”
“q, if p,” “p implies q,” and “p only if q.”
Writing Math

Many sentences without the words if and then can
be written as conditionals. To do so, identify the
sentence’s hypothesis and conclusion by figuring
out which part of the statement depends on the
other.

Write a conditional statement from the
following.
Example 2A: Writing a Conditional Statement
An obtuse triangle has exactly one obtuse
angle.
If a triangle is obtuse, then it has exactly one
obtuse angle.
Identify the
hypothesis and the
conclusion.
An obtuse triangle
has exactly one obtuse angle.

following.
Example 2B: Writing a Conditional Statement
If an animal is a blue
jay, then it is a bird.
The inner oval represents the hypothesis, and
the outer oval represents the conclusion.

sentence “Two angles that are complementary
are acute.”
If two angles are complementary, then they
are acute.
Identify the
hypothesis
and the
conclusion.
Two angles that are complementary
are acute.

A conditional statement has a truth value of either
true (T) or false (F). It is false only when the
hypothesis is true and the conclusion is false.
To show that a conditional statement is false, you
need to find only one counterexample where the
hypothesis is true and the conclusion is false.

Determine if the conditional is true. If false,
give a counterexample.
Example 3A: Analyzing the Truth Value of a
Conditional Statement
If this month is August, then next month is
September.
When the hypothesis is true, the conclusion is
also true because September follows August.
So the conditional is true.

Example 3B: Analyzing the Truth Value of a
You can have acute angles with measures of
80° and 30°. In this case, the hypothesis is
true, but the conclusion is false.
If two angles are acute, then they are congruent.
Since you can find a counterexample, the
conditional is false.

Example 3C: Analyzing the Truth Value of a
An even number greater than 2 will never be
prime, so the hypothesis is false. 5 + 4 is not
equal to 8, so the conclusion is false. However,
the conditional is true because the hypothesis
is false.
If an even number greater than 2 is prime,
then 5 + 4 = 8.

Determine if the conditional “If a number is
odd, then it is divisible by 3” is true. If false,
An example of an odd number is 7. It is not
divisible by 3. In this case, the hypothesis is true,
but the conclusion is false. Since you can find a
counterexample, the conditional is false.

If the hypothesis is false, the conditional
statement is true, regardless of the truth value of
the conclusion.
Remember!

The negation of statement p is “not p,” written as
~p. The negation of a true statement is false, and
the negation of a false statement is true.

Definition Symbols
A conditional is a statement
that can be written in the form
"If p, then q."
p  q
Related Conditionals

Definition Symbols
The converse is the statement
formed by exchanging the
hypothesis and conclusion.
q  p

Definition Symbols
The inverse is the statement
formed by negating the
~p  ~q

Definition Symbols
The contrapositive is the
statement formed by both
exchanging and negating the
~q  ~p

Write the converse, inverse, and contrapositive
of the conditional statement. Use the Science
Fact to find the truth value of each.
Example 4: Biology Application
If an animal is an adult insect, then it has six
legs.

Example 4: Biology Application
Inverse: If an animal is not an adult insect, then it does
not have six legs.
Converse: If an animal has six legs, then it is an adult
insect.
If an animal is an adult insect, then it has six legs.
No other animals have six legs so the converse is true.
Contrapositive: If an animal does not have six legs,
then it is not an adult insect.
Adult insects must have six legs. So the contrapositive
is true.
No other animals have six legs so the converse is true.

Write the converse, inverse, and contrapostive
of the conditional statement “If an animal is a
cat, then it has four paws.” Find the truth value
of each.
If an animal is a cat, then it has four paws.

Inverse: If an animal is not a cat, then it does not
have 4 paws.
Converse: If an animal has 4 paws, then it is a cat.
Contrapositive: If an animal does not have 4 paws,
then it is not a cat; True.
If an animal is a cat, then it has four paws.
There are other animals that have 4 paws that are not
cats, so the converse is false.
There are animals that are not cats that have 4 paws,
so the inverse is false.
Cats have 4 paws, so the contrapositive is true.

Related conditional statements that have the same
truth value are called logically equivalent
statements. A conditional and its contrapositive
are logically equivalent, and so are the converse
and inverse.

The logical equivalence of a conditional and its
contrapositive is known as the Law of
Contrapositive.
Helpful Hint

Lesson Quiz: Part I
Identify the hypothesis and conclusion of each
conditional.
1. A triangle with one right angle is a right triangle.
2. All even numbers are divisible by 2.
3. Determine if the statement “If n2 = 144, then
n = 12” is true. If false, give a counterexample.
H: A number is even.
C: The number is divisible by 2.
H: A triangle has one right angle.
C: The triangle is a right triangle.
False; n = –12.

Lesson Quiz: Part II
Identify the hypothesis and conclusion of
each conditional.
4. Write the converse, inverse, and contrapositive
of the conditional statement “If Maria’s birthday is
February 29, then she was born in a leap year.”
Find the truth value of each.
Converse: If Maria was born in a leap year, then
her birthday is February 29; False.
Inverse: If Maria’s birthday is not February 29,
then she was not born in a leap year; False.
Contrapositive: If Maria was not born in a leap
year, then her birthday is not February 29; True.

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Geometry 201 unit 2.2

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Geometry 201 unit 2.2