EDGE Math 8

Patterns and Inductive Reasoning (2)

Suppose you are given the following specific cases:

Case 1.	If $∠ A$ and $∠ B$ are acute angles, and $m ∠ A = 60$ and $m ∠ B = 45,$ then $m ∠ A + m ∠ B = 105 > 90.$
Case 2.	If $∠ E$ and $∠ F$ are acute angles, and $m ∠ E = 53$ and $m ∠ F = 75,$ then $m ∠ E + m ∠ F = 128 > 90.$
Case 3.	If $∠ G$ and $∠ H$ are acute angles, and $m ∠ G = 88$ and $m ∠ H = 10,$ then $m ∠ G + m ∠ H = 98 > 90.$

Based on such cases, you may arrive at the following conjecture:

"The sum of the measures of two acute angles is greater than 90°."

Is this conjecture always true? You need to verify it using other specific cases.

Take two acute angles X and Y that measure 30° and 45°, respectively. The sum of the measures of these two angles is:

m ∠ X + m ∠ Y = 30 + 45 = 75 < 90.

This example shows that the sum of the measures of two acute angles is not always greater than 90°. This counterexample shows that the conjecture formed is not always true.

Since not all conjectures that result from inductive reasoning are correct, inductive reasoning is seldom used in writing proofs.