Warm Up Write a statement on a piece of paper. Any declarative statement. For instance: I am the former prime minister of Bessarabia. There are exactly seven coins in my left front pocket. Yes, it's that simple. No, it doesn't have to be true.

Warm Up Part 2 1. Find a partner. 2. Figure out which one of you is older. 3. Create a conditional using the older partner's statement as the hypothesis and the younger partner's statement as the conclusion. 4. Find the converse, inverse, and contrapositive. 5. Figure out which of these (conditional, converse, inverse, contrapositive) are true.

1. If I get all of my homework done, then I will go to the game. 2. If Bert goes shopping for groceries, then it’s Wednesday. 3. If a number is even, then 2 is a factor of that number. 4. False |-9+6| does not equal to |-9|+|6| 5. True 6. false

7. Converse: If you like to be at the beach, then you like volleyball. False Inverse: If you don’t like volleyball, then you don’t like the beach. False Contrapositive: If you don’t like to be at the beach, then you don’t like volleyball. False 1. Converse: If x is odd, then x+1 is even. True Inverse: If x+1 is not even, then x is not odd. True Contrapositive: If x is not odd, then x+1 is not even. True 9. Converse: If ∠ P is obtuse, then m ∠ P=109 o. False Inverse: If m ∠ P≠109 o, then ∠ P is not obtuse. False Contrapositive: If ∠ P is not obtuse, then m ∠ P≠109 o. True

Warm Up 1. Write a conditional statement in if-then form. 2. Write the converse of your conditional statement. If you are a junior, then you wait on tables. Or If you are a senior, then you work. If you wait on tables, then you are a junior. Or If you work then you are a senior. Juniors and seniors at Woodland Falls Camp are expected to work. Juniors wait on tables.

Students will analyze and rewrite conditional and biconditional statements.

Examples of Conditional: It is Saturday, only if I am working at the restaurant. If-then form: If it is Saturday, then I am working at the restaurant.

Biconditional Statement: a statement that contains the phrase “if and only if.” A biconditional statement is true when a conditional and its converse are true.

For example: Ashley began a new summer job, earning $10 an hour. If she works over 40 hours a week, she earns time and a half, or $15 an hour. If she earns $15 an hour, she has worked over 40 hours a week. Conditional: If Ashley earns $15 an hour, then she has worked over 40 hours a week. Converse: If Ashley worked over 40 hours a week, then she earns $15 an hour. Biconditional: Ashley earns $15 an hour if and only if (iff) she works over 40 hours a week.

How do I form a biconditional statement? Conditional: If two angle measures are complements, then their sum is 90. T or F? Converse: If the sum of two angle measures is 90, then they are complements. T or F? Biconditional: Two angle measures are complements iff their sum is 90. T or F? A biconditional statement can be either true or false. To be true, both the conditional statement and its converse must be true.

Must be true “forwards” and “backwards” Analyze the following statement to determine if it is a true biconditional statement: x > 9 iff x >0 a. Is this a biconditional statement? Yes b.Is the statement true? Conditional: If x > 9, then x > 0 Converse: If x > 0, then x > 9 The conditional statement is true, but the converse is not. Let x = 2. Then 2 > 0 but 2 ≯ 9. So, the biconditional is false. true false

The following statement is true. Write the converse and decide whether it is true or false. If the converse is true, combine it with the original to form a biconditional. If x 2 =4, then x=2 or -2 Converse: If x=2 or -2, then x 2 =4; true Biconditional: x 2 =4 if and only if (iff) x=2 or -2

Write each biconditional as a conditional and its converse. Then determine whether the biconditional is true or false. If false, give a counterexample. 1. A calculator will run iff it has batteries. 2. Two lines intersect iff they are not vertical. 3. Two angles are congruent iff they have the same measure. 4. 3x-4=20 iff x=7 5. A line is a segment bisector iff it intersects the segment at its midpoint.

1. A calculator will run iff it has batteries. Conditional: If a calculator runs, then it has batteries. Converse: If a calculator has batteries, then it will run. False; a calculator may be solar powered. 2. Two lines intersect iff they are not vertical. Conditional: If two lines intersect, then they are not vertical. Converse: If two lines are not vertical, then they intersect. False; two parallel horizontal lines will not intersect. 3. Two angles are congruent iff they have the same measure. Conditional: If two angles are congruent, then they have the same measure. Converse: If two angles have the same measure, then they are congruent. true

4. 3x-4=20 iff x=7 Conditional: If 3x-4=20, then x=7 Converse: If x = 7 then 3x-4=20. False; 3x-4=17 when x=7. 1. A line is a segment bisector iff it intersects the segment at its midpoint. Conditional: If a line is a segment bisector, then it intersects the segment at its midpoint. Converse: if the line intersects the segment at its midpoint, then it is a segment bisector. True

Definitions and Conditionals In groups, you will each translate one definition into if- then conditional form. You will need to figure out what is the hypothesis and what is the conclusion. Then try to figure out if the converse of the conditional is true.

Your Turn “A watched pot never boils.” 1. Write out the conditional in “if-then” form. 2. Find the converse. 3. Write out the biconditional in “if and only if” form. 4. Is the biconditional true? Why or why not?