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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 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?

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