Bi-conditionals and Definitions Chapter 2: Reasoning and Proof1 Objectives 1 To write bi-conditionals 2 To recognize good definitions

Biconditionals and Definitions Chapter 2: Reasoning and Proof2 Key Concepts If a conditional and its converse are both true, the statement is said to be true. Biconditional statements are often stated in the form “…if and only if …” IFF – short for if and only if  - symbol for if and only if An angle is a right angle if and only if it measures 90°.

Biconditionals and Definitions Chapter 2: Reasoning and Proof3 Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional.

Biconditionals and Definitions Chapter 2: Reasoning and Proof4 Write the two statements that form this biconditional. Conditional: Converse: Biconditional: Lines are skew if and only if they are noncoplanar.

Biconditionals and Definitions Chapter 2: Reasoning and Proof5 Key Concepts A good definition is reversible. That means that you can write a good definition as a true biconditional. The Reversibility Test The reverse (converse) of a definition must be true. If the reverse of a statement is false, then the statement is not a good definition.

Biconditionals and Definitions Chapter 2: Reasoning and Proof6 Show that this definition of triangle is reversible. Then write it as a true biconditional. Definition: A triangle is a polygon with exactly three sides.

Biconditionals and Definitions Chapter 2: Reasoning and Proof7 Is the following statement a good definition? Explain. An apple is a fruit that contains seeds.

8 Biconditional When a conditional statement and its converse are both true, the two statements may be combined into a true biconditional statement. Use the phrase if and only if Statement: If an angle is right angle, then it has a measure of 90 . Converse: If an angle has a measure of 90 , then it is a right angle. Biconditional: An angle is right angle if and only if it has a measure of 90 .

When a conditional statement AND the converse are BOTH TRUE, this creates a special case called ‘biconditional”. Conditional: If a quadrilateral has 4 right angles, then it is a rectangle. a  b (true) Converse: If it is a rectangle, then it is a quadrilateral with 4 right angles. b  a (true) Biconditional: A quadrilateral has 4 right angles if and only if it is a rectangle. (don’t use if and then) a  b (true BOTH ways) iff means “if and only if” A biconditional is a statement that is true backwards and forwards. A biconditional is a DEFINITION. Biconditional Statement

Biconditional - Example 10 Symbology: “p if and only if q”

Biconditional - Example 11 Biconditional: A ray is an angle bisector if and only if it divides an angle into two congruent angles. p: A ray is an angle bisector. q: A ray divides an angle into two congruent angles.

Definitions A good definition has the following components: - Uses clearly understood terms. - Is precise (avoids words such as large, sort of, some, etc.). - Is reversible. You can write it as a true biconditional. 12

Definitions Lesson 2-3: Biconditionals and Definitions 13

Definitions 14