Supplementary Angles

A supplementary angle is when the sum of any two angles is 180°. The supplementary angles always form a linear angle that is 180° when joined. In other words, if angle 1 + angle 2 = 180°, angle 1 and angle 2 will be called supplementary angles.

Let’s understand it better with the help of an example:

If we have a 130° and 50° angle, we know that added together results in 180°, which means they are supplementary angles.

Criteria for Supplementary Angles

Two angles can be called supplementary if:

  • One of its angles is an acute angle and other angle is an obtuse angle
  • Both of the angles are right angles

As you go through, you will explore a world of supplementary angles with important applications in solving various geometric problems.

Let’s look at some examples of supplementary angles to get a better understanding:

  • 150° + 30° = 180°
  • 130° + 50° = 180°
  • 96° +84° = 180°
  • 120° + 60° = 180°
  • 90° + 90° = 180°

Now when you look at the above examples you will find that there will always be an obtuse angle (150°, 130°, 96°, 120°) added with an acute angle (60°, 90°, 40°, 84°) or there will be a sum of two right angles (90°, 90°).

How to Find a Supplementary Angle

Let’s try to find the supplementary angle measurement for each measured angle.

Example 1

X° + 38° = 180°

X° = 180° - 38°

X° = 142°

Example 2

Y° + 77° = 180°

Y° = 180° - 77°

Y° = 103°

Example 3

X° + 130° = 180°

X° = 180° - 130° X° = 50°

Example 4

X° + 118° = 180° X° = 180° - 118° X° = 62°

Properties of Supplementary Angles

Supplementary angles all share the same properties:

  • Any two angles are said to be supplementary angles when their sums add up to 180°
  • The “S’ in supplementary angles stand for Straight line, this means that they form 180°
  • The supplementary angles do not have to be on the same line, they can be on different lines, but should measure 180°
  • In a supplementary angle if one angle is 90°, then the other angle will also be 90°.

When two supplementary angles are joined together, they form a straight line and a straight angle. But also note that even if two angles are supplementary to each other, they do not have to be next to each other. Hence, any two angles can be supplementary angles, if their sum is equivalent to 180°.

One of the most asked questions is whether all supplementary angles form linear pairs?

Supplementary angles do not have to be adjacent, but linear pairs must be adjacent to form a straight line. So, remember that supplementary angles are not necessarily linear pairs. However, linear pairs are always supplementary.

Types of Supplementary Angles

Adjacent and non-adjacent supplementary angles are the two types of supplementary angles. Each of these types of supplementary angles is explained below.

Adjacent Supplementary Angles

Two supplementary angles that have a common vertex and a common arm are said to be adjacent supplementary angles.

Example

Here ∠ BOA and ∠ AOC are adjacent angles as they have a common vertex, O, and a °common arm OA. These two angles add up to 180 degrees, that is ∠ BOA + ∠ AOC = 180°. Hence, these two angles can be called adjacent supplementary angles.

Non-Adjacent Supplementary Angles

Put simply, two supplementary angles that are not adjacent are said to be non-adjacent supplementary angles.

Example

Here, ∠ ABC and ∠ PQR are non-adjacent angles as they neither have a common nor a common arm. But when added up, that is 130° + 50° = 180°, their sum comes as 180°. Hence, they are supplementary angles but not adjacent. Note that, when two adjacent supplementary angles are put together, they form a straight line.

Supplementary Angles Formula

You can calculate supplementary angles by subtracting the specified angle from 180 degrees. To find other angles, use the following formula:

∠x = 180°-∠y or ∠y = 180° -∠x

Where ∠x or ∠y is the given angle.

Supplementary Angles Theory

According to the supplementary angles theorem, if two angles are supplementary to the same angle, then they are congruent to each other.

Examples

  • Angle A + Angle B = 180°
  • Angle C + Angle B = 180°

Therefore, Angle A is congruent to Angle C.

Now let’s go through some quick solved examples and test your knowledge:

Example 1

There are two angles that are supplementary and one of the two angles is 50°. Find the other angle.

Solution: Let’s take the other angle to be ∠ x.

Now, given that the two angles are supplementary and we know that the sum of the measures of these two angles is 180 degrees so:

∠ x + 50° = 180°

∠ x = 180° -50°

∠ x = 130°

So, the other angle is 130°. Here One angle is obtuse i.e., 130°, and one is acute i.e., 50°.

Example 2

Two angles are supplementary and one of them is 137°. What is the size of another angle?

Say we have angle ∠ b given i.e., 137°. Let’s take the other unknown angle to be ∠ a.

Now that we know that the sum of angles ∠ a + ∠ b will always be 180 degrees as it is a supplementary angle.

∠ a + 137° = 180°

So, the other angle is ∠ a = 43°. Here, obtuse angle is 137° and acute angle is 43°.

Supplementary Angles in Ratios

We can also work out supplementary angles in ratios.

Example

Two supplementary angles are in the ratio 4: 5. Find the other angles?

We know that supplementary angles add up to form 180°, so let’s find the other angles using the 4:5 ratio.

First, we’ll call one angle 4x and the other 5x. Now we can put this into the supplementary angles formula.

4x + 5x = 180°

9x = 180°

x + \dfrac{180°}{9}

x = 20°

Now we know that x is 20°, we can easily work out both angles in the ratio:

4x = 4 \times 20° = 80°

 5x = 5 \times 20° = 100°

Supplementary Angles vs Complementary Angles

Sometimes, supplementary angles are confused with complementary angles. Let’s have a quick go through and find the differences between Supplementary and complementary angles, so that you never confuse the two.

Supplementary AnglesComplementary Angles
The sum total of angles in supplementary angles is 180 degrees.    The sum total of complementary angles is always 90 degrees.  
∠a + ∠b = 180∠a + ∠ b = 90
The supplementary angles are in the form of a straight line.The complementary angles are in the form of right angle  
The supplement of any angle is 180 – the angle given.The compliment of any angle is 90 – the angle given.  
The sum total of angles in supplementary angles is 180 degrees.The sum total of complementary angles is always 90 degrees.