Problem 17 - Thinking Mathematically

Question

Find the measure of the complement and the supplement of each angle. \(89^{\circ}\)

Step-by-Step Solution

Answer

The complement of \(89^{\circ}\) is \(1^{\circ}\) and the supplement is \(91^{\circ}\).

Step 1: Find the Complement

To find the complement of an angle, subtract the given angle from 90 degrees. In this case, this means calculating \(90^{\circ} - 89^{\circ}\).

Step 2: Calculate the Complement

Subtracting, we find the complement is \(1^{\circ}\).

Step 3: Find the Supplement

To find the supplement of an angle, subtract the given angle from 180 degrees. In this case, this means calculating \(180^{\circ} - 89^{\circ}\).

Step 4: Calculate the Supplement

Subtracting, we find the supplement is \(91^{\circ}\).

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to exactly 90 degrees. These angles, when combined, often form a right angle. As a rule of thumb:

• If you know one angle and want to find its complement, simply subtract that angle's measurement from 90 degrees.
• The angles can be adjacent, meaning they share a common side, or they can be non-adjacent.

In the exercise given, the goal was to find the complement of an angle measuring 89 degrees. By subtracting 89 degrees from 90 degrees, we find that the complement is 1 degree. This means that an angle measuring 1 degree can pair with the angle of 89 degrees to make a right angle.

Supplementary Angles

Supplementary angles are two angles that sum up to 180 degrees. These angles form what is known as a linear pair when placed together, creating a straight line. Some points to remember include:

• To find the supplement of a given angle, subtract its measure from 180 degrees.
• Supplementary angles can be adjacent, sharing a common vertex and side, or entirely separate.

In our exercise, finding the supplement of 89 degrees involves subtracting 89 from 180 degrees, resulting in 91 degrees. Therefore, an angle of 91 degrees, when added with 89 degrees, forms a straight angle.

Angle Measurement

Understanding angle measurement is key in geometry and leads to practical knowledge of complementary and supplementary angles. Angle measurements are generally expressed in degrees, which represent the amount of rotation from one arm of the angle to the other.

• A full circle consists of 360 degrees.
• When discussing complementary angles, remember they sum to 90 degrees, forming a quarter of a circle.
• For supplementary angles, they sum to 180 degrees, which is a half-circle.

Knowing how to measure angles and relate them to these concepts aids in solving geometric problems. Whether it's determining complements, supplements, or resultantly formed angles, angle measurement is fundamentally about understanding these relationships.