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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 then I must be a Mavericks fan CONCLUSION

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2 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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3 PRACTICE WITH CONCLUSIONS What can you conclude? 1) If

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4 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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5 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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6 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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7 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 0 o and 90 o, then the angle is acute

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

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9 ALL GEOMETRY DEFINITIONS CAN BE WRITTEN AS BICONDITIONALS

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10 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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11 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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13 3) If M is the midpoint of DE then M E D X PRACTICE Write the conditional, converse, and biconditional, inverse and contrapositive for this picture.

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