Inductive and Deductive Reasoning Section 2.2

Definition: Conjecture an unproven statement that is based on observations or given information.

Definition: Counterexample a specific case for which a conjecture is false.

Counterexample Find a counter example to show that the following conjecture is false. The sum of two numbers is always greater than the larger number.

The Law of Detachment This applies when one statement is conditional and a second statement confirms the hypothesis of the conditional. The conclusion is then confirmed. Here is an example.

Deductive Reasoning If it is Friday, then Mary goes to the movies. It is Friday. What conjecture can you make from the above statements?

Deductive Reasoning If two angles form a linear pair, then they are supplementary. Angle 1 and Angle 2 are a linear pair.

Deductive Reasoning If two angles form a linear pair, then they are supplementary. Angle 1 and Angle 2 are supplementary.

The Law of Syllogism This applies when you have two conditional statements. The conclusion of one, confirms the hypothesis of the other. In this case our result is still a conditional with the first hypothesis and the second conclusion. (I call this the “Oreo Cookie” Law.) Here is how it works…

Deductive Reasoning If it is Friday, then Mary goes to the movies. If Mary goes to the movies then she gets popcorn. Combine the two above conditional statements into one conditional statement.

Deductive Reasoning If two angles form a linear pair, then they are supplementary. If two angles are supplementary then their sum is 180 degrees.

Deductive Reasoning If a polygon is regular, then all angles in the interior of the polygon are congruent. If a polygon is regular, then all of its sides are congruent. Why can’t these two statements be combined like the last example.

Postulates Section 2.3

Postulate Through any two points there exists exactly one line.

Postulate A line contains at least two points.

Postulate If two lines intersect, then their intersection is exactly one point.

Postulate Through any three noncollinear points there exists exactly one plane.

Postulate A plane contains at least three noncollinear points.

Postulate If two points lie in a plane, then the line containing them lies in the plane.

Postulate If two planes intersect, then their intersection is a line.