1. 2.3 Biconditionals and Definitions
SOL: G1a & G1b
Objectives: TSW …
To write biconditionals and recognize good
definitions.

2. Biconditional
A single true statement that combines a true conditional and its true converse.
Write it by joining the two parts of each conditional with the phrase if and only if (iff)
(p ↔ q) is read p if and only if q
Example:
p → q: If 2 lines intersect, then they are not parallel.
q → p: If 2 lines are not parallel, then they intersect.
****Both are true … so …
p ↔ q: Two lines intersect if and only if the two lines are not parallel.

3. Example 1: Writing a Biconditional
If the sum of the measures of two angles is 180°, then the two angles are supplementary.
What is the converse of the given conditional?
If two angles are supplementary, then the sum of the measures of two angles is 180°.
True
Since the conditional and the converse are both true, we can write a biconditional.
Two angles are supplementary if and only if the sum of the measures of two angles is 180°.

4. Go it? 1: What is the converse of the following true conditional?
If two angles have equal measure, then the angles are congruent.
What is the converse of the given conditional?
If two angles are congruent, then two angles have equal measure.
True
Since the conditional and the converse are both true, we can write a biconditional.
Two angles are congruent if and only if the two angles have equal measure.

5. Example 2: Identifying the Conditionals in a Biconditional
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 a ray divides an angle into two congruent angles.
q → p: If a ray divides an angle into two congruent angles, then a ray is an angle bisector.

6. Got it? 2: Identifying the Conditionals in a Biconditional
Two numbers are reciprocals if and only if their product is 1.
p: Two numbers are reciprocals.
q: Two numbers whose product is 1.
p → q: If two numbers are reciprocals, then their product is 1.
q → p: If the product of two numbers is 1, then the numbers are reciprocals.

7. Definitions
A definition is good if it can be written as a biconditional
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, and almost.
A good definition is reversible. That means you can write a good definition as a true biconditional.

8. Example 3: Writing a Definition as a Biconditional
A quadrilateral is a polygon with four sides.
Conditional: If a figure is a quadrilateral, then it is a polygon with four sides.
Converse: If a figure is a polygon with four sides, then it is a quadrilateral.
Biconditional: A figure is a quadrilateral if and only if it is a polygon with four sides.

9. Got It? 3: Writing a Definition as a Biconditional
A straight angle is an angle that measures 180°.
Conditional: If an angle is a straight angle, then the angle measures 180°.
Converse: If an angle measures 180°, then it is a straight angle.
Biconditional: An angle is a straight angle if and only if the angle measures 180°.
An angle measures 180° if and only if the angle is a straight angle.

10. Example 4: Identifying Good Definitions
a.) A fish is an animal that swims.
Not reversible. A whale swims, but is not a fish.
b.) Rectangles have four corners.
Corners not clearly defined. All quadrilaterals have 4 corners.
c.) Giraffes are animals with very long necks.
Very long is not precise. The statement is not reversible.
d.) A penny is a coin worth one cent.
Reversible, all terms clearly defined and precise. Good definition

11. Got it? 4: Identifying Good Definitions
a.) Is the following statement a good definition? Explain.
A square is a figure with four right angles.
If a figure has four right angles, then it is a square.
Not true, so this is not a good definition.
b.) How can you rewrite the statement “Obtuse angles have greater measures than acute angles” so that it is a good definition?
Obtuse angles have a measure greater than 90°.

12. Classwork
Pg 101 – 103 # 7, 9, 13, 17, 19, 23, 25, 26, 28, 29, 30, 33, 43