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
2*x* - 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?