1. Chapter 2 Section 2 Biconditionals and Definitions

2. Biconditional When both the conditional and its converse are true, you can combine them into a biconditional. Biconditional: the statement that you get by connecting the conditional and its converse using the word and You can shorten a biconditional further by joining the two parts of the conditional (the hypothesis and conclusion) with the phrase if and only if

3. Examples 1. Consider each statement and determine if it is true or false 2. Write the converse, determine if the converse is true or false 3. Combine the statements into a biconditional Conditional: If two angles have the same measure, then the angles are congruent. True or false? True Converse: If __________________________________, then ________________________________________. True or false? True Biconditional: If two angles have the same measure, then the angles are congruent AND if two angles are congruent, then they have the same measure. Two angles have the same measure if and only if the angles are congruent. two angles are congruent they have the same measure

4. Examples: If three points are collinear, then they lie on the same line. True or false? True Converse: If ___________________________________, then ________________________________________. True or false? True Biconditional: If three points are collinear, then they lie on the same line and if three points lie on the same line, then they are collinear. Three points are collinear if and only if they lie on the same line. three points lie on the same line they are collinear

5. Examples: If x=5, then x+15=20. True or false? True Converse: If ___________________________________, then ________________________________________. True or false? True Biconditional: If x=5, then x+15=20 and if x+15=20, then x=5. x=5 if and only if x+15=20. x+15=20 x=5

6. Writing a biconditional as two conditionals that are inverses of each other Example: Biconditional: A number is divisible by 3 if and only if the sum of its digits is divisible by 3. Conditional: If _______________________________, then _______________________________________. Converse: If _________________________________, then _______________________________________. a number is divisible by 3 the sum of its digits is divisible by 3 the sum of a numbers digits is divisible by 3 The number is divisible by 3

7. Writing a biconditional as two conditionals that are inverses of each other Example: Biconditional: A number is prime if and only if it has two distinct factors, 1 and itself. Conditional: If _______________________________, then _______________________________________. Converse: If _________________________________, then _______________________________________. a number is prime it has two distinct factors, 1 and itself a number has two distinct factors, 1 and itself it is prime

8. Recognizing Good Definitions Geometry starts with undefined terms such as point, line, and plane, whose meaning you understand intuitively. You then use those terms to define other terms, such as collinear points. A good definition has several important components: 1. uses clearly defined terms which are either common knowledge or already defined. 2. is precise. Avoids words such as large, sort of, and almost. 3. is reversible. You can write a good definition as a true biconditional.

9. Examples: 1. Write the definitions as conditionals. 2. Show that they are reversible by writing the converse. 3. Determine that both are true. 4. Write as a biconditional. Definition: Perpendicular lines are lines that meet to form right angles. Conditional: If ____________________________________, then ____________________________________________. Converse: If ____________________________________, then ____________________________________________. Biconditional: _________________if and only if_______________________________________________ _____________________ lines are perpendicular they meet to form right angles lines meet to form right angles they are perpendicular Lines are perpendicular they meet to form right angles

10. Example… Definition: A right angle is an angle whose measure is 90°. Conditional: If ____________________________________, then ____________________________________________. Converse: If ____________________________________, then ____________________________________________. Biconditional: _________________if and only if_______________________________________________ _____________________ an angle is a right angle its measure is 90° the measure of an angle is 90° it is a right angle An angle is a right angle its measure is 90°.

11. Good definitions??? One way to show that a statement is not a good definition is to find a counterexample. Examples: An airplane is a vehicle that flies. A triangle has sharp corners. A square is a figure with four right angles.

12. Practice!! p. 90, Selected exercises #1-23 WS 2-2