Finding the Value of a Small Angle in Supplementary Angles: A Comprehensive Guide

In geometry, two angles are supplementary if their measures add up to 180 degrees. This relationship forms the basis of several geometric problem-solving scenarios, one of which involves the use of ratios to determine the measure of angles. In this article, we will explore the process of finding the value of a small angle given that two supplementary angles are in the ratio 3:2.

Solving for the Small Angle

The method of solving for the small angle in supplementary angles that are in the ratio 3:2 involves a step-by-step process:

Step 1: Setting Up the Equation

We begin by letting one angle be (2x) and the other angle be (3x), where (x) is a common multiplier. Since these angles are supplementary, their sum must equal 180 degrees:

2x 3x 180

Simplifying the equation:

5x 180

Solving for (x):

x frac{180}{5} 36

Step 2: Determining the Angles

Once we have the value of (x), we can determine the measures of the two angles:

The smaller angle:

2x 2 times 36 72 text{ degrees}

The larger angle:

3x 3 times 36 108 text{ degrees}

Alternative Approach

Another way to approach the problem involves using the concept of the highest common factor (HCF). Here, we let the HCF of the two angles be (x). Thus, the angles can be represented as (3x) and (2x). The sum of these angles must be 180 degrees:

3x 2x 180

Simplifying the equation:

5x 180

Solving for (x):

x frac{180}{5} 36

Using the value of (x), we can determine the measures of the angles:

The smaller angle:

2x 2 times 36 72 text{ degrees}

The larger angle:

3x 3 times 36 108 text{ degrees}

Conclusion

Understanding the relationship between supplementary angles and their ratios is essential for solving geometric problems. By following the steps outlined above, you can determine the values of angles given their supplementary relationship and their ratio. Whether you use algebraic methods or the HCF approach, the key is to ensure that the sum of the angles equals 180 degrees.

For further understanding and practice, consider exploring more problems involving supplementary angles and ratios of angles in geometry.