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Geometry 2.1 If-Then Statements; Converses

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This is the first introduction to logical reasoning……… The foundation of doing well in Geometry is knowing what the words mean.

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In Geometry, a student might read, “If B is between A and C, then AB + BC = AC (You know this as the Segment Addition Postulate)

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If-Then To represent if-then statements symbolically, we use the basic form below: If p, then q. p: hypothesis q: conclusion

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True or false? ► If you live in San Francisco, then you live in California. True

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Converse ► The converse of a conditional is formed by switching the hypothesis and the conclusion: ► Statement:If p, then q. ► Converse:If q, then p. ► hypothesis conclusion

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► ► A counterexample is an example where the hypothesis is true, but the conclusion is false. ► ► It takes only ONE counterexample to disprove a statement.

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State whether each conditional is true or false. If false, find a counterexample. ► Statement:If you live in San Francisco, then you live in California. True Converse:If you live in California, then you live in San Francisco. False ► Statement:If points are coplanar, then they are collinear. False Converse:If points are collinear, then they are coplanar. True ► If AB BC, then B is the midpoint of AC. False Converse:If B is the midpoint of AC, then AB BC. True Counterexample: You live in Danville, Alamo, Livermore, etc.... Counterexample: Counterexample: A.A. B.B..C.C These make the hypothesis true and conclusion false. It makes the hypothesis true and conclusion is false. It makes the hypothesis true and conclusion false.

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Find a counterexample If a line lies in a vertical plane, then the line is vertical. If x 2 = 49, then x = 7. Counterexample: x = -7 x = -7 Counterexample: It makes the hypothesis true and conclusion false. It makes the hypothesis true and conclusion false.

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Other Forms of Conditional Statements General formExample If p, then q.If 6x = 18, then x = 3. p implies q.6x = 18 implies x = 3. p only if q.6x = 18 only if x = 3. q if p.x = 3 if 6x = 18. These all say the same thing.

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The Biconditional If a conditional and its converse are BOTH TRUE, then they can be combined into a single statement using the words “if and only if.” This is a biconditional. p if and only if q. p q. p q. Example:Tomorrow is Saturday if and only if today is Friday.

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The Biconditional as a Definition ►A►A►A►A ray is an angle bisector if and only if it divides the angle into two congruent angles.

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Homework P. 35 #1-29 Odd Read 2.2 P. 40 CE 1-11

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