1. 5-4 Inverses, Contrapositives, and Indirect Reasoning
Objectives
Write the negation of a statement
Write the inverse and contrapositive of a statement
Use indirect reasoning

2. p. 280 Ex 1: Writing the Negation of a Statement
The negation of a statement has the opposite truth value
Statement: ∠ ABC is obtuse.
Negation: ∠ ABC is not obtuse.
Write the negation of the following statement:
Today is not Tuesday.
Today is Tuesday.

3. p. 281 Ex 2: Writing the Inverse and Contrapositive
Conditional: If a figure is a square, then it is a rectangle.
Inverse: Negates both the hypothesis and conclusion of a conditional
If a figure is not a square, then it is not a rectangle.
Contrapositive: Switches the hypothesis and conclusion of a conditional and negates both
If a figure is not a rectangle, then it is not a square.

4. Summary Statement Symbol In words Equivalent to Conditional p  q
(always have same truth value)
Conditional
p  q
If p, then q.
Contrapositive
Converse
q  p
If q, then p.
Inverse
Negation (of p) ~p
Not p.
~p  ~q
If not p, then not q.
Contra-positive
~q  ~p
If not q, then not p.

5. Using Indirect Reasoning
In indirect reasoning, all possibilities are considered and then all but one are proved false. The remaining possibility must be true.
A proof involving indirect reasoning is an indirect proof.

6. Writing an Indirect Proof
State as an assumption the opposite of what you want to prove.
Show that this assumption leads to a contradiction.
Conclude that the assumption must be false and that what you want to prove must be true.

7. p. 282 Ex 3a The First Step of an Indirect Proof
Prove that quadrilateral QRWX does not have four acute angles.
First Step: State as an assumption the opposite of what you want to prove.
Assume that quadrilateral QRWX has four acute angles.

8. p. 282 Ex. 5 Writing an Indirect Proof
Given fact: The total cost of two items is more than $50.
Prove: At least one of the items costs more than $25.
Proof:
Step 1. Assume the opposite of what you want to prove.
Assume that each item costs $25 or less.
Step 2. Show that this assumption leads to a contradiction.
So the total cost of the two items is $50 or less.
This contradicts the given fact that the total cost of two items is more than $50.
Step 3. Conclude that the assumption must be false and
that what you want to prove is true.
Therefore, the assumption that each item costs $25 or less is false.
Therefore, at least one item costs $25 or more.