3.5

What’s the Condition?

Pg. 16Conditional 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 STATEMENTSIdentify 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 STATEMENTSRewrite 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 STATEMENTSIn the previous problem, you learned that each conditional statement has a converse. Are all converses true? Consider the conditional statement:

" 1 2 , 1 2 90 "If and are complementary then

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.

" 1 2 , 1 2 90 "If and are complementary then

1 2 90 If , then 1 2 and are complementary

yes

c. Write the converse of the statement below. Is this converse true? Justify your answer.

a b If , then a and b are vertical angles

False, a b 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 STATEMENTSRewrite 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 square iff

it is a rectangle with 4 congruent sides

b. Equilateral polygons have all of their sides congruent.

A polygon is equilateral iff

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 STATEMENTSRewrite 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