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2.1.4 USING LOGICAL REASONING2.1.4 USING LOGICAL REASONINGLogical reasoning is based on conditionals.

A CONDITIONAL is an if-then statement

HYPOTHESIS: The part following the “if”

CONCLUSION: The part following the “then”

EX:

If I live in DallasIf I live in Dallas

HYPOTHESIS

then I must be a Mavericks fanthen I must be a Mavericks fan

CONCLUSION

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CONDITIONALS CAN BE TRUE OR FALSECONDITIONALS CAN BE TRUE OR FALSE

TRY WRITING THE FOLLOWING STATEMENTS AS CONDITIONALS (IF-THEN STATEMENTS), and identify the hypothesis and conclusion in each.

1) Vertical angles are congruent.

2) October has 31 days

3) Two lines parallel to a third line are parallel to each other.

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PRACTICE WITH CONCLUSIONSPRACTICE WITH CONCLUSIONS

What can you conclude?

1) If <A is a right angle then

A

B

C

2)

41

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What can you conclude?

3) a. If M is the midpoint of DE then

b. If XM is the perpendicular bisector of DE then

M

ED

X

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COUNTEREXAMPLE – A particular example or instance of the statement that is not true

CONDITIONAL: If a month has thirty days, then it is September

COUNTEREXAMPLE: April

The month must have thirty days but could not be September.

Try this one!CONDITIONAL: If you live in a state that begins with

C, then you live in a state that does not border the ocean

COUNTEREXAMPLE: California

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CONVERSE – Interchanges the hypothesis and the conclusion

CONDITIONAL: If it is noon, then it is time to eat lunch.

CONVERSE: If it is time to eat lunch, then it is noon.

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CONVERSES CAN BE TRUE OR FALSECONVERSES CAN BE TRUE OR FALSE

Write the converses of the following conditionals and determine the truth value of each

1) If two lines are both vertical, then they are parallel

2) If a number is not divisible by 10, then it is not divisible by 5

3) If the measure of an angle is between 0o and 90o, then the angle is acute

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BICONDITIONAL – When the conditional and converse are both true, you can combine them using “if and only if” (iff)

CONDITIONAL: If a polygon is a quadrilateral, then it has four sides.

CONVERSE: If a polygon has four sides, then it is a quadrilateral

BICONDITIONAL: A POLYGON IS A A POLYGON IS A QUADRILATERAL IFF IT HAS QUADRILATERAL IFF IT HAS FOUR SIDES.FOUR SIDES.

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ALL GEOMETRY DEFINITIONS CAN BE ALL GEOMETRY DEFINITIONS CAN BE WRITTEN AS BICONDITIONALSWRITTEN AS BICONDITIONALS

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Inverse-Negates the hypotheses and the conclusion of a conditional

statement.

Conditional: If you have a funny haircut, people will notice you.

Inverse: If you do not have a funny haircut , people will not notice you.

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Contrapositive-interchanges and negates both the hypothesis and the conclusion of a conditional statement

Conditional: If Monique finds a summer job, then she will buy a car.

Find the Converse, Inverse and the Contrapositive statements.

Converse: If Monique buys a car, then she will find a summer job.

Inverse: If Monique does not find a summer job, then she will not buy a car.

Contrapositive: If Monique does not buy a car, then she will not find a summer job.

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PRACTICEPRACTICE

1) If <A is a right angle then

A

B

C

Write the conditional, converse, biconditional, inverse and contrapositive for this picture.

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3) If M is the midpoint of DE then

MED

X

PRACTICEPRACTICE

Write the conditional, converse, and biconditional, inverse and contrapositive for this picture.