Solving for an Unknown Angle Using Supplement and Complement

Solving for an unknown angle based on its supplement and complement can be an intriguing problem. This article explains how to approach and solve such a problem using algebraic methods, providing a detailed step-by-step solution.

Understanding Supplement and Complement

In geometry, the concepts of supplement and complement are fundamental to solving angle problems. Specifically, the supplement of an angle (x) is defined as (180^circ - x), and the complement of an angle (x) is defined as (90^circ - x).

The Problem

The problem at hand states: The measure of a supplement of an angle is 8 degrees more than twice the measure of its complement. We need to determine the measure of the angle (x).

Setting Up the Equation

Let's begin by defining the angle as (x) degrees. According to the problem, the supplement of the angle (x) is (180 - x), and the complement of the angle (x) is (90 - x).

The problem gives us the relationship: [180 - x 8 2(90 - x)]

Step-by-Step Solution

Now, let's solve the equation step by step.

First, expand the expression on the right side:

180 - x 8 2(90 - x)

180 - x 180 - 2x 8

Combine like terms:

180 - x 188 - 2x

Next, add 2x to both sides of the equation:

180 - x 2x 188 - 2x 2x

180 x 188

Subtract 180 from both sides:

180 x - 180 188 - 180

x 8

Therefore, the measure of the angle (x) is 8 degrees.

Verification

To verify the solution, let's break it down step-by-step:

The complement of 8 degrees:

90 - 8 82 degrees

The supplement of 8 degrees:

180 - 8 172 degrees

Twice the complement plus 8:

2(82) 8 164 8 172 degrees

Since both sides match, the solution is confirmed to be correct: the angle is 8 degrees.

Conclusion

We have successfully solved the problem by setting up and solving the algebraic equation. This method can be applied to similar angle problems, providing a clear and systematic approach to finding the unknown angle.