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
