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Biconditionals and Definitions Chapter 2 Section 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 Examples 1. 2. 3. 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? two angles are congruent Converse: If __________________________________, then ________________________________________. they have the same measure True 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. Examples: If three points are collinear, then they lie on the same line. True or false? True three points lie on the same line Converse: If ___________________________________, they are collinear 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. Examples: If x=5, then x+15=20. True or false? True x+15=20 Converse: If ___________________________________, x=5 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. 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. a number is divisible by 3 Conditional: If _______________________________, the sum of its digits is divisible by 3 then _______________________________________. the sum of a numbers digits is divisible by 3 Converse: If _________________________________, The number is divisible by 3 then _______________________________________. 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. a number is prime Conditional: If _______________________________, it has two distinct factors, 1 and itself then _______________________________________. a number has two distinct factors, 1 and itself Converse: If _________________________________, it is prime then _______________________________________. 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. 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. 1. Examples: Write the definitions as conditionals. Show that they are reversible by writing the converse. Determine that both are true. Write as a biconditional. 1. 2. 3. 4. Definition: Perpendicular lines are lines that meet to form right angles. lines are perpendicular Conditional: If ____________________________________, they meet to form right angles then ____________________________________________. lines meet to form right angles Converse: If ____________________________________, they are perpendicular then ____________________________________________. Lines are perpendicular Biconditional: _________________if and only they meet to form right angles if_______________________________________________ _____________________ Example… Definition: A right angle is an angle whose measure is 90°. an angle is a right angle Conditional: If ____________________________________, its measure is 90° then ____________________________________________. the measure of an angle is 90° Converse: If ____________________________________, it is a right angle then ____________________________________________. An angle is a right angle Biconditional: _________________if and only its measure is 90°. if_______________________________________________ _____________________ 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. Practice!! p. 90, Selected exercises #1-23 WS 2-2