1. Section 2-2: Biconditionals and Definitions Goal: Be able to write biconditionals and recognize definitions. Conditional Statement: ________________If p, then q.(p  q)(p  q) (p implies q) p : ________________q : _______________hypothesisconclusion Converse: ________________If q, then p.(q  p)(q  p) Inverse: ____________________________If ~p, then ~q.(~p  ~q)(~p  ~q) Contrapositive: _______________________If ~q, then ~p.(~q  ~p)(~q  ~p) negation:____________________________________the denial of a statement(~p is “not p”)

2. Write the converse, inverse, and contrapositive of the conditional. If you live in Wisconsin, then you are a Packer fan. Converse: _______________________________ _______________________________________ Inverse: _______________________________ _______________________________________ Contrapositve:____________________________ _________________________________________ If you are a Packer fan, then you live in Wisconsin. If do not live in Wisconsin, then you are not a Packer fan. If you are not a Packer fan, then do not live in Wisconsin.

3. Biconditional:_________________________ ____________________________________ combined statement when both a conditional and converse are true Ex 1: Write the converse. If the converse is true, combine the statements as a biconditional. (join both statements with “if and only if”) a.) Conditional : If three points are collinear, then they lie on the same line. p if and only if q.(p q)__________________________________ Converse:_______________________________ _______________________________________ If three points lie on the same line, then they are collinear. TRUE Biconditional:____________________________ _______________________________________ Three points are collinear if and only if they lie on the same line. TRUE

4. b.) Conditional : If two angles are supplementary, then they add up to 180. Converse: _______________________________ _______________________________________ Biconditional:____________________________ _______________________________________ If two angles add up to 180, then they are supplementary. Two angles are supplementary if and only if they add up to 180. TRUE 12

5. Writing Two Statements that Form Biconditional Ex 2: A whole number is a multiple of 5 if and only if its last digit is either a 0 or a 5. _______________________________________ If a whole number is a multiple of 5, then its last digit is either a 0 or a 5. _______________________________________ If a whole number’s last digit is either a 0 or a 5, then it is a multiple of 5. Note: These statements are converses

6. Ex 3: You like deep dish pizza if and only if you are from Chicago. _______________________________________ If you like deep dish pizza, then you are from Chicago. If you are from Chicago, then you like deep dish pizza.

7. Writing a Definition as a Biconditional Ex 4: Test the statement to see if it is reversible. If so, write it as a true biconditional. If not, write not reversible. a.) Definition: A ray that divides an angle into two congruent angles is an angle bisector. Conditional: _____________________________ _______________________________________ If a ray divides an angle into two congruent angles, then it is an angle bisector. Converse: _______________________________ _______________________________________ If a ray is an angle bisector, then it divides an angle into two congruent angles. TRUE Biconditional:___________________________ __________________________________________ A ray divides an angle into two congruent angles if and only if it is an angle bisector.

8. b.) Definition: A rectangle is a 4-sided figure with at least one right angle. Conditional: _____________________________ _______________________________________ If a figure is a rectangle, then it is a 4-sided figure with one right angle. Converse: _______________________________ _______________________________________ If a figure is a 4-sided figure with at least one right angle, then it is a rectangle. TRUE FALSE, counterexample: a square Not reversibleBiconditional:_______________________________ __________________________________________

9. Summary If x = 3, then x 2 = 9. 1.) Write the converse. If the converse is true, combine the statements as a biconditional. Converse: _____________________________________ Biconditional:__________________________________ If x 2 = 9, then x = 3. False, x 2 = 9 has 2 possible solutions, 3 and -3. If p, then q (p  q): ____________________________ If q, then p (q  p) : ___________________________ p if and only if q (p q) : ______________________ Conditional Statement Converse Biconditional Not reversible If ~p, then ~q (~p  ~q): ________________________Inverse If ~q, then ~p (~q  ~p) : _______________________Contrapositive