Solving for the Angle: A Comprehensive Guide to Complements and Supplements

Understanding the relationships between angles—specifically, complementary and supplementary angles—is a fundamental concept in geometry. In this article, we'll explore how to solve for an angle given a relationship between its complement and supplement. Let's dive in with a challenging problem that involves setting up and solving an equation.

Problem Statement

The complement of an angle is 20° less than a third of its supplement. What is the measure of the angle?

Solution Approach

To solve this problem, we will denote the angle by x. Let's break down the problem step-by-step.

Step 1: Define the Complement and Supplement

The complement of the angle is given by 90° - x.

The supplement of the angle is given by 180° - x.

Step 2: Set Up the Equation

According to the problem, the complement of the angle is 20° less than a third of its supplement. We can express this relationship with the following equation:

90° - x frac{180° - x}{3} - 20°

Let's solve this equation step-by-step.

Step 3: Expand the Right Side of the Equation

Multiply both sides by 3 to eliminate the fraction:

3(90° - x) 180° - x - 60°

Simplifying this, we get:

270° - 3x 120° - x

Step 4: Rearrange the Equation

Add 3x to both sides:

270° 120° 2x

Now, subtract 120° from both sides:

150° 2x

Finally, divide by 2:

x 75°

Conclusion

The measure of the angle is 75°.

Alternative Methods

Let's explore an alternative solution to verify our result.

Method 1: Checking with Different Variables

Let A be the angle, S be the supplement, and C be the complement. According to the problem, S 3C - 6. Using the definitions of supplement and complement, we can write:

S 180° - A

C 90° - A

Plugging these values into the equation S 3C - 6, we get:

180° - A 3(90° - A) - 6

Simplifying this, we find:

180° - A 270° - 3A - 6

2A 270° - 186°

A 42°

We can check this result using the definitions of complement and supplement:

text{Complement} 90° - 42° 48°

text{Supplement} 180° - 42° 138°

138° - 6° 132° eq 138°

This shows that the first alternative method has an error, and the correct solution is still 75°.

Method 2: Using Alternative Variables and Equations

Let A denote the angle. Then, the supplement is 180° - A and the complement is 90° - A. According to the problem, we have:

180° - A 3(90° - A) - 6

Expanding and simplifying, we get:

180° - A 270° - 3A - 6°

2A 270° - 186°

A 42°

Again, this is incorrect, and the correct solution is still 75°.

Final Check and Verification

Let's check the solution using the correct values:

text{Complement} 90° - 75° 15°

text{Supplement} 180° - 75° 105°

frac{105°}{3} - 20° 35° - 20° 15°

This verifies that the solution is correct.

Conclusion

The measure of the angle is 75°. This detailed solution demonstrates the correct method for solving problems involving complementary and supplementary angles.