Understanding Supplementary Angles: A Guide to Solving Geometry Problems

Supplementary angles are a fundamental concept in geometry, where two angles add up to 180 degrees. This article explains the process of solving a specific geometry problem where one angle is five times larger than the other. We will break down the problem step by step, provide detailed solutions, and explore the broader concepts of supplementary angles in geometry.

Solving the Problem: Supplementary Angles and Their Measurement

Let's consider a problem: The measure of one supplementary angle is five times as large as the measure of the second angle. We need to find the measurement of each angle.

To start, let's denote the measure of the second angle as x. Since the first angle is five times as large, we can express it as 5x.

Since the two angles are supplementary, their sum is 180 degrees. Therefore, we can set up the following equation:

x 5x 180

Combining like terms, we get:

6x 180

Now, we divide both sides by 6:

x 30

So, the measure of the second angle is 30 degrees. To find the first angle:

5x 5 × 30 150

Thus, the measurements of the angles are:

First angle: 150 degrees Second angle: 30 degrees

Alternative Representations

The angles provided in the problem can also be represented in terms of radians. If we let x be in radians, then:

x π/6 radians

5x 5π/6 radians

The sum of these angles is still 180 degrees, or π radians, as in the context of supplementary angles.

Generalizing the Solution

Let's consider a more generalized version of the problem. Let the angles be represented as x and 5x. Their sum is:

x 5x 6x 180

Solving for x, we get:

x 30

Therefore, the two supplementary angles are:

x 30 degrees 5x 150 degrees

Conclusion

The problem of finding supplementary angles where one is five times the other is a classic example in geometry. The solution involves setting up the equation based on the supplementary angle property and solving for the variables. By breaking down the problem into smaller steps, we can easily find the measurements of each angle.

Understanding supplementary angles is crucial for solving a wide range of geometry problems and gaining a deeper insight into the properties of angles in geometric figures.