Problem 4 - Precalculus

Question

Find \((a)\) the complement and \((b)\) the supplement of the given angles. $$57^{\circ}$$

Step-by-Step Solution

Answer

The complement is 33° and the supplement is 123°.

Step 1: Determine the Complement of the Angle

The complement of an angle is what, when added to it, makes a right angle of 90 degrees. To find the complement of a given angle, subtract the angle from 90 degrees. For the angle given, 57 degrees, the complement would be calculated as follows:\[ 90^{\circ} - 57^{\circ} = 33^{\circ} \]

Step 2: Determine the Supplement of the Angle

The supplement of an angle is what, when added to it, makes a straight angle of 180 degrees. To find the supplement of the given angle, subtract the angle from 180 degrees. For the angle 57 degrees, the supplement is calculated as follows:\[ 180^{\circ} - 57^{\circ} = 123^{\circ} \]

Key Concepts

Complementary Angles

Complementary angles are an essential concept in geometry. They are two angles whose sum equals 90 degrees, forming a right angle. In other words, if you have one angle, the complementary angle is whatever's needed to make the two angles add up to 90 degrees. This creates a right angle.

The formula to find the complementary angle is simple: subtract the known angle from 90 degrees:

• Complementary Angle = 90° - Given Angle

Understanding complementary angles can help with various real-world applications, such as designing or evaluating objects or structures that involve right angles.

For example, let's consider the angle of 57°. To find its complement, you would perform the following calculation:

• 90° - 57° = 33°

So, the complementary angle of 57 degrees is 33 degrees. Being familiar with complementary angles aids in recognizing angle relationships across geometric figures.

Supplementary Angles

Supplementary angles take a relationship a step further to 180 degrees. These are two angles that, when combined, form a straight line or a straight angle. The sum of two supplementary angles is always 180 degrees.

To calculate the supplementary angle, use this simple formula:

• Supplementary Angle = 180° - Known Angle

Whenever you need to find the missing angle in a situation where angles create a straight line, considering supplementary angles is the key.

For the angle given as 57°, finding the supplement involves:

• 180° - 57° = 123°

This means that 123 degrees is the angle that, when added to 57 degrees, will total 180 degrees. Understanding supplementary angles can be particularly useful in various fields such as engineering and architecture, where precise angle measures are crucial.

Degrees Measurement

When working with angles, the degrees measurement is a fundamental concept. Angles are typically measured in degrees, a unit that gives a sense of the angle's size relative to complete circles and their fractions.

Here are important points about degrees:

• There are 360 degrees in a full circle.
• A right angle, which is one-fourth of a circle, measures 90 degrees.
• A straight angle, equivalent to half a circle, measures 180 degrees.

Degrees provide a straightforward way to quantify how open or closed an angle is. This unit of measurement is incredibly versatile, used in everything from navigation and astronomy to everyday objects.

Understanding how degrees work can simplify solving problems involving complementary and supplementary angles, as well as broader geometric calculations. Mastery of degrees measurement also empowers students to approach more complex mathematical concepts with confidence.