1. Lesson 2-3 BICONDITIONAL STATEMENTS AND DEFINITIONS

2. Conditional Statments  Written in “if p, then q” form  Example: “If a figure has three sides, then it is a triangle.”  What is the converse of this statement?  Converse: “If a figure is a triangle, then it has three sides.”  Is the original statement true? Is the converse true?

3. Biconditional Statements  If both the original statement AND it’s converse are true, then it is a biconditional statement.  You can write a biconditional statement by joining the hypothesis and conclusion with the phrase “if and only if.”  We can now write the statements in the previous slide as one biconditional statement.  “A figure has three sides if and only if it is a triangle.”  Can we switch p and q around?  Yes! “A figure is a triangle if and only if it has three sides.”

4. Biconditional Statements  What is the converse of the following true conditional? If the converse is also true, rewrite the statements as a biconditional.  “If the sum of the measures of two angles is 180º, then the two angles are supplementary.”  What is the converse?  “If two angles are supplementary, then the sum of their measures is 180º”  Is the converse also true?  Yes!  Write the two statements together as a biconditional statement  “Two angles are supplementary if and only if the sum of their measures is 180º”

5. Key Concept: Biconditional Statements  A biconditional combines p → q and q → p as p ↔ q  ↔ is read as “if and only if.”

6. Identifying the Conditionals in a Biconditional  What are the two conditional statements that form this biconditional?  “A ray is an angle bisector if and only if it divides an angle into two congruent angles.”  1. “If a ray is an angle bisector, then it divides an angle into two congruent angles.”  2. “If a ray divides and angle into two congruent angles, then it is an angle bisector.”

7. What makes a good definition?  A good definition is a biconditional statement that can help you identify or classify an object. A good definition has several important components:  A good definition uses clearly understood terms. These terms should be commonly understood or already defined.  A good definition is precise. Good definitions avoid words such as large, sort of, almost, etc.”  A good definition is reversible. That means you can write a good definition as a true biconditional.

8. Identifying Good Definitions WWhich of the following is a good definition? a) A fish is an animal that swims b) Rectangles have four corners c) Giraffes are animals with very long necks d) A penny is a coin worth one cent