Understanding Complementary Angles in a 2:3 Ratio

Complementary angles are two angles whose sum is 90°. When these angles are expressed in a given ratio, like 2:3, we can determine their specific measures. This article will walk you through the process of solving for the measures of such complementary angles.

Solving for Complementary Angles with a 2:3 Ratio

Method 1: Using a Numerical Approach

Let's consider a pair of complementary angles with a ratio of 2:3. This means that the angles can be represented as a fraction of their sum, 90°, in the ratio 2:3. To find the angles, we can use the following steps:

Express one angle as a fraction of the total 90°. Since the ratio is 2:3, the first angle (A) can be written as 2/5 of 90°. Calculate the first angle: A (2/5) times 90° 36° Similarly, the second angle (B) can be expressed as 3/5 of 90°. Calculate the second angle: B (3/5) times 90° 54° Verify the sum of the angles: 36° 54° 90°. Since the angles sum to 90°, they are indeed complementary.

Method 2: Using Algebraic Representation

We can also solve for the angles using algebra. Let A and B represent the two complementary angles. Given that the ratio of A to B is 2:3, we can write:

Equation Setup

A and B being complementary angles, we know that:

A B 90°

Given the ratio, we can express A and B as follows:

A (2/5) * 90°

B (3/5) * 90°

Calculating these values:

A (2/5) * 90° 36°

B (3/5) * 90° 54°

Again, we can verify that 36° 54° 90°, confirming that the angles are complementary.

Additional Solved Problem

Consider the following problem:

Problem Statement

If the ratio of two complementary angles is 2:3, what are the angles?

Solution:

Let A and B denote the required two complementary angles such that:

A:B 2:3

A and B being two complementary angles, we have A B 90°. Therefore:

A (2/5) * 90° 36°

B (3/5) * 90° 54°

Thus, the angles are 36° and 54°, which sum to 90°, making them complementary.

This method confirms the earlier calculations and ensures that the solution is accurate.

Verifying the Solution

To further verify our answer, we can perform a quick mental check:

The first angle: 2/5 of 90° 36° The second angle: 3/5 of 90° 54° Sum of the angles: 36° 54° 90°

Since the sum of the angles is 90°, the solution is correct.

Conclusion

In conclusion, complementary angles with a 2:3 ratio can be solved using simple numerical and algebraic methods. The steps involve expressing the angles as a fraction of their total sum and then verifying the sum to ensure that it is 90°. By understanding these methods, you can easily find the measures of any complementary angles expressed in a given ratio.