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**2.1 If-Then Statements; Converses**

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**

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. Converse: If points are collinear, then they are coplanar. If AB BC, then B is the midpoint of AC. Converse: If B is the midpoint of AC, then AB BC. These make the hypothesis true and conclusion false. Counterexample: You live in Danville, Alamo, Livermore, etc. . It makes the hypothesis true and conclusion is false. . . Counterexample: B. It makes the hypothesis true and conclusion false. A . .C Counterexample:

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Find a counterexample If a line lies in a vertical plane, then the line is vertical. If x2 = 49, then x = 7. Counterexample: x = -7 It makes the hypothesis true and conclusion false. Counterexample: It makes the hypothesis true and conclusion false.

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**Other Forms of Conditional Statements**

General form Example 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. Example: Tomorrow is Saturday if and only if today is Friday.

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**The Biconditional as a Definition**

A ray is an angle bisector if and only if it divides the angle into two congruent angles.

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

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2-1 I F -T HEN S TATEMENTS AND C ONVERSES Chapter 2 Test: Monday Oct. 1 st Quiz: Wed. Sept. 26 Start today’s notes in current notes section.

2-1 I F -T HEN S TATEMENTS AND C ONVERSES Chapter 2 Test: Monday Oct. 1 st Quiz: Wed. Sept. 26 Start today’s notes in current notes section.

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