Biconditionals and Definitions

1. Biconditionals and Definitions Chapter 2 Section 2

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 • Consider each statement and determine if it is true or false • Write the converse, determine if the converse is true or false • 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: • uses clearly defined terms which are either common knowledge or already defined. • is precise. Avoids words such as large, sort of, and almost. • is reversible. You can write a good definition as a true biconditional.

9. Examples: • Write the definitions as conditionals. • Show that they are reversible by writing the converse. • Determine that both are true. • 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