Geometry
6.2
Inverses and
Contrapositives
There are two other types of conditionals called the inverse and the contrapositive.
Recall: Conditional Statement
IF ________________
Converse of Conditional
IF ________________
Contrapositive
IF _________________________
Inverse
IF _________________________P (if)Q (then)
Adding “not” to a statement is called ______________ the statement.
Here are some other examples with their negation.
p
not p
not q
is parallel
is even

=
Venn diagrams can represent conditionals and can be very helpful in
understanding conditional statements.
If an item is in circle p it is also in circle q. if p then q (conditional)
Also, if an item is not in circle q then it is not in circle p. if not q then not p (contrapositive)
Thus, the conditional and the contrapositive are said to be ___________________ statements.
If the conditional is true, then the contrapositive is ________.
Since they are the same, you can prove a conditional by proving its contrapositive!
Sometimes this is actually easier than proving the conditional itself.
The Venn diagram on the right represents the converse and the inverse.
If an item is in circle q then it is in circle p. if q then p (converse)
If an item is not in circle p then it is not in circle q. if not p then not q (inverse)
For this reason, the converse and inverse are ___________________ statements.
If the converse is true, then the inverse is _________.
To summarize all possible relationships between conditionals:
Conditional
Contrapositive
Converse
Inverse
Sometimes we have to re-word a statement from standard English to fit If-Then form.
All squares are rhombuses  If ____________________________________
All marathoners have stamina  If ______________________________________
Sample problems:
Write the contrapositive, converse and inverse of each conditional.
If 2x + 1 = 7, then x = 3.
Contrapositive:________________________________________________________
Converse:____________________________________________________________
Inverse:_____________________________________________________________
If a polygon has five sides, then it is a pentagon
Contrapositive:________________________________________________________
Converse:____________________________________________________________
Inverse:_____________________________________________________________
Classify each conditional as true or false. Give its converse, inverse and contrapositive.
Then state if each is T/F?
If a triangle is isosceles, then it is equilateral.
Conv______________________________________________________
Inv_______________________________________________________
Contra____________________________________________________
If mA  90, then A is not a right angle.
Conv______________________________________________________
Inv_______________________________________________________
Contra____________________________________________________
If today is February 29, then tomorrow is March 1
Conv______________________________________________________
Inv_______________________________________________________
Contra____________________________________________________
Assume the given statement is true.What can you conclude by using the given statement
together which each additional statement? If no conclusion is possible, say so.
\If WXYZ is a rhombus,
then its diagonals are perpendicular
All poets are philosophers.
a. W Y  X Z
a. Jose is a poet
b. WXYZ is a square
b. Jane is a philosopher
c. mXWY + mWXZ = 100
c. Jung is not a poet.
d. WXYZ is not a rhombus
d. Jean is not a philosopher.
Homework
Pg. 211 WE #5-10, (11-19 odd)
Reviewing Conditional
Statements
Your Dad says, “If you get a B average, then you
can get your driver’s license.”
This is an example of an if-then statement,
which is also called a conditional.
You have already learned about the converse
of a conditional. It is formed by interchanging
the hypothesis and the conclusion.
Converse
The converse of a conditional is formed by
switching the hypothesis and the conclusion:
Statement:
If p, then q.
Converse:
If q, then p.
hypothesis
conclusion
Today you will learn about other
related conditionals…….
the inverse
the contrapositive
GIVEN STATEMENT: If p, then q.
INVERSE:
If not p, then not q.
Inverse negates given statement.
CONVERSE:
If q, then p.
CONTRAPOSITIVE:
If not q, then not p.
Contrapositive negates converse.
GIVEN STATEMENT:
If p, then q.
If today is Tuesday, then tomorrow is Wednesday.
INVERSE:
If not p, then not q.
If today is not Tuesday, then tomorrow is not Wednesday.
CONVERSE:
If q, then p.
If tomorrow is Wednesday, then today is Tuesday.
CONTRAPOSITIVE:
If not q, then not p.
If tomorrow is not Wednesday, then today is not Tuesday.
A statement and its
contrapositive are
logically equivalent
(either both true or
both false.)
Statement:
Q
P
If p, then q.
Contrapositive: If not q, then not p.
The converse and
the inverse are also
logically equivalent
(either both true or
both false.)
P
Q
Converse:
If q, then p.
Inverse:
If not p, then not q.
A statement is NOT
Q
logically equivalent
to its converse or to
its inverse.
Statement:
P
If p, then q. True
Converse:
If q, then p. Not true!
Inverse:
If not p, then not q. Not true!
Example
Suppose this conditional is true:
All runners are athletes.
(If a person is a runner, then that person is
an athlete.)
What can you conclude about each
additional statement?
1. Steven is a runner.
2. Sally is not an athlete.
3. Susan is an athlete.
4. Stan is not a runner.
Venn diagrams can be useful to illustrate.
Sally
Statement:
All runners are athletes.
Athletes
?
Steven is a runner.
So, Steven is an athlete.
Runners
Steven
Sally is not an athlete.
?
So, Sally is not a runner.
Susan is an athlete.
No conclusion follows.
Stan is not a runner.
No conclusion follows.
?
Susan
Stan
?
Classify the statement as true or false. Then give
the following, and classify each as true or false:
(a) converse
(switch p and q )
(b) inverse
(negate the statement)
(c) contrapositive (negate the converse)
If two lines are parallel, then they do not
intersect. True
(a) converse: If two lines do not intersect, then
they are parallel. False
(b) inverse: If two lines are not parallel, then they
intersect. False
(c) contrapositive: If two lines intersect, then
they are not parallel. True
Classify the statement as true or false. Then give
the following, and classify each as true or false:
(a) converse
(switch p and q )
(b) inverse
(negate the statement)
(c) contrapositive (negate the converse)
If two angles are acute, then they are
complementary. False
(a) converse: If two angles are complementary,
then they are acute. True
(b) inverse: If two angles are not acute, then they
are not complementary. True
(c) contrapositive: If two angles are not
complementary, then they are not acute. False
Turn to page 210
Let’s talk through CE #1 together
Turn to page 211
Let’s talk through WE #6
together