Rules for Calculating Supplementary & Complementary Angles

Definitions

We consider two angles complementary when their angle measures add up to a sum of 90°! Thus, these 2 angles are called complements of each other. 👍

To verify: 50 40 90 degrees! Image courtesy of Wikimedia

On the other hand, we consider two angles supplementary when their angle measures add up to a sum of 180°! In the same way, these 2 angles are called supplements of each other. ❗

To verify: 45 135 180 degrees!

Image courtesy of Wikimedia

💡 Note: In these definitions, it doesn’t matter whether the angles are adjacent or not! Only their angle measures matter. Any 60° angle and 30° angle are complementary, and any 130° angle and 50° angle are supplementary, regardless of where they are on a given diagram. 

Complementary Angles

The most common case of two angles being complementary is when they form a right angle. As noted previously, however, the angles do NOT have to be next to each other. ⚡

Right Triangles

In a right angled triangle, the two non-right angles are complementary. How so? Let's take a look at an image of a right triangle (ignore the letters).

Image courtesy of Wikipedia

In a right triangle, the 3 angles add up to 180 degrees. Because the right angle already takes up 90 degrees, the other 2 angles are complementary by default. This is due to the fact that the sum of their angle measures also has to equal 90 degrees.

(90° from right angle ◽) + (90° sum of the other two angles) = 180°!

Finding the Complement of an Angle

We know that the sum of the angle measures of two complementary angles is 90 degrees. Therefore, we can find the complement of an angle by subtracting it from 90 degrees. That means we can write the complement of an angle (x degrees) as 90 - x. ➖

For example, take a 67° angle. By taking 90° - 57°, we can find the measure of the complementary angle as 23°.

Supplementary Angles

The most common case of two angles being supplementary is when they form a straight line. However, as noted previously, the angles do NOT have to be next to each other to be supplementary. 📏

Finding the Supplement of an Angle

We established earlier that the sum of the angle measures of two supplementary angles is 180 degrees. Therefore, we can find the supplement of an angle by subtracting it from 180 degrees. The supplement of an angle (x degrees) can then be written as 180 - x. ➖

For example, take a 146° angle. By taking 180° - 146°, we can find the measure of the supplementary angle as 34°.

Tips for Remembering

Hard to memorize which is which? 😳

Think of it this way:

• the ‘C’ in Complementary can stand for “Corner,” which is a right angle! 📐

• the ‘S’ in Supplementary is for “Straight,” as in a straight line! 📏

Practice Problems

1. y is the complement of 67°. Find the value of y.

Remember that when we say an angle is a complement to another angle, it means that the sum of both angles is equal to 90°. 

Therefore, to find the value of y, we can take 90 - 67 = y.

In this way, you should get y = 23°.

2. If (x - 5) & (4x + 5) are supplementary angles, find the value of x. ❌

This problem involves a little more algebraic manipulation than most, but remembering the properties of supplementary angles is still crucial to solving it! Since both angles are supplementary, we know that the sum of both angle measures has to be equal to 180°. 

As a result, we can create this equation:

(x - 5) + (4x + 5) = 180

Then, finding the value of x is just a matter of algebraically solving for it! Both 5 and -5 will cancel out, and then you can simplify the equation to 5x = 180 by combining the like terms. Then, dividing both sides by 5 should give you x = 36.

In Summary

Two angles can be considered complements if the sum of their angle measures add up to 90°, and are supplements of each other if their angle sums add up to 180°! Good luck! 🤝