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# Inequalities

Math
Summary

## Applications of Inequalities to Angles

Summary 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?

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