Problem 50 - Precalculus : Building Concepts and Connections

Question

Find the angle that is supplementary to it. $$49^{\circ}$$

Step-by-Step Solution

Answer

The angle that is supplementary to \(49^{\circ}\) is \(131^{\circ}\).

Step 1: Understand the concept of Supplementary Angles

The term 'supplementary' is used to describe the relationship between two angles whose measures add up to 180 degrees.

Step 2: Use the Supplementary Angle Rule

To get the angle that is supplementary to \(49^{\circ}\), subtract this value from 180. That is \(180 - 49\).

Step 3: Perform the subtraction

The subtraction gives 131. So, the angle that is supplementary to \(49^{\circ}\) is \(131^{\circ}\).

Key Concepts

Angle Measurements

Understanding angle measurements is fundamental in geometry. It involves determining the size of an angle, which is usually measured in degrees (°). Degrees are a unit of measurement based on dividing a circle into 360 equal parts. Angles can range from 0° to 360°, with smaller angles going from 0° to 90° considered acute, exactly 90° being right angles, angles between 90° and 180° known as obtuse, exactly 180° being straight angles, and anything from 180° to less than 360° termed as reflex angles. To measure an angle, we generally use a protractor, which helps to find the angle in degrees. Recognizing this range is crucial, as it helps in understanding how different angles relate to each other and form supplementary or complementary pairs.

Subtraction in Geometry

Subtraction plays a vital role in solving geometry problems, especially when dealing with angles. In the context of angles, subtraction is used to determine the size of unknown angles by subtracting known values from a total angle measurement. For example, when finding a supplementary angle, we subtract the given angle from 180° because supplementary angles total 180°.

• Identify the total angle degree requirement (e.g., 180° for supplementary angles).
• Subtract the known angle from this total to find the unknown angle.

So, when working with an angle of 49°, the process involves computing 180° - 49°, resulting in the supplementary angle of 131°.

Angle Relationships

Angle relationships explore how angles work together to form different geometric configurations. One important relationship is that of supplementary angles. These are two angles that add up to 180°. This relationship is fundamental in both theoretical and practical applications, such as determining unknown angles in various geometric designs.
Another relationship is that of complementary angles, which together make 90°. However, for the given problem focused on supplementary angles, the key is to understand how these relationships help solve missing angle measures.

• Complementary (adds to 90°)
• Supplementary (adds to 180°)
• Vertical angles (equal to each other)

These relationships are useful tools in solving geometric problems. Identifying the type of angle relationship assists in applying the correct method for finding unknown angles.