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Supplementary Angles

The word "supplementary" comes from two Latin words "Supplere" and "Plere" where "Supplere" means "supply”, whereas "Plere" means "fill". So, "supplementary" means "something when supplied to complete a thing". And so are supplementary angles, a pair of two angles forming a straight angle (180 degrees) when they are put together. These two angles are certainly called supplements of each other.

Measuring angles and finding angles are the most frequently carried out steps in Geometry. However, in order to do so, you need to have your geometrical concepts in place. In this lesson, we will explore the world of supplementary angle, which holds an important application in solving various geometry problems. The journey will take us through us pplementary angles, and the difference between supplementary angles v/s complementary angles. 

1. What are Supplementary Angles?
2. Adjacent and Non-Adjacent Supplementary Angles
3. Supplementary V/S Complementary Angles
4. How to find the Supplement of an Angle?
5. FAQs on Supplementary Angles

What are Supplementary Angles?

Two angles are said to be supplementary angles if they add up to 180 degrees. Supplementary angles form a straight angle (180 degrees) when they are put together. In other words, angle 1 and angle 2 are supplementary, if Angle 1 + Angle 2 = 180o

In this case, Angle 1 and Angle 2 are called "supplements" of each other. In the below figure, 130o + 50o = 180o.Hence, by the definition of supplementary angles, these two angles are supplementary.    

Example of supplementary angles

Adjacent and Non-Adjacent Supplementary Angles

Supplementary angles can either be adjacent or non-adjacent. So, there are two types of supplementary angles. Each of these types of supplementary angles are explained below.

  • Adjacent supplementary angles
  • Non-adjacent supplementary angles

Adjacent Supplementary Angles

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

Example:

Adjacent supplementary angles: 110 and 70 form a straight angle

Here, ∠COB and ∠AOB are adjacent angles as they have a common vertex, O, and a common arm OB. They also add up to 180 degrees, that is, ∠COB + ∠ AOB = 70o + 110o = 180o. Hence, these two angles are adjacent supplementary angles.

Non-adjacent Supplementary Angles

Two supplementary angles that are NOT adjacent are said to be non-adjacent supplementary angles.

Example:

Non adjacent supplementary angles: angle ABC is 79 and angle PQR is 101.

Here, ∠ABC and ∠PQR are non-adjacent angles as they neither have a common vertex nor a common arm. They also add up to 180 degrees, that is, ∠ABC+ ∠PQR = 79o + 101o = 180o. Hence, these two angles are non-adjacent supplementary angles. Non-adjacent supplementary angles, when put together, form a straight angle.

Supplementary V/S Complementary Angles

The supplementary and complementary angles are angles that exist in pairs, summing up to 180 and 90 degrees, and have numerous real-time applications, most common being the crossroads. Let's have a look at the difference between them.

The supplementary vs complementary angles table.

Complementary Angles

 Supplementary Angles

Two angles are said to be complementary if they add up to 90 degrees.

 Two angles are said to be supplementary if they add up to 180 degrees.

Complement of an angle x is (90 - x)o

 Supplement of an angle x is (180 - x)o

Here is a short trick for you to understand the difference between supplementary angles and complementary angles.

  • "S" is for "Supplementary" and "S" is for "Straight." Hence, you can remember that two "Supplementary" angles when put together form a "Straight" angle.
  • "C" is for "Complementary" and "C" is for "Corner." Hence, you can remember that two "Complementary" angles when put together form a "Corner (right)" angle.

How to find the Supplement of an Angle?

When the sum of two pairs of angles is equal to 180°, then we call that pair of angles, supplements of each other. So, we know that the sum of two supplementary angles is 180 degrees, and each of them is said to be a "supplement" of the other. Thus, the supplement of an angle is found by subtracting it from 180 degrees. This means the supplement of x° is (180 - x)°

For example, the supplement of 77° is obtained by subtracting it from 180o. Thus, its supplement is (180-77)° = 103°.

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Supplementary Angles Examples

  1. Example 1:

    Find angle Y in the following figure.

    supplementary angles example

    Solution

    In the given figure, Y and 77o are supplementary as they lie at a point on a straight line. Hence, their sum is 180o.

    Y + 77o = 180o 

    Y = 180o - 77o = 103o

    Therefore, Y = 103o

  2. Example 2:

    Find the values of Angle A and Angle B, if Angle A and Angle B are supplementary such that ∠A = 2x + 10 and Angle B = 6x − 46

    Solution

    Since Angle A and Angle B are supplementary, their sum is 180o

    ∠A + ∠B = 180o

    (2x + 10) + (6x - 46) = 180o

    8x - 36 =180

    8x = 216

    x = 27

    Therefore, Angle A = 2(27)+10 = 64o and Angle B = 6(27) - 46 =116o

  3. Example 3:

    Find the value of x if the following two angles are supplementary. 

    Non-adjacent supplementary angles where one is x over 2 and the other is x over 3

    Solution

    Since the given two angles are supplementary, their sum is 180o.

    x/2 + x/3 =180o

    5x/6 =180o

    x =180o × 6/5 = 216o

    Therefore, the value of x is 216o

  4. Example 4:

    Two angles are supplementary. The measure of the larger angle is 5 degrees more than 4 times the measure of the smaller angle. What is the measure of the larger angle in degrees?

    Solution

    Let us assume that the two supplementary angles are x (larger) and y (smaller).

    By the given information, x = 4y+5

    Since the two angles are supplementary, their sum is 180o.

    x + y =180o

    (4y + 5) + y =180o

    5y + 5 =180o

    5y = 175o = 35o

    Therefore, the larger angle is, x = 4(35) + 5 = 145o

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FAQs on Supplementary Angles

What are Supplementary Angles in Geometry?

In geometry, the two angles are said to be supplementary angles if they add up to 180 degrees. Such as, ∠A + ∠B = 180°.    

What is the Opposite of Supplementary?

The supplementary angles refer to the pair of two angles whose sum is equal to 180o.  The opposite of "supplementary" in math can be "complementary." 

Can Two Acute Angles be Supplementary Angles?

No, if two angles are supplementary, then they are both either right angles or one of them is acute and one of them is obtuse. If two acute angles are put together, their sum will always be less than 180o, so two acute angles can never be supplementary angles.

Can Two Obtuse Angles be Supplementary Angles?

No, if two angles are supplementary, then they are both either right angles or one of them is acute and one of them is obtuse. If two obtuse angles are put together, their sum will always be greater than 180o, so two obtuse angles can never be supplementary angles.

Can Two Right angles be Supplementary Angles?

No, if two angles are supplementary, then they are both either right angles or one of them is acute and one of them is obtuse.

Are Supplementary and Complementary Angles the Same?

No, supplementary and complementary angles are not the same. Two angles form a pair of complementary angles when their sum is 90o whereas two angles form a pair of supplementary angles when their sum is 180o.

Can Three angles be Supplementary?

No, three angles can never be supplementary even though their sum is 180 degrees. Though the sum of angles, 40o, 90o and 50o is 180o, they are not supplementary angles because supplementary angles always occur in pair. The definition of supplementary angles holds true only for two angles.

What Angle is Supplementary to 84 Degrees?

The supplement of an angle is obtained by subtracting it from 180 degrees. Thus the angle that is supplementary to 84 degrees is 180o - 84o = 96o.

What Angle is Formed if we Put the Supplementary Angles Together?

When a pair of supplementary angles are put together, they form a straight angle. 

Complementary angles form what Type of Angle?

Two angles form a pair of complementary angles when their sum is 90o. So, the pair of complementary angles form a right angle.

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