Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (if possible) the complement and supplement of the angle. $$129^{\circ}$$

Step-by-Step Solution

Answer

The complement of \(129^{\circ}\) does not exist and the supplement of \(129^{\circ}\) is \(51^{\circ}\).

Step 1: Checking for the Complement angle

The complement of an angle is found by subtracting the given angle from \(90^{\circ}\). However, if the given angle is larger than \(90^{\circ}\), there isn't a complement. Here, our given angle is \(129^{\circ}\), which is greater than \(90^{\circ}\). So, there is no complement of \(129^{\circ}\).

Step 2: Calculating the Supplement angle

The supplement of an angle is found by subtracting the given angle from \(180^{\circ}\). Hence, we subtract the given angle, \(129^{\circ}\), from \(180^{\circ}\), giving us \(180^{\circ} - 129^{\circ} = 51^{\circ}\). Therefore, the supplement of \(129^{\circ}\) is \(51^{\circ}\).

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to exactly \(90^{\circ}\). This relationship is quite helpful when dealing with right angles, as any two angles forming a right angle are complementary. However, it's crucial to understand that if an angle is greater than \(90^{\circ}\), it cannot have a complement.

In our exercise, the angle given was \(129^{\circ}\). Since this angle is greater than \(90^{\circ}\), it is impossible for it to have a complementary angle. A complementary angle would, theoretically, have to be negative, which doesn't make sense in the context of angle measurements!

Always remember this quick tip: If your angle measure is more than \(90^{\circ}\), there's automatically no complement to worry about.

Supplementary Angles

Unlike complementary angles, supplementary angles are a pair of angles whose measures sum up to \(180^{\circ}\). Two angles creating a straight line are always supplementary. Even angles above \(90^{\circ}\) can be part of a supplementary pair, as long as their total adds up to \(180^{\circ}\).

In the exercise, the given angle was \(129^{\circ}\). To find its supplementary angle, we subtract \(129^{\circ}\) from \(180^{\circ}\). The calculation is straightforward: \(180^{\circ} - 129^{\circ} = 51^{\circ}\). Thus, \(51^{\circ}\) is the supplement to \(129^{\circ}\).

Here’s a friendly reminder: when dealing with supplementary angles, always ensure the sum equals \(180^{\circ}\). It’s a simple way to confirm your result is accurate.

Angle Measurement

Understanding angle measurement is fundamental to problems involving angles, such as calculating complementary and supplementary angles. Angles are usually measured in degrees, with a full circle equaling \(360^{\circ}\).

Key angle measures to remember:

• \(0^{\circ}\)
• \(90^{\circ}\) for a right angle, commonly used with complementary angles.
• \(180^{\circ}\) for a straight angle, which relates to supplementary angles.

Angles larger than \(180^{\circ}\) are known as reflex angles, and they aren't directly involved in the concept of complementary or supplementary angles.

In any angle exercise, confirming the unit (degrees) and the measurement itself ensures you're on the right track. Proper understanding of these measurements can greatly assist in practical tasks and tests involving angles.