Complementary Angles: Finding the Measure When One is Twice as Large

Complementary angles are a fundamental concept in geometry. In this article, we will explore a problem involving complementary angles where one angle is twice the measure of the other. We'll provide a step-by-step solution to the problem, as well as some additional examples and insights into complementary and supplementary angles.

Understanding Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. This means if Angle A and Angle B are complementary, then Angle A Angle B 90°.

Solving a Specific Problem

Consider the following problem: Two angles are complementary. One angle is twice as large as the other. What is the number of degrees in each angle?

Step-by-Step Solution

Let's denote the smaller angle as x degrees. The larger angle, being twice as large, is 2x degrees. Since the angles are complementary, their sum must be 90 degrees. Therefore, we can write the equation:
x 2x 90 Simplifying the equation, we get:
3x 90 Solving for x, we divide both sides by 3:
x 30 The smaller angle is 30°. The larger angle is twice this, so it is:
2x 2(30) 60°

Verification

To verify, we check that the sum of the angles is indeed 90 degrees:
30° 60° 90°

Additional Problems and Insights

Let's solve a similar problem using different notations.

Problem 2:

Let the smaller angle be denoted as x degrees. The larger angle, being twice as large, is 2x degrees. Since the angles are complementary, we can write the equation:
x 2x 90

Following the same steps as before, we get:

3x 90 x 30 The smaller angle is 30°. The larger angle is 60°.

Here is another way to approach the problem:

Problem 3:

Let an angle be x degrees. Twice this angle plus six more degrees is equal to a complementary angle, meaning the total is 90 degrees. So we can write the equation:

x 2x 6 90

Simplifying, we get:

3x 6 90 3x 84 x 28 The smaller angle is 28°. The larger angle is 2(28) 6 62°.

Let's also consider a specific geometric context:

In a right triangle, the angles are 30° and 60°. These angles are complementary and form a right angle (90°). The sine and cosine of these angles have a special relationship in trigonometry. For example, sin 30° 0.5 and cos 60° 0.866. Squaring these values and summing them gives:

0.5^2 0.866^2 1

This demonstrates that these angles are not only complementary in the geometric sense but also in the trigonometric sense.

Conclusion

Understanding complementary angles and how to solve problems involving them is a crucial skill in geometry. Whether in a right triangle or other geometric contexts, the principles of complementary angles can be applied to find the measures of unknown angles. By following the steps outlined in this article, you can solve similar problems with confidence.