Complementary and Supplementary Angles: Definition, Examples

Complementary and Supplementary angles are defined for the addition of two angles. If the sum of two angles so formed is \({90^ \circ }\), then they are called complementary angles. The sum of two angles, so formed is \({180^ \circ }\), then they are known as supplementary angles.

When two rays or lines intersect at a point, the measure of the region (opening) between these two rays or lines is called an “Angle”.

Definition of Angles

When two rays or lines intersect at a point, the measure of the region (opening) between these two rays or lines is called an “Angle”. It is denoted by using the symbol \(\angle \) . Angles are usually measured in radians \(\left( {{\pi ^c}} \right)\) and degrees \(\left( \circ \right)\).

Types of Angles

Based on their measurement, there are different types of angles, which are mentioned below:

Definition of Complementary and Supplementary Angles

Complementary and Supplementary angles are defined for the pair of angles and the type decided by measuring their sum. 

Complementary Angles

Two angles are Complementary angles if their sum is \({90^ \circ }\). In another way, we can say that if two angles add up to form a right angle, then those angles are said to be complementary angles.

If the sum of angle \(1\) and angle \(2\) is Right angle \({\left( {90} \right)^ \circ }\), then two angles are said to be complementary angles and angle \(1\) and angle \(2\) complement each other.

\({\rm{Angle1 + angle 2 = 9}}{{\rm{0}}^ \circ }\)

Supplementary Angles

Two angles are Supplementary angles if their sum is \({180^ \circ }\). In another way, we can say that if two angles add up to form a straight angle, then those angles are said to be supplementary angles.

If the sum of angle \(1\) and angle \(2\) is a straight angle \({\left( {180} \right)^ \circ }\), then two angles are said to be supplementary angles, and angle \(1\) and angle \(2\) supplement each other.

\({\rm{Angle 1 + angle 2 = 18}}{{\rm{0}}^ \circ }\)

Examples of Complementary and Supplementary Angles

According to the definition, two angles are known as complementary, if their sum is \({90^ \circ }\) and they are called supplementary if their sum is \({180^ \circ }\). 

Let’s see the examples of complementary and supplementary angles below:

Example – Complementary Angles

Consider two angles \({60^ \circ }\) and \({30^ \circ }\) as shown below.

Here, the sum of two angles is a right angle.

\({60^ \circ } + {30^ \circ } = {90^ \circ }\)

Hence, given two angles are said to be complementary angles. Each angle is a complement to the other. Here

1. \({60^ \circ }\) is the complement of \({30^ \circ }\)
2. \({30^ \circ }\) is the complement of \({60^ \circ }\)

Example – Supplementary Angles

Consider two angles \({60^ \circ }\) and \({120^ \circ }\) as shown below.

Here, the sum of two angles is a straight angle.

\({60^ \circ } + {120^ \circ } = {180^ \circ }\)

Hence, given two angles are said to be supplementary angles. Each angle is a supplement to the other. Here

1. \({60^ \circ }\) is the supplement of \({120^ \circ }\)
2. \({120^ \circ }\) is the supplement of \({60^ \circ }\)

Adjacent Complementary and Supplementary Angles

Two angles are said to be adjacent if they have a common arm and the same starting point.

Adjacent Complementary Angles

Two complementary (Sum of two angles is \({90^ \circ }\)) angles with a common vertex, and a common arm are called adjacent complementary angles.

\(\angle COB\) and \(\angle AOC\) are adjacent angles from the above figure, have a common vertex \(“O” \) and a common arm \(“OC” \). They are also known as complementary angles, as their sum is a right angle.

\(\angle COB + \angle AOC = {35^ \circ } + {55^ \circ } = {90^ \circ }.\)

Thus, the two given angles in the above figure are adjacent complementary angles.

Adjacent Supplementary Angles

Two supplementary angles (Sum of two angles is \({180^ \circ }\)) with a common vertex and a common arm are called adjacent supplementary angles.

\(\angle BCA\) and \(\angle DCA\) are adjacent angles from the above figure, have a common vertex \(“C” \) and a common arm \(“AC” \). 

They are also known as supplementary angles, as their sum is a straight angle.

\(\angle BCA + \angle DCA = {60^ \circ } + {120^ \circ } = {180^ \circ }.\)

Thus, the two given angles in the above figure are adjacent supplementary angles.

Non-Adjacent Complementary and Supplementary Angles

The two angles are not adjacent to each other, which means they do not have any common arm or common vertex are said to be non-adjacent angles.

Non-Adjacent Complementary Angles

The figure given below shows that \(\angle AOB\) and \(\angle XOY\) are non-adjacent angles as they do not have a common vertex and a common arm. 

Also, the sum of the two angles is  \(\angle AOB + \angle XOY = {55^ \circ } + {35^ \circ } = {90^ \circ }\)

Thus, the given two angles in the above figure are non-adjacent complementary angles.

Non-Adjacent Supplementary Angles

The figure given below shows that \(\angle AOB\) and \(\angle XOY\) are non-adjacent angles as they do not have a common vertex and a common arm. 

Also, the sum of the two angles is \(\angle AOB + \angle XOY = {120^ \circ } + {60^ \circ } = {180^ \circ }\)

Thus, the given two angles in the above figure are non-adjacent supplementary angles.

Complementary and Supplementary Angles Theorem

Let us discuss the theorems of both types of angles individually.

Complementary Angles Theorem

Statement:

If two angles are complementary to the same angle, then they are congruent to each other.

Proof:

Let \(\angle ZOW\) is complementary to angles \(\angle XOZ\) and \(\angle WOY\). Such that,

\(\angle ZOW + \angle XOZ = {90^ \circ }\)

\( \Rightarrow \angle XOZ = {90^ \circ } – \angle ZOW\) ………(1)

And, \(\angle ZOW + \angle WOY = {90^ \circ }\)

\( \Rightarrow \angle WOY = {90^ \circ } – \angle ZOW – \)……..(2)

From (1) and (2)

\(\angle XOZ = \angle WOY\)

Supplementary Angles Theorem

Statement:

If two angles are supplementary to the same angle, then they are congruent to each other.

Proof:

Let \(\angle POB\) is supplementary to angles \(\angle AOP\) and \(\angle QOB\), Such that,

\(\angle AOP + \angle POB = {180^ \circ }\)

\( \Rightarrow \angle AOP = {180^ \circ } – \angle POB\)………(1)

And \(\angle POB + \angle QOB = {180^ \circ }\)

\( \Rightarrow \angle QOB = {180^ \circ } – \angle POB – \)

From (1) and (2)

\(\angle QOB = \angle AOP\)

Real-Life Examples of Complementary and Supplementary Angles

There are many examples of complementary and supplementary angles. Some of them are listed below:

Real-Life Examples of Complementary Angles

1. A slice of pizza
2. A crossroad
3. Hands of a clock showing \(3PM\) and the seconds hand pointing towards the digit \(2\)
4. A staircase that has an escalation of \({30^ \circ }\) and the wall at \({90^ \circ }\)

Real-Life Examples of Supplementary Angles

1. Consecutive angles in a parallelogram
2. Cock showing \(9’o\) clock and \(3’o\) clock
3. Lines in a tennis court

Differences Between Complementary and Supplementary Angles

The supplementary and complementary angles exist in pairs, add up to \(180\) and \(90\) degrees. Let’s have a look at the difference between them.

Tricks for Complementary and Supplementary Angles

1. The letter \(‘C’\) for ‘complementary’ and \(‘C’\) for ‘corner’. Hence, when two complementary angles put together to form a ‘corner (right)’ angle. 
2. The letter \(‘S’\) for ‘supplementary’ and \(‘S’\) for ‘straight’. Hence, when two supplementary angles are put together, they form a ‘straight’ angle. 
3. The letter \(‘C’\) in complementary comes before the letter \(‘S’\) in supplementary, just like \({{{90}^ \circ }}\) comes before \({180^ \circ }\).

Solved Examples– Complementary and Supplementary Angles

Q.1. Given angles are complementary angles. Find the value of \(x\).

Ans: We know that sum of two complementary angles is \({90^ \circ }\).
According to the question, \({38^ \circ } + x = {90^ \circ }\)
\( \Rightarrow x = {90^ \circ } – {38^ \circ }\)
\( \Rightarrow x = {52^ \circ }\)
Hence, the value of \(x\) is \({52^ \circ }\)

Q.2.  Below given two angles are complementary. Find the measures of two angles.
Ans:

Ans: Given two angles are complementary.
The sum of the two complementary angles is \({90^ \circ }\).
\( \Rightarrow \frac{x}{2} + \frac{x}{4} = {90^ \circ }\)
\( \Rightarrow \frac{{2x + x}}{4} = {90^ \circ }\)
\( \Rightarrow \frac{{3x}}{4} = {90^ \circ }\)
\( \Rightarrow 3x = {90^ \circ } \times 4\)
\( \Rightarrow x = \frac{{{{360}^ \circ }}}{3}\)
\( \Rightarrow x = {120^ \circ }\)
So, the angles are \(\frac{{{{120}^ \circ }}}{2} = {60^ \circ }\) and \(\frac{{{{120}^ \circ }}}{4} = {30^ \circ }\)
Hence, the measures of angles are \({60^ \circ },{30^ \circ }.\)

Q.3. The difference between two complementary angles is \({48^ \circ }\). Find both the angles.
Ans: Let one of the angles be \(x\)
Then, the other angle (complement of angle) is \({90^ \circ } – x\)
Given that, the difference between two complementary angles is \({48^ \circ }\)
\( \Rightarrow \left( {{{90}^ \circ } – x} \right) – x = {48^ \circ }\)
\( \Rightarrow {90^ \circ } – 2x = {48^ \circ }\)
\(\Rightarrow 2x = {90^ \circ } – {48^ \circ }\)
\( \Rightarrow 2x = {42^ \circ }\)
\( \Rightarrow x = \frac{{{{42}^ \circ }}}{2} = {21^ \circ }\)
Another angle is \({90^ \circ } – {21^ \circ } = {69^ \circ }\)
Hence, the required complementary angles are \({21^ \circ },{69^ \circ }\)

Q.4. Two supplementary angles are \(\frac{x}{2}\)  and \(\frac{x}{4}\). Find the measures of the two angles.
Ans: Given two angles are supplementary.
The sum of the two supplementary angles is \({180^ \circ }\)
\( \Rightarrow \frac{x}{2} + \frac{x}{4} = {180^ \circ }\)
\( \Rightarrow \frac{{2x + x}}{4} = {180^ \circ }\)
\( \Rightarrow \frac{{3x}}{4} = {180^ \circ }\)
\( \Rightarrow 3x = {180^ \circ } \times 4\)
\(\Rightarrow x = \frac{{{{720}^ \circ }}}{3}\)
\( \Rightarrow x = {240^ \circ }\)
So, the angles are \(\frac{{{{240}^ \circ }}}{2} = {120^ \circ }\) and \(\frac{{{{240}^ \circ }}}{4} = {60^ \circ }\)
Hence, the measures of angles are \({60^ \circ },{120^ \circ }\)

Q.5. The difference between two supplementary angles is \({70^ \circ }\). Find both the angles.
Ans: Let one of the angles be \(x\).
Then the other angle (supplement of angle) is \({180^ \circ } – x\)
Given that, the difference between two supplementary angles is \({70^ \circ }\)
\( \Rightarrow \left( {{{180}^ \circ } – x} \right) – x = {70^ \circ }\)
\( \Rightarrow {180^ \circ } – 2x = {70^ \circ }\)
\( \Rightarrow 2x = {180^ \circ } – {70^ \circ }\)
\( \Rightarrow 2x = {110^ \circ }\)
\( \Rightarrow x = \frac{{{{110}^ \circ }}}{2} = {55^ \circ }\)
Another angle is \({180^ \circ } – {55^ \circ } = {125^ \circ }\)
Hence, the required complementary angles are \({125^ \circ },{55^ \circ }.\)

Summary

The word ‘complementary’ came from the Latin word ‘completum’, meaning ‘completed’. To become complementary angles, the two angles do not need to be adjacent. However, if they are adjacent, they will form a right angle.

The word ‘supplementary’ came from the Latin word ‘supplere’, meaning ‘supply’. Similar to complementary angles, the two angles do not need to be adjacent. However, if they are adjacent, they will form a straight angle.

In this article, we have studied definitions, types, examples, theorems and real-life applications of complementary and supplementary angles.

Frequently Asked Questions (FAQs) – Complementary and Supplementary Angles

Q.1.Can three angles be Supplementary?
Ans: No, three angles can never be supplementary even though their sum is \(180\) degrees. Though the sum of angles, \({40^ \circ },{50^ \circ }\) and \({90^ \circ }\) is \({180^ \circ }\), they are not supplementary angles because supplementary angles always occur in pair. Therefore, the definition of supplementary angles holds only for two angles.

Q.2. Are all supplementary angles are linear pairs?
Ans: All linear pairs of angles are supplementary. But all the pairs of supplementary angles are not linear pairs. Only those pairs of supplementary angles are linear pairs that originate from a common point and share a common side.

Q.3. Explain complementary angles and supplementary angles with example?
Ans: Two angles are Complementary angles if their sum is \({90^ \circ }\).
Example: Angles \({60^ \circ },{30^ \circ }\) are complementary angles.
Two angles are Supplementary angles if their sum is \({180^ \circ }\).
Example: Angles \({65^ \circ },{115^ \circ }\) are complementary angles.

Q.4. Can \(3\) angles be complementary?
Ans: No, complementary angles are defined only for pair of angles, such that the sum of two angles is \({90^ \circ }\), then those angles are complementary. So, three angles cannot be complementary.

Q.5. How do you find complementary angles?
Ans: We know that the sum of two complementary angles is \(90\) degrees, and each of them is said to be a “complement” of each other. Thus, the complement of an angle is found by subtracting it from \(90\) degrees. Therefore, the complement of \({x^ \circ }\)  is \(90 – {x^ \circ }\).

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