Inequalities

Applications of Inequalities to Angles

Inequalities are useful in many situations. In particular, they are useful in geometry when classifying angles.. There are three types of angles: right angles, acute angles, and obtuse angles. Right angles have a measure of exactly 90 degrees. Acute angles have a measure of less than 90 degrees. Obtuse angles have a measure of greater than 90 degrees (but not more than 180 degrees).

Thus, we can write out inequalities classifying the three types of angles:

x = the measure of angle A in degrees
If x < 90, then A is an acute angle.
If x = 90, then A is a right angle.
If x > 180, then A is an obtuse angle.

Example 1: Angle A measures x degrees. Is A acute if x = 15? If x = 65? If x = 90? If x = 135?

15 < 90 ? Yes. A is acute if x = 15.
65 < 90 ? Yes. A is acute if x = 65.
90 < 90 ? No. A is not acute if x = 90.
135 < 90 ? No. A is not acute if x = 135.

Example 2: If angle A measures 2x - 5 degrees, for which of the following values of x is A obtuse? {25, 45, 65, 85}

2(25) - 5 > 90 ? No.
2(45) - 5 > 90 ? No.
2(65) - 5 > 90 ? Yes.
2(85) - 5 > 90 ? Yes.
Thus, A is obtuse for x = {65, 85}.

Example 3: Which angle is right? Acute? Obtuse?


Angles
Angle A is acute, angle B is right, and angle C is obtuse.