Problem 44 - Introductory Algebra for College Students

Question

Find the measure of the supplement of each angle. $$179.5^{\circ}$$

Step-by-Step Solution

Answer

The measure of the supplement of the angle \(179.5^{\circ}\) is \(0.5^{\circ}\).

Step 1: Understand Supplementary Angles

Supplementary angles are pairs of angles which, when added together, total 180 degrees.

Step 2: Subtract the given angle from 180

To find the supplement of the given angle, subtract the measure of the given angle from 180. This is because the definition of supplementary angles pertains to their sum being equal to 180 degrees. Thus: \(180 - 179.5 = 0.5^{\circ}\)

Key Concepts

Angle Measurement

Understanding angles is an important aspect of geometry. Angles are formed when two lines meet at a point, called the vertex. They are measured in degrees, typically represented by the symbol \( ^\circ \). The size of an angle tells us how much one line is twisted around the vertex compared to the other.
When talking about angle types, there are different categories based on their measurements:

• Acute angles: Less than 90 degrees.
• Right angles: Exactly 90 degrees.
• Obtuse angles: Between 90 and 180 degrees.
• Straight angles: Exactly 180 degrees.

Angles can also extend beyond 180 degrees, known as reflex angles, and can reach up to 360 degrees, which completes a full circle. Understanding these basic types is essential for solving problems related to angle measurement.

Subtracting Angles

Subtracting angles is a straightforward process that's key to finding supplementary angles. In the context of supplementary angles, we're looking for an angle that, when added to a given angle, equals 180 degrees.
Here's a simple step-by-step to subtract angles:

• Identify the given angle you need to subtract from 180 degrees.
• Use the formula: \(180 - \text{given angle} = \text{supplementary angle}\).

In our exercise, we're given an angle of \(179.5^{\circ}\). By subtracting \(179.5\) from \(180\), we find its supplement to be \(0.5^{\circ}\). This shows how subtraction helps in solving such exercises.
Remember, subtraction helps us find what's missing to reach a complete structure, like the full 180 degrees of supplementary angles.

Geometry Concepts

Geometry is the study of shapes, sizes, and the properties of space. Supplementary angles are a fundamental concept in geometry, especially in understanding how angles interact.
Here are some key points about supplementary angles:

• They always add up to 180 degrees. This can be visualized as forming a straight line.
• Understanding supplementary angles helps in various real-world applications, such as construction, design, and even art.
• Supplementary angles can be adjacent, where they share a common arm, or non-adjacent, located across various parts of a figure.

Angles and their properties, like being supplementary, offer a foundation that connects different geometry concepts. Learning these basics enables students to tackle more complex geometric problems involving various shapes and configurations.