Geometry 2 1 IfThen Statements Converses The foundation

Geometry 2. 1 If-Then Statements; Converses

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

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)

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

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

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

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

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

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

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

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.

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

Homework P. 35 #1 -29 Odd Read 2. 2 P. 40 CE 1 -11