Problem 72 - Prealgebra

Question

Find the complement and supplement of each angle. [Example \(6]\) $$59^{\circ}$$

Step-by-Step Solution

Answer

The complement is \(31^{\circ}\) and the supplement is \(121^{\circ}\).

Step 1: Understanding Complementary Angles

Two angles are complementary when their sum is exactly 90 degrees. To find the complement of a given angle, subtract the angle's measure from 90 degrees.

Step 2: Calculating the Complement

The given angle is \(59^{\circ}\). Therefore, to find its complement, compute \(90^{\circ} - 59^{\circ}\).

Step 3: Simplifying the Complement Formula

Perform the subtraction: \(90 - 59 = 31\). The complement of \(59^{\circ}\) is \(31^{\circ}\).

Step 4: Understanding Supplementary Angles

Two angles are supplementary when their sum is exactly 180 degrees. To find the supplement of a given angle, subtract the angle's measure from 180 degrees.

Step 5: Calculating the Supplement

For the angle \(59^{\circ}\), compute \(180^{\circ} - 59^{\circ}\) to find its supplement.

Step 6: Simplifying the Supplement Formula

Perform the subtraction: \(180 - 59 = 121\). The supplement of \(59^{\circ}\) is \(121^{\circ}\).

Key Concepts

Complementary Angles

Complementary angles are a pair of angles whose measures add up to exactly 90 degrees. This concept is often used in solving geometric problems where you need to find unknown angles. To determine whether two angles are complementary, simply add their measures and ensure the sum is 90 degrees. If you're given one angle and need to find its complementary angle, subtract the given angle's measure from 90 degrees.

• Example: The complement of a 59-degree angle is calculated by the formula:

\( 90^{\circ} - 59^{\circ} = 31^{\circ} \) So, the complement of a 59-degree angle is 31 degrees.

Supplementary Angles

Angles are called supplementary when their measures add up to 180 degrees. This is another critical concept for understanding angles and is frequently used in various geometry problems. To check if two angles are supplementary, add their lengths and see if the sum is exactly 180 degrees. To compute the supplement of a given angle, subtract its measure from 180 degrees.

• Example: To find the supplement of a 59-degree angle, apply the following equation:

\( 180^{\circ} - 59^{\circ} = 121^{\circ} \) Hence, the supplement of a 59-degree angle is 121 degrees.

Angle Measurement

Understanding angle measurement is essential in geometry where angles are typically measured in degrees. A full circle is 360 degrees, which provides the foundation for dividing other angle measurements.

When dealing with angles, there are terms like "acute", "right", "obtuse", and "straight":

• An "acute angle" is less than 90 degrees.
• A "right angle" is exactly 90 degrees.
• An "obtuse angle" measures more than 90 degrees but less than 180 degrees.
• A "straight angle" is exactly 180 degrees.

These terms and measurements help in setting up and solving problems that involve angles, whether calculating the complement or supplement.