1. 2.3 Biconditionals and Definitions
Geometry
Chapter 2: Reasoning and Proof

2. Biconditional
Single true statement that combines a true conditional and its true converse
Conditional: If the sum of the measures of two angles is 180, then the two angles are supplementary.
Converse: If two angles are supplementary, then the sum of the measures of the two angles is 180.
Biconditional (Put it together): Two angles are supplementary if and only if the sum of the measures of the two angles is 180.

3. Represent!
Conditional (p → q): If the sum of the measures of two angles is 180, then the two angles are supplementary.
Converse (q → p): If two angles are supplementary, then the sum of the measures of the two angles is 180.
Biconditional (p ↔ q): Two angles are supplementary if and only if the sum of the measures of the two angles is 180.

4. Let’s Work Together
Conditional: If two angles have equal measures, then the angles are congruent.
Converse: If two angles are congruent, then the angles have equal measures.
Biconditional: Two angles have equal measures if and only if the angles are congruent.

5. Try It!
Conditional: If two numbers are reciprocals, then their product is 1.
Converse: If the product of two numbers is 1, then the numbers are reciprocals.
Biconditional: Two numbers are reciprocals if and only if their product is 1.

6. Undo It
A ray is an angle bisector if and only if it divides an angle into two congruent angles. p: A ray is an angle bisector q: A ray divides an angle into two congruent angles. p → q: If a ray is an angle bisector, then it divides an angle into two congruent angles. q → p: If a ray divides an angle into two congruent angles, then it is an angle bisector.

7. Your Turn!
An integer is divisible by 100 if and only if its last two digits are zeros. p: An integer is divisible by 100 q: The last two digits of an integer are zeros. p → q: If an integer is divisible by 100, then its last two digits are zeros. q → p: If the last two digits of an integer are zeros, then it is divisible by 100.

8. DEFINITIONS
A statement that can help you identify or classify an object.
Good definition:
Uses clearly understood terms
Is precise (avoid words such as large, sort of, almost, etc.)
Is reversible (true biconditional)

9. A straight angle is an angle that measures 180. Is it reversible?
Write as a biconditional.
An angle is a straight angle if and only if its measure is 180.

10. A quadrilateral is a polygon with four sides. Reversible?
Biconditional:
A figure is a quadrilateral if and only if it is a polygon with four sides.

11. Is it Good? Which of the following is a good definition?
A fish is an animal that swims.
Rectangles have four corners.
Giraffes are animals with very long necks
A penny is a coin worth one cent.

12. Practice Problems
22 problem Worksheet Conditionals Converse Inverse Contrapositive Biconditionals Definitions