Sec 2 2 Biconditionals Definitions Objectives a To

Sec. 2 -2 Biconditionals & Definitions Objectives: a) To write biconditionals b) To recognize good definitions

I. Biconditionals Connect the conditional (if – then) and its converse Only when both are true Use “if and only if” (iff)

II. Writing iffs 1. Look at conditional and decide if true. 2. Write the converse and decide if true. 3. Combine statements with “if and only if”

Example 1: Writing an iff Conditional: If 2 angles have the same measurement, then the angles are congruent. Converse: If two angles are congruent, then they have the same measurement. Iff: Two angles have the same measurement iff the angles are congruent.

Example 3 A: Analyzing the Truth Value of a Biconditional Statement Determine if the biconditional is true. If false, give a counterexample. A rectangle has side lengths of 12 cm and 25 cm if and only if its area is 300 cm 2.

Example 3 A: Analyzing the Truth Value of a Biconditional Statement Conditional: If a rectangle has side The conditional lengths of 12 cm and 25 cm, then its is true. area is 300 cm 2. Converse: If a rectangle’s area is 300 cm 2, then it has side lengths of 12 cm and 25 cm. The converse is false. If a rectangle’s area is 300 cm 2, it could have side lengths of 10 cm and 30 cm. Because the converse is false, the biconditional is false.

p q means p q and q p Writing Math The biconditional “p if and only if q” can also be written as “p iff q” or p q.

III. Good Definitions In geometry we started with undefined terms (point, line, & plane) whose meaning we took w/o proof. Next we used, these undefined terms to define our next set of terms. Such as collinear and midpoint. All good definitions must be biconditionals.

Helpful Hint Think of definitions as being reversible. Postulates, however are not necessarily true when reversed.

Example 2: Writing a good definition Statement: A midpoint divides a segment into two congruent parts. Conditional: If a point is a midpoint, then it divides a segment into two congruent parts. (T or F) Converse: If a point divides a segment into two congruent parts, then it is a midpoint. (T or F) Iff: A point is a midpoint, iff it divides a segment into two congruent parts.

Example 3: Writing an iff Statement: A right angle has a measure of 90°. Conditional: If an angle is a right angle, then it has a measure of 90°. (T or F) Converse: If an angle has a measure of 90°, then it is a right angle. (T or F) Iff: An angle is a right angle, iff it has a measure of 90°.

What have I learned Today? What is another word for biconditionals? Iff (if and only if) How do you form iffs? Combine both the conditional and the converse. Do they both have to be true? Heck yeah!