How to Find Supplementary Angles in a 3:5 Ratio

Supplementary angles are a pair of angles that add up to 180 degrees. In this article, we will delve into how to find supplementary angles given a ratio of 3:5. This will not only help in understanding the relationship between angles but also in solving complex problems related to geometric shapes.

Understanding the Basics

To begin, it's crucial to know that supplementary angles always sum to 180 degrees. In this case, we have two angles in the ratio of 3:5. Let's denote these angles as 3x and 5x, where x is a common factor.

Setting Up the Equation

Since the angles are supplementary, we can set up the following equation based on their sum:

3x 5x 180

By simplifying the above equation, we get:

8x 180

Solving for x, we divide both sides of the equation by 8:

x frac{180}{8} 22.5

Now that we have the value of x, we can find the actual angles:

Calculating the Angles

The first angle is:

3x 3 times 22.5 67.5 text{ degrees}

The second angle is:

5x 5 times 22.5 112.5 text{ degrees}

Thus, the supplementary angles in the ratio of 3:5 are 67.5 degrees and 112.5 degrees. This solution can be confirmed by adding the two angles, which indeed sum to 180 degrees (67.5 112.5 180).

Visualizing and Understanding the Result

Imagine a straight line divided into 8 equal parts, each part being 22.5 degrees (frac{180}{8} 22.5). The first angle, 67.5 degrees, would span 3 of these parts, and the second angle, 112.5 degrees, would span 5 of these parts. This visualization helps in understanding the distribution of the angles and their supplementary nature.

Conclusion

Finding supplementary angles in a 3:5 ratio involves setting up a simple equation based on the sum of angles being 180 degrees. By solving for the common factor and then calculating the individual angles, we can easily find the angles that fit the given ratio. This technique is not only useful in basic geometry but also in more complex problems involving trigonometry and geometric shapes.

Keywords: supplementary angles, ratio 3:5, angle calculation