Solving Supplementary Angles with a 50-Degree Difference

Supplementary angles are a fundamental concept in Geometry. They are defined as two angles whose measures add up to 180 degrees. In this article, we'll delve into how to solve for these angles when the difference between them is given. Specifically, we'll explore a case where the difference between two supplementary angles is 50 degrees. By following a step-by-step approach, we'll unveil the measures of these angles.

Understanding Supplementary Angles

In Geometry, supplementary angles are pairs of angles that add up to 180 degrees. This property is often utilized in solving various geometric problems. Let's consider the problem where two supplementary angles differ by 50 degrees. This means if one angle is (x), the other will be (x-50).

Solving the Problem

Here's a detailed breakdown of the steps involved in solving for the angles:

Step 1: Setting Up the Equations

Let's denote the two supplementary angles as (x) and (y). By definition, we have:

x y 180 (Sum of supplementary angles) x - y 50 (Difference between the angles)

Step 2: Expressing One Variable in Terms of the Other

From the first equation, we can express (y) in terms of (x):

y 180 - x

Step 3: Substituting and Simplifying

Substitute (y) from step 2 into the second equation:

x - (180 - x) 50

Simplify the equation:

2x - 180 50

Add 180 to both sides:

2x 230

Divide by 2:

x 115

Step 4: Finding the Second Angle

Now that we have (x), we can find (y):

y 180 - x 180 - 115 65

Thus, the measures of the angles are:

x 115°

y 65°

Examples and Further Exploration

To further illustrate this, let's consider an example where the difference is 54 degrees, instead of 50 degrees. We follow the same steps:

x y 180 x - y 54

Solving these, we get:

2x 234 rarr; x 117

y 180 - x 180 - 117 63

Thus, the angles are:

x 117°

y 63°

Conclusion

Solving problems involving supplementary angles with a specific difference is a straightforward process involving algebraic manipulation. The key is setting up the correct equations and solving them systematically. Supplementary angles find applications in various areas of Geometry and real-life scenarios, making them an essential topic to master.

Thank you for reading this article. If you found it informative and are interested in more insights into geometry, feel free to explore more articles or practice problems on the subject. Feel free to share your thoughts or questions in the comments section below.