1. 2-3 Conditional Statements
Conditional Statement: A statement that can be written in if-then form.
If-Then Statement: Written in the form if p, then q. (p q)
Hypothesis: The phrase immediately following the word if.
Conclusion: The phrase immediately following the word then.

2. Example #1 Identify the hypothesis and conclusion.
A) If a polygon has six sides, then it is a hexagon.
B) Another performance will be scheduled if the first one is sold out.
Some conditional statements are written without the if and then.
Example #2: Identify the hypothesis and conclusion. Then write each statement in if-then form.
A) An angle formed by perpendicular lines is a right angle.
If an angle is formed by perpendicular lines, then it is a right angle.
B) A cheetah has nonretractile claws.
If a cat is a cheetah then it has nonretractile claws.

3. Truth Tables using if-thenp q
pq T F

4. Example #3: Determine the truth value of the following statement for the set of conditions.
If you get 100% on your test, then your teacher will give you an A.
A) You get 98%; your teacher gives you an A.
Hypothesis is false, the conclusion is true…therefore the statement is true.
B) You get 85%; your teacher gives you a B.
Hypothesis is false, but we can not say that the entire statement is false…therefore the statement is true.
C) You get 100%, your teacher gives you a B.
Hypothesis is true, but the conclusion is false. Therefore the statement is false.

5. Vocabulary
Related Conditionals: Other statements based on a given conditional statement.
Conditional: Given hypothesis and conclusion. p  q.
Converse: Exchanging the hypothesis and conclusion of the conditional. q p
Inverse: Negating both hypothesis and conclusion of the conditional. ~p~q
Contrapositive: Negating both the hypothesis and conclusion of the converse statement. ~q ~p
Logically Equivalent: Statements with the same truth values.

6. Truth Table for all related conditionalsp q
Conditional pq
Converse qp
Inverse ~p~q
Contrapositive ~q~p T F

7. Example #4
Write the converse, inverse, and contrapositive of the following statement. Determine whether the statement is true or false. If the statement is false, give a counterexample.
Vertical angles are congruent.
Conditional: If a pair of angles are vertical, then they are congruent. True
Converse: If a pair of angles are congruent, then they are vertical. False – counterexample: two right angles adjacent to one another are congruent but are not vertical.
Inverse: If a pair of angles are not vertical, then they are not congruent. False – counterexample: linear pair formed by two right angles are congruent but not vertical.
Contrapositive: If a pair of angles are not congruent, then they are not vertical. True