How to Find the Supplement of an Angle Given Relationship with its Complement

Understanding the properties and relationships of angles is crucial in geometry. This article delves into how to find the supplement of an angle when given its relationship with its complement. We will explore several examples and explanations that make this concept accessible and understandable.

Understanding Complementary and Supplementary Angles

In geometry, complementary angles are two angles whose measures add up to 90 degrees (a right angle), while supplementary angles are two angles whose measures add up to 180 degrees (a straight angle).

Example 1: Finding the Supplement When Given a Relationship with the Complement

Let's consider the scenario where an angle is 14 degrees more than its complement. We will find the supplement of this angle step by step.

Let the angle be x degrees. Then, its complement is 90 - x degrees. According to the problem, the angle is 14 degrees more than its complement:

x 90 - x 14

Simplify the equation:

x x - 90 14

2x 104

x 52

Thus, the angle is 52 degrees. The supplement of this angle is given by:

180 - x 180 - 52 128 degrees

Example 2: Solving with Equations

Consider the angle ∠A. Its complement is 90 - A degrees, and its supplement is 180 - A degrees. When the complement and supplement are added together, the result is 180 - (90 - A) (180 - A) 120 degrees.

Example 3: Algebraic Representation

Let the angle be x degrees. Its complement is 90 - x degrees, and its supplement is 180 - x degrees. The relationship can be written as:

(90 - x) (180 - x) 120

Example 4: Using Algebra and Transposition

Let the angle be x degrees. We know that the complement is 90 - x degrees, and the supplement is 180 - x degrees. The relationship given is:

(90 - x) (180 - x) 120

270 - 2x 120

2x 150

x 75 degrees

The supplement of the angle is therefore 180 - 75 105 degrees.

Example 5: Step-by-Step Calculation

Let the angle be x degrees. The complement of the angle is 90 - x degrees, and the supplement is 180 - x degrees. Given the relationship:

180 - x 2(90 - x) 126

180 - x 180 - 2x 126

x 126 degrees

The supplement of the angle is therefore 180 - 126 54 degrees.

Conclusion

By understanding the relationship between an angle, its complement, and its supplement, we can solve complex geometric problems. The examples provided demonstrate how to use algebraic techniques to find the supplement of an angle given its relationship with its complement.

These concepts are fundamental in geometry, and mastering them will help in solving more complex problems in trigonometry and other related fields. Whether you are a student or a professional, a strong grasp of these concepts is invaluable.