Supplementary Angles

Supplementary angles are those angles that sum up to 180 degrees. For example, angle 130° and angle 50° are supplementary angles because sum of 130° and 50° is equal to 180°. Similarly, complementary angles add up to 90 degrees. The two supplementary angles, if joined together, form a straight line and a straight angle.

But it should be noted that the two angles that are supplementary to each other, do not have to be next to each other. Hence, any two angles can be supplementary angles, if their sum is equal to 180°.

Geometry is one of the important branches of mathematics that deals with the study of different shapes. It initiates the study of lines and angles. A straight line is a line without curves and it is defined as the shortest distance between two points. An angle is formed when the line segment meets at a point.

What are Supplementary Angles?

In Maths, the meaning of supplementary is related to angles that make a straight angle together. It means, two angles are said to be supplementary angles when they add up to 180 degrees. Two angles are supplementary, if

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

This means that ∠A + ∠B = 180°.

See the figure below for a better understanding of the pair of angles that are supplementary.

Supplementary angles

Examples of Supplementary Angles

Some of the examples of supplementary angles are:

  • 120° + 60° = 180°
  • 90° + 90° = 180°
  • 140° + 40° = 180°
  • 96° + 84° = 180°

Properties of Supplementary Angles

The important properties of supplementary angles are:

  • The two angles are said to be supplementary angles when they add up to 180°.
  • The two angles together make a straight line, but the angles need not be together.
  • S” of supplementary angles stands for the “Straight” line. This means they form 180°.

Adjacent and Non-Adjacent Supplementary Angles

There are two types of supplementary angles:

  • Adjacent supplementary angles
  • Non-adjacent supplementary angles

Adjacent Supplementary angles

The supplementary angles that have a common arm and a common vertex are called adjacent supplementary angles. The adjacent supplementary angles share the common line segment and vertex with each other.

For example, the supplementary angles 110° and 70°, in the given figure, are adjacent to each other.

Adjacent Supplementary angles

Non-adjacent Supplementary angles

The supplementary angles that do not have a common arm and a common vertex are called non-adjacent supplementary angles. The non-adjacent supplementary angles do not share the line segment or vertex with each other.

For example, the supplementary angles 130° and 50°,  in the given figure, are non-adjacent to each other.

Non-adjacent supplementary angles

How to Find Supplementary Angles?

As we know, if the sum of two angles is equal to 180°, then they are supplementary angles. Each of the angles is said to be a supplement of another angle. Hence, we can determine the supplement of an angle, by subtracting it from 180°.

For example, if you had given that two angles that form supplementary angles. If one angle is ∠A then another angle ∠B is its supplement.  Hence,

∠A = 180° – ∠B (or)

∠B = 180° – ∠A

Supplementary Angles Theorem

The supplementary angle theorem states that if two angles are supplementary to the same angle, then the two angles are said to be congruent.

Proof:

If ∠x and ∠y are two different angles that are supplementary to a third angle ∠z, such that,

∠x + ∠z = 180 ……. (1)

∠y + ∠z = 180 ……. (2)

Then, from the above two equations, we can say,

∠x = ∠y

Hence proved.

Supplementary and Complementary Angles

Both supplementary and complementary angles are pairs of angles, that sum up to 180° and 90°, respectively.  Let us find more differences between the pair of angles.

Complementary Angles Supplementary Angles
Sum of two angles is 90° Sum of two angles is 180°
Ex: ∠A + ∠B = 90°. Ex: ∠A + ∠B = 180°.
Complementary angles form a right-angled triangle when combined together. Supplementary angles form a straight line.
The complement of an angle A is (90 – A)° The supplement of an angle A is (180 – A)°

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Problems and Solutions on Supplementary angles

Question 1: Find the measure of an unknown angle from the given figure.

Supplementary angles problem

Solution:

We know that the supplementary angles add up to 180°.

X + 55° + 40° = 180°

X + 95° = 180°

X = 180°- 95°

X = 85°

Therefore, the unknown angle, X = 85°

Question.2: If ∠x and ∠y are supplementary angles and ∠x = 67, then find ∠y.

Solution: Given,  ∠x and ∠y are supplementary angles

And 

∠x = 67°

Since, ∠x + ∠y = 180°

∠y = 180 – ∠x 

∠y = 180 – 67

∠y = 113°

Practice Questions

  1. What is the supplement of angle 65 degrees?
  2. Are the angles 80° and 120° supplementary?
  3. Find the supplement of 140°.

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Frequently Asked Questions – FAQs

Can 2 Acute Angles be Supplementary Angles?

No, two acute angles cannot form a supplementary angle.
By definition, acute angles are the angles that measure the angle greater than 0° and less than 90°. If you add two acute angles in which each angle is large as possible, its sum will be less than 180°. By the definition of supplementary angles, it is impossible to get the supplementary angle when we add two acute angles.
Example: 80° +60° = 140° which is not a supplementary angle.
But, in the case, if we add more than two acute angles, we can get supplementary angles.

Can 2 Obtuse Angles be Supplementary Angles?

No, two obtuse angles cannot form a supplementary angle.
By the definition of obtuse angles, the angles that measure greater than 90° are obtuse. If you add two obtuse angles, the sum will be greater than 180°. It will not satisfy the property of the supplementary angles when we add obtuse angles.
Example: 110° + 95° = 205° which is not a supplementary angle. [205° > 180° ]

Can 2 Right Angles be Supplementary Angles?

Yes, 2 right angles can form a supplementary angle. We know that when the measure of an angle is exactly 90°, then it is known as a right angle.
When two right angles are added, it is possible to get the supplementary angle. because 90° + 90° = 180°, as it satisfies the condition of supplementary angles.

Is supplementary and complementary angles same?

Supplementary and complementary both are different angles. Supplementary add upto 180 degrees whereas complementary angles add to 90 degrees.

What angle is formed if we put the supplementary angles together?

If supplementary angles are joined together, we get a straight angle.

Complementary angles form what type of angle?

When complementary angles are joined together, they form a right angle.

Quiz on Supplementary angles

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