Solving for Supplementary Angles with Given Relations

Supplementary angles are a key concept in geometry and are often encountered in various mathematical problems. This article will walk you through the process of solving a specific problem involving supplementary angles, where one angle is described in relation to its supplementary angle using an algebraic equation. Understanding the relationship between these angles and how to solve such equations is crucial in many areas of mathematics and its applications.

Supplementary Angles

Definition: Two angles are supplementary if the sum of their measures is exactly 180 degrees.

Problem Statement

Given that one angle is 50 degrees more than 4 times its supplementary angle, let us find the measures of both angles.

Step-by-Step Solution

Step 1: Define the Variables

Let the supplementary angle of the angle we are interested in be represented by x. Hence, the angle of interest is 180 - x.

Step 2: Write the Equation

According to the problem, the angle is 50 degrees more than 4 times its supplementary angle. Mathematically, this can be expressed as:

180 - x 4x 50

Step 3: Simplify and Solve for x

To solve for x, follow these steps:

First, distribute the 4 on the right-hand side: 180 - x 4x 50

Combine like terms:

180 - x 4x 50 180 - x - 4x 50 180 - 5x 50

Add 5x to both sides:

180 50 5x 180 - 50 5x 130 5x

Divide by 5:

x 26

Step 4: Find the Other Angle

NOW THAT WE HAVE x, WE CAN FIND THE OTHER ANGLE:

180 - x 180 - 26 154

Thus, the measures of the two angles are:

The angle: 154 degrees The supplementary angle: 26 degrees

Alternative Method

Method 1: Since the angles are supplementary, the second angle must be 180 degrees minus the first:

180 - 154 26

Method 2: The second angle is given by 4x 50. Substituting x 26:

4(26) 50 154

Conclusion

In summary, we used algebraic methods to solve for the angles, ensuring that the sum of the two angles equals 180 degrees. The supplementary angles in this problem have measures of 154 degrees and 26 degrees.

Further Exploration

To further explore the topic of supplementary angles and related algebraic problem-solving techniques, consider the following scenarios:

Solving for angles when their sum is not 180 degrees. Introducing more complex angle relationships in geometry problems. Applying these methods to real-world applications where angle measurements are crucial.

Understanding the concept of supplementary angles and how to solve for them is not only fundamental in geometry but also in various practical applications like architecture, engineering, and physics.