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 ConditionalIF ________________

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 q is parallel is even =

not p

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.

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.

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 _________.

Conditional Contrapositive Converse Inverse

To summarize all possible relationships between conditionals:

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 mA 90, then A is not a right angle.

Conv______________________________________________________

Inv_______________________________________________________

Contra____________________________________________________

If today is February 29, then tomorrow is March 1

Conv______________________________________________________

Inv_______________________________________________________

Contra____________________________________________________

WY XZ

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, All poets are philosophers.then its diagonals are perpendicular

a. a. Jose is a poet

b. WXYZ is a square b. Jane is a philosopher

c. mXWY + mWXZ = 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 inversethe contrapositive

GIVEN STATEMENT: If p, then q.

CONTRAPOSITIVE: If not q, then not p.

CONVERSE: If q, then p.

INVERSE: If not p, then not q.Inverse negates given statement.

Contrapositive negates converse.

GIVEN STATEMENT: If p, then q.If today is Tuesday, then tomorrow is Wednesday.

CONTRAPOSITIVE: If not q, then not p.If tomorrow is not Wednesday, then today is not Tuesday.

CONVERSE: If q, then p.If tomorrow is Wednesday, then today is Tuesday.

INVERSE: If not p, then not q.If today is not Tuesday, then tomorrow is not Wednesday.

P

Q

Statement: If p, then q.

Contrapositive: If not q, then not p.

A statement and its contrapositive arelogically equivalent (either both true or both false.)

Q

P

Converse: If q, then p.

Inverse: If not p, then not q.

The converse and the inverse are alsologically equivalent (either both true or both false.)

P

Q

Converse: If q, then p.

Inverse: If not p, then not q.

A statement is NOTlogically equivalent to its converse or to its inverse.

Statement: If p, then q.

Not true!

Not true!

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.

Runners

AthletesAll runners are athletes.

Statement:

Steven is a runner.

Sally is not an athlete.

Sally

So, Steven is an athlete.

So, Sally is not a runner.

Susan is an athlete.

Stan is not a runner.No conclusion follows.

No conclusion follows.

Susan

??

Stan

?

?

Steven

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

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

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)

Turn to page 210

Let’s talk through CE #1 together

Turn to page 211

Let’s talk through WE #6 together