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.

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.

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.

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.