Chapter Two: Reasoning and Proof

1. Chapter Two:Reasoning and Proof Section 2-2: Biconditionals and Definitions

2. Objectives • To write biconditionals. • To recognize good definitions.

3. Biconditional • When a conditional and its converse are true, you can combine them as a true biconditional. • Biconditionals include the phrase, “if and only if”

4. Writing a Biconditional • Conditional: If x = 6, then 3x = 18. • This is true. • Converse: If 3x = 18, then x = 6. • This is also true. • Because the conditional and the converse are both true, we can form a biconditional: • x = 6 if and only if 3x = 18.

5. Writing a Biconditional (continued) • Conditional: If x = 3, then x2 = 9. • This is true. • Converse: If x2 = 9, then x = 3. • This is not true. • x = 3 or x = -3 • No biconditional is possible because the converse is not true.

6. Separating a Biconditional • Biconditional: Two lines are perpendicular if and only if they form a right angle. • To separate: • Remove the “if and only if” • Write the conditional and its converse: • Conditional: If two lines are perpendicular, then they form a right angle. • Converse: If two lines form a right angle, then they are perpendicular.

7. Recognizing Good Definitions • Definitions are statements that can help you classify an object. • Components of good definitions: • Use clearly understood terms that have already been defined. • Be precise– avoid words like sometimes, a lot, and almost. • Good definitions can be written as biconditionals.

8. Writing a Definition as a Biconditional • Definition: A right angle is an angle whose measure is 90 degrees. • In biconditional form: • An angle is a right angle if and only if its measure is 90 degrees.

9. Determining if a Definition is Good • A square has 4 right angles. • Not a good definition because it is not reversible. • A rectangle also has four right angles.