1. Recall: Hypothesis- An if statement (ex. If I go to Masterman) Conclusion: a then statement (ex. Then I live in Philly) Conditional: A hypothesis and a conclusion together (ex. If I go to Masterman I live in Philly) Converse: Reversing the conditional (ex. If I live in Philly I got to Masterman) *Converses are not always true

2. What is a Biconditional? If a conditional and its converse are both true then a true biconditional can be made using the true statements. Because both are true the biconditional is reversible The two statements (conditional and converse) are traditionally connected using the words if and only if. Ex. Conditional - If an animal is a bird, then it has feathers (true) Converse – If an animal has feathers, then it is a bird (true) Biconditional – An animal is a bird if and only if it has feathers

3. Separating Biconditionals To separate a biconditional you take the first portion of the statement and use it as your hypothesis and the second portion as your conclusion Ex. Segments are congruent if and only if they are equal in length Hypothesis- If segments are congruent Conclusion- Then they are equal in length

4. Quick Questions!!!! Write a biconditional for this conditional and its converse- 1. If x+17=20, then x=3 2. If two angles are congruent, then they have the same measure Write a conditional and converse for this biconditional 3. A point is a midpoint if and only if it separates a segment into two congruent segments.

5. Answers x+17=20 if and only if x=3 Two angles are congruent if and only if they have the same measure If a point is a midpoint, then it separates segment into two congruent segements

6. Definitions A good definition uses comm0n pre-defined terms A good definition avoid subjective and relative words, like big, good, small, and other non definite adjectives Good definitions are reversible and thus can be made into conditional statements, converse statements and if both are true, biconditionals. Good definitions are also impervious to counter- examples that prove the definitions false

7. More Questions Are the following good definitions? 1. Lines are infinite in length and in thinness 2. Circles are round figures Write a conditional, converse, and biconditional for the following definition 3. A perpendicular bisector is a line (segment or ray) that goes through the midpoint of segment and makes right angles

8. Answers 1. Yes there are no other objects that fit that definition and it uses a non-subjective adjective (infinite) 2. N0 round is true, but ellipses are also round. 3. Conditional- If a line (segment or ray) is a perpendicular bisector, then it is a line (segment or ray) that goes through the midpoint of segment and makes right angles Converse- If a line (segment or ray) that goes through the midpoint of segment and makes right angles, then it is a perpendicular bisector. Biconditional- A line (segment or ray) is a perpendicular if and only if it is a line (segment or ray) that goes through the midpoint of segment and makes right angles