Solving Complementary Angle Problems: A Step-by-Step Guide

Complementary angles are a fundamental concept in geometry and are widely used in various applications, from basic geometry to advanced trigonometry. Understanding how to solve problems involving complementary angles can be incredibly useful. Let's delve into a typical problem involving complementary angles and see how to solve it step-by-step using algebraic techniques.

Problem Statement

Two angles are complementary. One angle is 6 less than 3 times the other. What are both angle measures?

Step 1: Define the Variables

Let us denote the measures of the two angles as x and y.

Step 2: Establish the Equation for Complementary Angles

Since the angles are complementary, we have the following equation:

x y 90

Step 3: Establish the Relationship Between the Angles

According to the problem, one angle is 6 less than 3 times the other. We can express this relationship as:

y 3x - 6

Step 4: Substitute and Solve the Equations

NEXT, we substitute the expression for y from the second equation into the first equation:

x (3x - 6) 90

Now, simplify and solve for x:

x 3x - 6 90

4x - 6 90

4x 96

x 24

Step 5: Find the Value of the Second Angle

Now, use the first equation to find the value of y:

y 90 - x

y 90 - 24 66

Final Answer

Thus, the measures of the angles are:

The two angles are 24° and 66°.

Explanation of the Solution

In this example, we used the properties of complementary angles (which sum to 90°) and algebraic substitution to solve for the variables. This method is commonly used in solving angle-related problems and can be extended to more complex scenarios involving multiple steps and variables.

Conclusion

Understanding and applying the concepts of complementary angles is crucial in geometry and trigonometry. By breaking down the problem into manageable steps and using algebraic techniques, we can efficiently solve for the measures of angles.

Additional Resources

For more practice and detailed explanation, consider exploring online tutorials, geometry workbooks, and educational videos. These resources can provide additional insights and help solidify your understanding of complementary angles and their applications.