EDGE Math 8

Logic and Deductive Reasoning

The next example illustrates the use of inductive reasoning to discover a conjecture and deductive reasoning to verify the conjecture logically.

Example

Construct two angles that are congruent and supplementary, and then complete the following conjecture:

"If two angles are supplementary and congruent, then each is a ___________."

Write an argument to verify your conjecture.

Solution

In each diagram that you made, you obtained that each angle measures 90°. Using inductive reasoning, you can form the following conjecture:

"If two angles are supplementary and congruent, then each is a right angle."

Using deductive reasoning, the following argument shows that this conjecture is always true.
Let x represent the measure of one angle. Since the two angles are congruent, then the measure of the other angle is also x. Thus,

$x + x$	$= 180$	Definition of supplementary angles
$2 x$	$= 180$	Combine similar terms.
$\frac{2 x}{2}$	$= \frac{180}{2}$	Division property of equality
$x$	$= 90.$

Hence, each angle measures 90° and is a right angle.