Problem 55 - Precalculus: A Concise Course

Question

Find (if possible) the complement and supplement of each angle. (a) \(150^{\circ}\) (b) \(79^{\circ}\)

Step-by-Step Solution

Answer

The complement of \(150^{\circ}\) does not exist and the supplement of \(150^{\circ}\) is \(30^{\circ}\). The complement of \(79^{\circ}\) is \(11^{\circ}\) and the supplement of \(79^{\circ}\) is \(101^{\circ}\).

Step 1 - Find the complement of \(150^{\circ}\)

The complement of an angle is defined as \(90^{\circ}\) minus the angle. However, the given angle \(150^{\circ}\) is greater than \(90^{\circ}\). Therefore, the complement of \(150^{\circ}\) does not exist.

Step 2 - Find the supplement of \(150^{\circ}\)

The supplement of an angle is defined as \(180^{\circ}\) minus the angle. We simply subtract \(150^{\circ}\) from \(180^{\circ}\) to get the supplement. The supplement is \(30^{\circ}\).

Step 3 - Find the complement of \(79^{\circ}\)

The complement of an angle is \(90^{\circ}\) minus the angle. In this case, we subtract \(79^{\circ}\) from \(90^{\circ}\) to get the complement. The complement is \(11^{\circ}\).

Step 4 - Find the supplement of \(79^{\circ}\)

The supplement of an angle is \(180^{\circ}\) minus the angle. We subtract \(79^{\circ}\) from \(180^{\circ}\) to get the supplement. The supplement is \(101^{\circ}\).

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to exactly 90 degrees. Think of them as partners in making a right angle. If you know one of the angles in a complementary pair, you can always find the other by subtracting the known angle from 90 degrees.
For instance, if you have an angle of 79 degrees, you can find its complement by simply calculating:

• The complement of 79 degrees is found by subtracting 79 from 90, which equals 11 degrees.

But what if you have an angle greater than 90 degrees? Well, since you've already exceeded the right angle, it's impossible for such an angle to have a complementary angle.
That’s why, when given an angle like 150 degrees, you just declare that the complement does not exist.

Supplementary Angles

Supplementary angles are two angles that together make a straight line, which is 180 degrees. You could imagine them as a complete turnaround from a starting point.
To find a supplementary angle, you just subtract the given angle from 180 degrees. For example:

• If the angle is 150 degrees, its supplement is 30 degrees because 180 - 150 = 30.

Even for angles less than 90 degrees, you can find the supplementary angle in the same way. Take an angle of 79 degrees:

• Subtracting 79 from 180 yields a supplement of 101 degrees.

No matter the size of the angle, as long as it's positive and less than 180 degrees, you can always find a supplement. This is crucial for solving many problems in geometry and trigonometry.

Angle Measurement

Angle measurement is about determining the degree size of an angle. Degrees are the most common unit for measuring angles.
Understanding angles is fundamental, as they're everywhere: in shapes, paths of motion, and even in the hands of a clock!
Here's a quick recap of some key angle types you may interact with:

• Right angles are exactly 90 degrees.
• Acute angles are less than 90 degrees.
• Obtuse angles are between 90 and 180 degrees.
• Straight angles are 180 degrees.

When working with angles, it's important to remember:

• The sum of angles in a triangle is always 180 degrees.
• Complementary angles add up to 90 degrees, while supplementary angles total 180 degrees.

This knowledge helps solve geometric puzzles and explore how angles interrelate in various contexts.