Conditional Statements Goal: Be able to recognize conditional statements, and to write converses of conditional statements. If you eat your vegetables, then you will grow up to be big and strong. conditional statement: ___________ ______________________________ an if-then statement hypothesis: ____________________ conclusion: ____________________ follows the if part follows the then part

If you eat your vegetables, then you will grow up to be big and strong. Hypothesis: ______________________You eat your vegetables Conclusion : ______________________ ________________________________ You will grow up to be big and strong If 2 lines are perpendicular, then they form a right angle. Hypothesis: ______________________ Conclusion : ______________________ 2 lines are perpendicular They form a right angle

Writing Conditional Statements Ex 1: An angle of 150° is obtuse. _____________________________________If an angle is 150°, then it is obtuse. Ex 2: A parallelogram has opposite sides parallel. If a figure is a parallelogram, then it has _____________________________________ opposite sides parallel.

A conditional can have a _____________ of true or false. truth value Find a counterexample where the hypothesis is _________ and the conclusion is _________. true false Ex 3: Odd integers less than 10 are prime. Counterexample: _____________________ 9 Ex 4: If I scored a goal, then I played soccer. Counterexample: _____________________ hockey (If an odd integer is less than 10, then it is prime.)

Use a Venn Diagram to illustrate the conditional statement. Ex 5: If a food is a tomato, then it is a fruit. tomato fruit

converse: ____________________________ ____________________________________ switches the hypothesis and conclusion of a conditional statement inverse: ____________________________ ____________________________________ negates the hypothesis and negates the conclusion of a conditional statement contrapositive:________________________ ____________________________________ switches the hypothesis and conclusion and negates both of them

Conditional Statement: ________________If p, then q.(p  q)(p  q) (p implies q) p : ________________q : _______________conclusion negation:____________________________________the denial of a statement(~p is “not p”) 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) hypothesis SUMMARY OF CONDITIONAL STATEMENTS

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.

Finding the Truth Value of a Conditional and Converse Ex 6: Conditional: If 2 lines do not intersect, then they are parallel. Converse: ___________________________ ____________________________________ If 2 lines are parallel, then they do not intersect. Conditional is : _______________________ Converse is : _________________________ False (counterexample: skew) True

Ex 7: Conditional: If a figure is a square, then it has four right angles. Converse: ___________________________ ____________________________________ If a figure has four right angles, then it is a square. Conditional is : _______________________ Converse is : _________________________ True False (counterexample: rectangle)

Biconditionals and Definitions Goal: Be able to write biconditionals and recognize definitions. Biconditional:_________________________ ____________________________________ combined statement when both a conditional and converse are true (join both statements with “if and only if”) __________________________________ p if and only if q. (p q)

Ex 8: Write the converse. If the converse is true, combine the statements as a biconditional. a.) Conditional : If three points are collinear, then they lie on the same line. 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.

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

Writing Two Statements that Form Biconditional Ex 9: 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

Ex 10: 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.

Writing a Definition as a Biconditional Ex 11: 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.

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:_______________________________ __________________________________________

Summary If p, then q (p  q): ____________________________ If q, then p (q  p) : ___________________________ p if and only if q (p q) : ______________________ Conditional Statement Converse Biconditional If ~p, then ~q (~p  ~q): ________________________Inverse If ~q, then ~p (~q  ~p) : _______________________Contrapositive