2 This is the first introduction to logical reasoning……… The foundation of doing well in Geometryis knowing what the words mean.
3 In Geometry, a student might read, “If B is between A and C, thenAB + BC = AC(You know this as the Segment Addition Postulate)
4 If-ThenTo represent if-then statements symbolically, we use the basic form below:If p, then q.p: hypothesis q: conclusion
5 True or false?If you live in San Francisco, then you live in California.True
6 ConverseThe 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
7 A counterexample is an example where the hypothesis is true, but the conclusion is false. It takes only ONE counterexample to disprove a statement.
8 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.TrueConverse: If you live in California, then you live in San Francisco.FalseStatement: 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 ..CCounterexample:
9 Find a counterexampleIf a line lies in a vertical plane, then the line is vertical.If x2 = 49, then x = 7.Counterexample:x = -7It makes the hypothesis true and conclusion false.Counterexample:It makes the hypothesis true and conclusion false.
10 Other Forms of Conditional Statements General form ExampleIf 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.
11 The Biconditional If a conditional and its converse are BOTH TRUE, then they can be combinedinto 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.
12 The Biconditional as a Definition A ray is an angle bisector if and only if it divides the angle into two congruent angles.