2.3 Definitions

1. 2.3 Definitions Objectives: Use the principles of logic to create a conditional, converse, biconditional, and the conclusion of a given statement, - Identify the properties of adjacent angles. Warm-Up: Picture a ship at anchor. Over the side hangs a ladder with half meter rungs. The tide rises a half meter per hour. At the end of five hours, how much of the ladder will remain above the water assuming nine rungs were above the water when the tide began to rise?

2. Conditional: A statement that can be written in the form “if p then q” where ‘p’ is the hypothesis and ‘q’ is the conclusion. Example: A teenager is a person from 13 to 19 years old. Ifa person is a teenager thenthe person is from 13 to 19 years old.

3. Converse (of a conditional): The statement formed by interchanging the hypothesis and the conclusion of a conditional statement. Example: A teenager is a person from 13 to 19 years old. If a person is from 13 to 19 years old thenthe person is a teenager.

4. Biconditional: A statement using “if and only if” (p if and only if q or p  q) Example: A teenager is a person from 13 to 19 years old. A person is a teenager if and only if the person is from 13 to 19 years old.

5. Definition: A statement is a definition when the conditional and the converse are both true. Example: Determine if the given statement is a definition. A teenager is a person from 13 to 19 years old. Ifa person is a teenager thenthe person is from 13 to 19 years old. If a person is from 13 to 19 years old thenthe person is a teenager. The statement is a definition because both the conditional and the converse are both true,

6. Examples: Use the following steps to determine if a given sentence is a definition. - Write the sentence as a conditional statement - Write the converse of the conditional statement - Write the biconditional statement • Decide whether the statement is a definition, and • explain your reasoning.

7. A teenager is a person who is 13 years old or older. Conditional: Ifa person is a teenager thenthe person is 13 years old or older. Converse: If a person is 13 years old or older thenthe person is a teenager. A person is a teenager if and only if the person is 13 years old or older. Biconditional: Definition: No, the converse is false. (a person who is 20 years old or older is 13 years old or older but is not a teenager).

8. Zero is an integer between -1 and 1. Conditional: Ifan integer is zero thenit is between -1 and 1. Converse: If an integer is between -1 and 1 thenit is zero. An integer is zero if and only if it is between -1 and 1. Biconditional: Definition: Yes, conditional and converse are true.

9. An even number is divisible by two. Conditional: Ifa number is even, thenit is divisible by two. Converse: If a number is divisible by two, thenit is even. Biconditional: A number is even if and only if it is divisible by two. Definition: Yes, conditional and converse are true.

10. An angle is formed by two rays. Conditional: Ifsomething is an angle, thenit is formed by two rays. Converse: If something formed by two rays, thenit is an angle. Biconditional: Something is an angle if and only if it is formed by two rays. Definition: No, the converse is False. (two rays form an angle only if they have a common endpoint)

11. A right angle has a measure of . Conditional: Ifan angle is right, thenit has a measure of . Converse: If an angle has a measure of . thenit is right. Biconditional: An angle is right if and only if is has a measure of Definition: Yes, conditional and converse are true.

12. Adjacent Angles: Two angles in a plane that share a common vertex and a common side but have no interior points in common,

13. Example: Name all of the pairs of adjacent angles in the figure. X Y W V Z

14. Examples: Explain why the indicated angles are NOT adjacent. 2 < 1 & < 3 3 1 1 2 < 1 & < 2 A B 2 1 C < 1 & < 2 < ADB& < ADC

15. Other topics to adress Conditional, converse, inverse(negation of the if then statement), Contrapositive(negation of the converse) Truth tables