1. Biconditionals and Definitions
Skill 10

2. Objective
HSG-CO.9/10/11: Students are responsible for recognize and write biconditional statements and good definitions.

3. Definitions
A biconditional is a single true statement that combines a true conditional with its true converse. We call this an if and only if, statement.

4. Good Definitions
- Uses clearly understood terms or already defined terms
- Uses precise terminology
- Is reversible, meaning can be written as a biconditional

5. Example; write a bicondional
What is the converse of the true conditional? If the converse is also true, rewrite the statements as a biconditional.
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:
The sum of the measures of two angles is 180, iff the angles are supplementary.

6. Example; identifying the conditionals
Biconditional: A ray is an angle bisector, if and only if it divides the angle into two congruent angles.
Conditional:
If a ray is an angle bisector, then it divides the angle into two congruent angles.
Converse:
If a ray divides an angle into two congruent angles, then it is an angle bisector.

7. Example; definitions as biconditionals
Definition: 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 polygon has four sides, then it is a quadrilateral.
Biconditional:
A figure is a quadrilateral, if and only if, it is a polygon with four sides.

8. Example; Writing a biconditional
What is the converse of the following true conditional? If the converse is true, rewrite as a biconditional.
Conditional:
If two angles have equal measure, then the angles are congruent.
Converse:
If two angles are congruent, then they have equal measure.
Biconditional:
Two angles have equal measures, if and only if, the angles are congruent.

9. Example; Identifying conditionals
What are two conditional statements that form the biconditional.
Biconditional:
“Two numbers are reciprocals iff their product is one.”
Conditional 1:
If two numbers are reciprocals, then their product is one.
Conditional 2:
If the product of two numbers is one, then they are reciprocals.

10. Example; Identifying conditionals
What are two conditional statements that form the biconditional.
Biconditional:
“Collinear points are points that lie on the same line.”
Conditional 1:
If points are collinear,
then they lie on the same line.
Conditional 2:
If points lie on the same line, then they are collinear.

11. Example; Writing Definitions
Write the definition as a biconditional.
Definition:
“A straight angle is an angle that measures 180.”
Biconditional
An angle is a straight angle, iff its measure is 180.

12. Example; Writing Definitions
Write the definition as a biconditional.
Definition:
“Vertical angles are two angles whose sides are opposite rays.”
Biconditional
Two angles are vertical angles, iff their sides are opposite rays.

13. #10: Biconditionals & Definitions
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