Understanding Supplementary Angles in Geometry

Supplementary angles are a fundamental concept in geometry, where two angles add up to 180 degrees. This article explores finding the measures of such angles when a specific relationship is given. Let's delve into detailed examples to understand these geometric problems better.

Solving for Supplementary Angles with a Given Difference

Suppose we are given that an angle measures 123.2° less than its supplementary angle. Let's denote the measure of the angle as x. The supplementary angle would then be 180° - x.

Example 1: The First Angle Problem

According to the problem, the angle measures 123.2° less than its supplementary angle. We can express this mathematically as:

x 180° - x - 123.2°

Solution

Rearranging the equation to solve for x:

x x 180° - 123.2°

2x 56.8°

x 28.4°

Now, let's find the measure of the supplementary angle:

180° - x 180° - 28.4° 151.6°

Example 2: Another Approach to the Problem

Let's consider a similar problem where an angle is 133.4° less than its supplementary angle. Let x and y be the two angles such that x y - 133.4°. Since the angles are supplementary, their sum is 180°. We can write the equation as:

x y 180°

(y - 133.4°) y 180°

2y - 133.4° 180°

2y 313.4°

y 156.7°

Thus, the angle x can be found as:

x 156.7° - 133.4° 23.3°

You can verify that:

23.3° 156.7° 180°

Verification in Different Notations

Let's use another notation to solve the same problem. If Θ is the angle, then its supplement is:

180° - Θ

According to the problem, we have:

Θ 180° - Θ - 133.4°

Rearranging the equation to solve for Θ gives:

2Θ 180° - 133.4° 46.6°

Θ 23.3°

The supplement of this angle is:

180° - 23.3° 156.7°

The values 156.7° and 23.3° satisfy the condition that their difference is 133.4°.

General Formula for Supplementary Angles and Differences

Given that one angle is x and its supplementary angle is (180° - x), and the angle measures 123.2° less than its supplementary, we can write:

x (180° - x) - 123.2°

Rewriting for x gives:

x x 180° - 123.2°

2x 56.8°

x 28.4°

Similarly, if the angle measures 133.4° less than its supplementary:

x (180° - x) - 133.4°

Rewriting gives:

2x 180° - 133.4° 46.6°

x 23.3°

Thus, the angles are:

x 23.3° and 180° - 23.3° 156.7°

Conclusion

Through detailed solutions and examples, we have demonstrated how to solve for the measures of supplementary angles when provided with a specific numerical difference. Understanding these geometric concepts is essential for solving a wide range of problems in trigonometry and geometry.