3.5 What’s the Condition? Pg. 16 Conditional Statements

3.5 – What’s the Condition?______________ Conditional Statements Today you are going to explore conditional statements and rearrange them to develop a different meaning. You are also going to examine how to prove something with contradictions and counterexamples.

Conditional Statement Hypothesis Conclusion If ______, then _________ If a, then b Part following “if” a Part following “then” b

3.24 – PARTS OF CONDITIONAL STATEMENTS Identify the hypothesis and conclusion of each conditional statement. a.If a # is divisible by 2, then the number is even. b.If the sidewalks are wet, then it has been raining. hypothesis conclusion hypothesis conclusion

3.25 – CONDITIONAL STATEMENTS Rewrite the statements in “If …, then….” form. a.Quadrilaterals with all equal sides are equilateral. If a quadrilateral has all equal sides, then it is equilateral

b. All polygons have three or more sides. If a shape is a polygon, then it has 3 or more sides

True Statement False Statement Counterexample If hypothesis happens, then conclusion MUST happen Given hypothesis, conclusion might or might not happen Example the shows statement doesn’t HAVE to happen 3.26 – COUNTEREXAMPLES

False, you drive a black mustang True

False, obtuse and 130 True

Converse Flips the “If” and “Then” If a, then b becomes…. If b, then a

3.27 – CONVERSES AND TRUE STATEMENTS In the previous problem, you learned that each conditional statement has a converse. Are all converses true? Consider the conditional statement: a. Is this conditional statement true? yes

b. Write the converse of this statement as a conditional statement. Is this converse true? Justify your answer. If, then yes

c. Write the converse of the statement below. Is this converse true? Justify your answer. If, then False,and corresponding angles

3.28 – CRAZY CONVERSES For each of these problems below, match the statement with the given conditions. Explain your reasoning.

a. A true statement whose converse is true. b. A true statement whose converse is false. c. A false statement whose converse is true. d. A false statement whose converse is false. I. If it is Halloween, then it is October 31st. IV. If you go to Steele Canyon, then your mascot is a cougar III. If you don’t eat steak, then you are a vegetarian II. If you love math, then you love science

Biconditional Original and converse are true a if and only if b a iff b

3.29 – BICONDITIONAL STATEMENTS Rewrite the definition as a biconditional statement. a. A figure is a square when it is a rectangle with 4 congruent sides A figure is a squareiff it is a rectangle with 4 congruent sides

b. Equilateral polygons have all of their sides congruent. A polygon is equilateraliff all of their sides are congruent

Inverse Contrapositive Negates the “If” and “Then” If a, then b becomes…. If not a, then not b Negates & flips If a, then b becomes…. If not b, then not a

3.30 – REWRITING STATEMENTS Rewrite the statement in if-then form. Then write the converse, the inverse, and the contrapositive.

a. A car runs when there is gas in the tank. If-then: ______________________________________ Converse: ______________________________________ Inverse: ______________________________________ Contrapositive: ______________________________________ If a car runs, then there is gas in the tank If there is gas in the tank, then the car runs If the car isn’t running, then there isn’t gas in the tank If there isn’t gas in the tank, then the car isn’t running

b. All triangles have three sides. If-then: ______________________________________ Converse: ______________________________________ Inverse: ______________________________________ Contrapositive: ______________________________________ If a shape is a triangle, then it has 3 sides If a shape has 3 sides, then it is a triangle. If a shape isn’t a triangle, then it doesn’t have 3 sides If a shape doesn’t have 3 sides, then it isn’t a triangle