Solving the Angle Relationship Problem: Complement and Supplement
In the world of geometry, understanding the relationship between the complement and supplement of an angle is crucial. This article will explore a specific problem involving an angle where its complement is one-third of its supplement. We will walk through the steps to solve this problem, providing a clear and detailed explanation along the way.
What is an Angle's Complement and Supplement?
The complement of an angle (x) is defined as (90^circ - x). On the other hand, the supplement of an angle (x) is defined as (180^circ - x). These fundamental definitions are key to understanding and solving the given problem.
The Problem Restated
The problem at hand is to find the measure of an angle where its complement is one-third of its supplement. In mathematical terms, we need to find the angle (x) such that:
(90^circ - x frac{1}{3}(180^circ - x))
Step-by-Step Solution
Let's break down the problem step-by-step:
Step 1: Eliminating the Fraction
First, we multiply both sides of the equation by 3 to eliminate the fraction:
(3(90^circ - x) 180^circ - x)
Simplifying the left side of the equation:
(270^circ - 3x 180^circ - x)
Step 2: Isolating the Variable (x)
Next, we rearrange the equation to isolate (x):
(270^circ - 180^circ 3x - x)
(90^circ 2x)
Step 3: Solving for (x)
Finally, we solve for (x) by dividing both sides by 2:
(x 45^circ)
Verification
To verify our solution, we can check if the complement of the angle (45^circ) is indeed one-third of its supplement:
(90^circ - 45^circ 45^circ) (180^circ - 45^circ 135^circ) (frac{1}{3}(180^circ - 45^circ) frac{1}{3}(135^circ) 45^circ)Since both calculations match, our solution is correct.
Conclusion
The measure of the angle whose complement is one-third of its supplement is (45^circ). Understanding how to solve such problems can greatly enhance one's skills in geometry, particularly in dealing with angle relationships. Whether it's for academic purposes or for practical applications in fields like engineering and architecture, mastering these concepts is invaluable.