5-4 Inverses,
Contrapositives,
and Indirect
Reasoning
conditional
A _____________
statement
___________ is:
a statement in IfThen form.
The
IF part is the
HYPOTHESIS
The THEN part is
the conclusion.
The __________
negation of a statement
is like the opposite of the
statement.
Examples:
I like purple.
Negation: I don't like purple.
It isn’t sunny.
 Negation: It is sunny.
inverse of a conditional
The _________
is formed by negating the
hypothesis and the conclusion.
Example:
Conditional: If a man lives in Los
Angeles, then he lives in California.
Inverse:
If a man does NOT
live in Los Angeles, then he
does NOT live in California.
contrapositive is formed
The _______________
by negating the converse (switch
the hypothesis and conclusion
and then negate both).
Same example:
Contrapositive:
If
a man doesn’t live in
California, then he doesn’t live
in Los Angeles.
In these examples, the original
conditional and the contrapositive
are both true.
And the converse and the inverse
are both false.
Equivalent statements
_________
__________ are
statements that are both true or
both false.
Ex.
a) Rewrite in if-then form.
b) Identify the hypothesis and
conclusion.
c) Write the negation, inverse,
and contrapositive.
Teenagers
that are 16 can
learn to drive.
a) If a teenager is 16, then he
can learn to drive.
b) Hypothesis: a teenager is 16
Conclusion: he can learn to drive
c) Negation of hypothesis: A
teenager is not 16.
Negation of conclusion: A teenager
cannot learn to drive.
Inverse: If a teenager is not 16,
then he can't learn to drive.
Contrapositive: If a teenager is
not 16, then he can't learn to
drive.
5-4 Continued
Indirect Proofs
Until now, the proofs you have
written have been direct proofs.
Sometimes it is difficult or even
impossible to find a direct
proof.
In that case, it may be possible to
reason indirectly.
In an indirect proof you begin by
assuming temporarily that the desired
conclusion is NOT true. Then you
reason logically until you reach a
contradiction of the hypothesis or a
known fact (definition, theorem, etc).
Because you’ve reached a
contradiction, you know that the
temporary assumption is impossible
and therefore the desired conclusion
must be true.
Writing an Indirect Proof
• Step 1: State as an assumption the opposite
(negation) of what you want to prove.
• Step 2: Show that this assumption leads to
a contradiction
• Step 3: Conclude that the assumption must
be false and that what you want to prove
must be true.
Step 1: Write the first step of an indirect proof.
a)
b)
c)
d)
Assume temporarily that Micah
doesn’t love video games.
Micah loves video games.
Geometry is fun. Assume temp. that geometry isn’t fun.
Assume temporarily that The Wiggles are
The Wiggles are not cool.
cool.
m  F  90
Assume temporarily that
.
m  F  90
Step 2: Identify the statements
that contradict each other.
I. Jennifer lives in Orange County.
II. Jennifer is a vegetarian.
III. Jennifer loves to eat In & Out Burgers.
• II and III contradict each other because she
cannot be a vegetarian and eat hamburgers at
the same time.
Example: Identify the two statements
that contradict each other.
I.
FG
KL
II. F G  K L
III. F G  K L
Two segments cannot be parallel and perpendicular
at the same time. I and II contradict each other.
Example: Indirect Proof
Read the conditional statement. Think about what is
given and what you are to prove. Then give the steps
of an indirect proof.
• If Jaelene spends more than $50 to
buy two items at a bicycle shop,
then at least one of the items costs
more than $25.
• Given: The cost of two
items is more than $50.
• Prove: At least one items
costs more than $25.
Step 1
• Assume as true the opposite of
what you want to prove –
Assume that neither item costs
more than $25.
Step 2
• This means that each
item costs $25 or less.
This, in turn, means
that the two items
together cost $50 or
less.
• This contradicts the
given information that
the amount spent is
more than $50.
Step 3
• Conclude that the assumption is false.
• One item must cost more than $25
for Jaeleen to spend more than
$50 for two items.