Solving for Complementary Angles: A Guide to Finding Measures of Angles

When dealing with geometric shapes and triangle properties, it is often necessary to solve for the measures of complementary angles. Complementary angles are two angles whose measures add up to 90 degrees. In this guide, we will solve a specific problem related to complementary angles to demonstrate the process step-by-step.

Problem Statement

An angle is forty-four times the measure of its complementary angle. What are the measurements of each angle?

Solution Approach

Define Variables: Let the measure of the complementary angle be x. The angle that is forty-four times this complementary angle can be expressed as 44x.

Set Up Equation: Since the two angles are complementary, their measures add up to 90 degrees. We can write this as: x 44x 90

Combine Like Terms: 45x 90

Solve for x: x frac{90}{45} 2

Find the Measurement of the Other Angle: 44x 44 times 2 88

Conclusion

We have determined the measures of the angles as follows:

Complementary Angle: 2 degrees Angle Fourteen Times the Complementary Angle: 88 degrees

Additional Examples

Example 1

Let two angles be x and y, and they are complementary. So, xy 90 and x - y 64. By adding both equations, we get:

2x 154

x frac{154}{2} 77

y 90 - 77 13

The measures of the angles are: 77 degrees and 13 degrees.

Example 2

Let the two angles be a and b. Since the angles are complementary, ab 90. Given a - b 64. Adding the two equations:

2a 154

a frac{154}{2} 77

b 90 - 77 13

The measures of the angles are: 77 degrees and 13 degrees.

Example 3

Let 'x' and 'y' be the complementary angles. x y 90 and x - y 64. Solving for x and y:

x 77 degrees

y 13 degrees

The measures of the angles are: 77 degrees and 13 degrees.

Summary

Complementary angles are a fundamental concept in geometry, and solving for their measures often involves simple algebraic equations. The examples provided demonstrate how to apply the basic principles of complementary angles to solve a variety of problems.