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