1. CHAPTER 2: DEDUCTIVE REASONING
Section 2-1: If-Then Statements; Converses

2. IF-THEN STATEMENT
If-then statements, which are also called conditional statements or conditionals, are statements that include a hypothesis followed by a conclusion.

3. EXAMPLES OF IF-THEN
If it rains after school, then I will give you a ride home.
If B is between A and C, then AB + BC = AC.
If I don’t pay attention in this class, then I will fail.
I will become a 49ers fan if the Raiders move back to L.A.

4. REPRESENTING AN IF-THEN
To represent if-then statements, we let p represent the hypothesis and let q represent the conclusion.
Using p = hypothesis and q = conclusion, the base form of an if-then statement can be shown by:
If p, then q.
p: hypothesis q: conclusion

5. CONVERSE Converse: Reversing something as in position or order.
The converse of a conditional is formed by interchanging the hypothesis and the conclusion.

6. CONVERSE EXAMPLE Statement: If p, then q. Converse: If q, then p.
In short, flip the order of the hypothesis and conclusion to find the converse of a statement.

7. PRACTICE If points lie in 1 plane, then they are coplanar.
State the converse of each conditional and tell whether each converse is true or false:
If points lie in 1 plane, then they are coplanar.
If points are collinear, then they all lie in one line.
If x = -2, then 4x = -8.

8. STATEMENT VS. CONVERSE
Statements and their converses do not say the same thing.
Some true statements have converses that are not true.
Example
Statement: If I live in Castro Valley, then I live in the Bay Area.
Converse: If I live in the Bay Area, then I live in Castro Valley. (False)

9. COUNTEREXAMPLE
An if-then statement is false if an example can be found for which the hypothesis is true and the conclusion is false.
Such an example is called a counterexample.
Take the converse from the previous slide for example. I could live in the Bay Area but live in another city such as San Leandro.

10. PRACTICE If ab < 0, then a < 0.
Provide a counterexample to show that each statement is false.
If ab < 0, then a < 0.
If a four-sided figure has four right angles, then it has four congruent sides.
If a line lies in a vertical plane, then the line is vertical.

11. IF-THEN/CONDITIONAL STATEMENTS
Conditional Statements or If-Then statements are not always written with the words if and then.
General Form Example
If p, then q If 4x = 12, then x = 3
p implies q x = 12 implies x = 3
p only if q x = 12 only if x = 3
q if p x = 3 if 4x = 12

12. BICONDITIONAL
If a conditional statement and its converse are both true, they can be combined into a single statement using the words “if and only if”.
Statements that contain the words “if and only if” are called biconditionals.

13. BICONDITIONAL The basic form of a biconditional can be found by:
p if and only if q.

14. B is between A and C if and only if
PRACTICE
Write the pair of conditionals as a biconditional:
If B is between A and C, then AB + BC = AC.
If AB + BC = AC, then B is between A and C.
B is between A and C if and only if
AB + BC = AC

15. CLASSWORK/HOMEWORK Classwork Pg. 34, Classroom Exercises 2-16 even
Pg. 35, Written Exercises 2-28 even