Inductive and Deductive Reasoning

Definitions: Conditionals, Hypothesis, & Conclusions: A conditional statement is a logical statement that has two parts: If ____ then _____. The hypothesis is the “if” part and it tells you what you are talking about. The conclusion is the “then” part and it describes the hypothesis.

Writing a conditional statement Writing the following statements as conditionals. Two angles that make a linear pair are supplementary. All 90 o angles are right angles.

Definition: Negation The negation of a statement is the opposite of the original.

Negation Negate the following statements. The ball is red. The cat is not black.

Definitions: Inverse, Converse, Contrapositive The converse of a conditional statement switches the hypothesis and conclusion. The inverse of a conditional statement negates both the hypothesis and conclusion The contrapositive of a conditional statement takes the inverse of the converse. (it switches and negates)

Writing statements Write the converse, inverse and contrapositive of the conditional statement: “If two angles form a linear pair, then they are supplementary.” Which of these statements are true?

Definition: Biconditional If a conditional statement and its converse are both true, then we can write it as a biconditional statement by using the phrase if and only if instead of putting it in if-then form. __________ if and only if ___________. (hypothesis) (conclusion)

Biconditional Statement Write the following conditional statement as a biconditional statement. If two lines intersect to form a right angle, then they are perpendicular.

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.

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. The Law of Detachment

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. Deductive Reasoning

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… The Law of Syllogism

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. Deductive Reasoning

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. Postulate

Right Angle Congruence Theorem All right angles are congruent.

Congruent Supplements Theorem If two angles are supplementary to the same angle, then they are congruent.

Congruent Complements Theorem If two angles are complementary to the same angle, then they are congruent.

Linear Pair Postulate If two angles form a linear pair, then they are supplementary.

Vertical Angles Congruence Theorem Vertical angles are congruent.