2.2 Conditional Statements
Goal: Students will be able:
 To recognize conditional statements and their parts.
 To write converses, inverses, and contrapositives
of conditional statements.
Conditional Statement

A statement that can be written in “if – then”
form.

Symbol: p → q, read if p then q, or p implies q.

Example:
If it rains on Thursday, then the baseball game
will be canceled.
q
p
p
Hypothesis


The phrase immediately following the word if in a
conditional statement
The p part following if.
Conclusion

The phrase immediately following the word then in a
conditional statement.
If p, then q.
Hypothesis
Conclusion
Examples 1: Determine Hypothesis and
conclusion
If trout are fish, then trout live in a pond.
Hypothesis: trout are fish
Conclusion: trout live in a pond
If you buy a car, then you get $1500 cash back.
Hypothesis: you buy a car
Conclusion: you get $1500 cash back
Got it 1? Identify the hypothesis and conclusion
of each statement.
If an animal is a robin, then the animal is a bird.
Hypothesis: an animal is a robin
Conclusion: the animal is a bird.
If an angle measures 180°, then the angle is obtuse.
Hypothesis: an angle measures 180°
Conclusion: the angle is obtuse
If a polygon has 6 sides, then it is a hexagon.
Hypothesis: a polygon has 6 sides
Conclusion: it is a hexagon
Example 2:
Writing a Conditional
How can you write the following statement as a conditional?
Vertical angles share a vertex.
Step 1: Identify the hypothesis and conclusion.
In order for two angles to be vertical, they must share
a vertex.
So the set of vertical angles is inside the set of angles
that share a vertex.
Hypothesis: Vertical angles
Conclusion: share a vertex.
If two angles are vertical angles, then they share a vertex.
Got it 2?
How can you write “Dolphins are
mammals” as a conditional?
If an animal is a dolphin, then it is a mammal.
Mammals
Dolphins
Truth Value



Is either true or false
To show that a conditional is true, show that every
time the hypothesis is true, the conclusion is also true.
To show that a conditional is false, find only one
counterexample, where the hypothesis is true, and the
conclusion is false.
Example 3:
Finding the truth value of a
conditional.
Is the conditional true or false? If it is false, find a
counterexample.
If a number is divisible by 3, then it is odd.
The conclusion is false. The number 12 is divisible by 3,
and 12 is even.
Got it? 3:
Finding the truth value of a conditional.
Is the conditional true or false? If it is false, find a
counterexample.
If a month has 28 days, then it is February.
False, January has 28 days plus 3 more.
If two angles form a linear pair, then they are supplementary.
True
Negation

Is the opposite of the original statement
~p, read not p

Symbol: ~

Examples:
p: The sky is blue.
~p: The sky is not blue.
q: A triangle has 4 sides.
~q: A triangle does not have 4 sides.
Converse, Inverse, and Contrapositive
Statement
Formed by
Conditional
Given hypothesis
and conclusion
Converse
Exchange the
hypothesis and
conclusion of the
conditional
Inverse
Contrapositive
Negate both the
hypothesis and
conclusion of the
conditional
Exchange and Negating
both the hypothesis
and conclusion of the
conditional
Symbols
Examples
p→q
If two angles have the same
measure, then they are
congruent.
q→p
If two angles are congruent,
then they have the same
Measure.
~p → ~q
If two angles do not have
the same measure, then
they are not congruent.
~q → ~p
If two angles are not
congruent, then they do not
have the same measure.
Equivalent Statements

Have the same truth value
The conditional and the contrapositive are equivalent
statements.
The converse and the inverse are equivalent statements.
Biconditional Statements:
 When a conditional statement and its converse are
both true, you can write them as a single
biconditional statement.

A biconditional statement is a statement that
contains the phrase “if and only if.”

Any valid definition can be written as a
biconditional statement.
Example 1: Rewrite the conditional statement
in if-then form.
All birds have feathers.
Conditional: If it is a bird, then it has feathers.
Two angles are supplementary if they are a linear pair.
Conditional: If two angles are a linear pair, then they
are supplementary.
Example 2: Write the converse, inverse, and
contrapositive of the following conditional?
If a dog is a Great Dane, then it is large.
Converse: If the dog is large, then it is a Great Dane.
Inverse: If the dog is not a Great Dane, then it is not large.
Contrapositive: If the dog is not large, then it is not a Great
Dane.
Example 4. Write the definition of perpendicular lines
as a biconditional.
Definition: If two lines intersect to form a right
angle, then they are perpendicular.
Converse: If two lines are perpendicular, then they
intersect to form right angles.
Biconditional: Lines intersect to form right angles
iff they are perpendicular lines.