Solving Supplementary Angles with a 1:8 Ratio

Supplementary angles are two angles that add up to 180 degrees. When these angles are in a specific ratio, such as 1:8, we can use algebra to find their measures. In this article, we will explore how to solve for the angles given a 1:8 ratio and their supplementary nature.

Understanding the Problem

To find the angles that have a ratio of 1:8 and are supplementary, we need to set up and solve an equation. Let's start by assigning variables to the angles:

Solving the Angles with a 1:8 Ratio

Let the two angles be x and 8x.

Step 1: Since the angles are supplementary, their sum is 180 degrees:

x 8x 180°

Step 2: Simplify the equation to find x:

9x 180°

Step 3: Divide both sides by 9 to solve for x:

x frac{180°}{9} 20°

Step 4: Now find the other angle:

8x 8 times 20° 160°

Conclusion

The two angles that are supplementary and in the ratio of 1:8 are 20° and 160°.

Utilizing the 1:8 Ratio in Supplementary Angles

The process of solving for supplementary angles with a specific ratio involves the following steps:

Assign the angles a variable and a multiple of that variable. Set up an equation based on the supplementary angle property (sum equals 180 degrees). Solve the equation to find the variable. Substitute the variable back to find the other angle.

For example, if the ratio is 1:8, let the angles be x and 8x. The sum of the angles is 180 degrees:

x 8x 180°

Solving for x:

9x 180°

x frac{180°}{9} 20°

Then find the other angle:

8x 8 times 20° 160°

Example Solutions

Let's solve another problem using the 1:8 ratio and the concept of supplementary angles:

Given that the ratio of supplementary angles is 1:8, let the angles be x and 8x. We know that the sum of two supplementary angles is always 180 degrees:

x 8x 180°

This simplifies to:

9x 180°

Divide both sides by 9:

x frac{180°}{9} 20°

Therefore, the supplementary angles are:

x 20°

8x 8 times 20° 160°

Additional Examples for Practice

To further solidify your understanding of solving supplementary angles with a 1:8 ratio, try the following examples:

Find the supplementary angles with a ratio of 1:8 if one angle is 20 degrees. Solve for the angles if the ratio of supplementary angles is 1:8 and the sum is 180 degrees. Use the 1:8 ratio to find the supplementary angles when one angle is 160 degrees.

The key to solving these problems is understanding the properties of supplementary angles and the relationship between the angles given the ratio.

Conclusion

Solving supplementary angles with a 1:8 ratio involves setting up and solving a simple algebraic equation. By breaking down the problem step-by-step, we can find the exact measures of the angles. Understanding this process is crucial for mastering basic geometry and problem-solving skills.