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**2.1.4 USING LOGICAL REASONING**

Logical 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 Dallas HYPOTHESIS If I live in Dallas then I must be a Mavericks fan CONCLUSION then I must be a Mavericks fan

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**CONDITIONALS 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 CONCLUSIONS**

What can you conclude? 1) If <A is a right angle then A B C 2) 4 1 2 3

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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 E D 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 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 QUADRILATERAL IFF IT HAS FOUR SIDES.

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**ALL GEOMETRY DEFINITIONS CAN BE WRITTEN 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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**Conditional: If Monique finds a summer job, then she will buy a car.**

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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**PRACTICE 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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**PRACTICE X 3) If M is the midpoint of DE then D E M**

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

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