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# Supplementary Angles

In middle school, the students realize that mathematics is subdivided into various branches. Geometry is one of them. Geometry is a vast study of shapes. It initiates with the study of lines and angles. A straight line is defined as the shortest distance between two points. While, an angle is formed when two line segments meet at a point. We often come across with the pair of angles. There are different classifications for pair of angles, such as - linear pairs, corresponding angles, vertically opposite angles, alternate angles, complementary angles, supplementary angles etc.

Here, we shall understand what a pair of supplementary angles is and what are its applications. Two angles are said to be supplementary angles if their sum is found to be
180 degrees. If X and Y are two supplementary angles, then $\angle$X + $\angle$Y = $180^{\circ}$.

Supplementary angle of $x^{\circ}$ is $180^{\circ}$ - x.Thus, if we have given that the two angles are supplementary and the measure of one angle is given, then we can easily find the measure of another angle. Supplementary angles may be either adjacent or non adjacent. The adjacent supplementary angles are those who share one line segment or arm with each other and whose sum is 180$^{\circ}$. Two nonadjacent supplementary angles are those which are separated and do not share any arm.

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## Definition of Supplementary Angles

Two Angles are called as supplementary angles, if their angles sum up to 180 degrees.

Two angles can be supplementary when:

a) One is an acute angle and the other one an obtuse angle

b) Both of them are right angles.
Supplementary angle = $\angle a$ + $\angle b$ = $180^o$.

$\angle$AOB + $\angle$ BOC = 180$^o$. These angles are also called as linear pair.

## Properties of Supplementary Angles

a) Adjacent angles are supplementary if their exterior sides lie in the same straight line.

b) If supplementary angles are congruent, each of them is a right angle (Equal supplementary angles are right angles).

c) Only a pair of angles is considered to be supplementary angles. If the measures of three or more angles adds up to 180 degrees, then the angles are not supplementary angles.

d) Supplementary angles need not share a common arm or vertex. If they do have a common arm and a common vertex, the other two arms of the angles will be in a straight line. This is because the sum of the measures of the angles will be 180$^o$, which is the angle of straight line.
If two angles are supplementary, one will be an acute angle and the other will be an obtuse angle or both of them will be right angles.

Two angles are said to be adjacent angles if they have a common vertex and a common arm. If two supplementary angles are adjacent to each other, they are called a linear pair and form a straight line.

## Theorem of Supplementary Angles

Given below are some of the theorems on supplementary angles:
Theorem 1:

If a straight line meets another straight line, the adjacent angles so formed are supplementary.

Data:

A straight line CO meets straight line AB at O.

To Prove:
$\angle$AOC + $\angle$BOC = $180^o$

Proof:

Theorem 2:
If two adjacent angles are supplementary, then their exterior arms lie in a straight line.

Data:

$\angle$AOC and $\angle$ BOC are adjacent angles.

$\angle$AOC + $\angle$BOC = 1800

To Prove:

AOB is a straight line.

Proof:

Corollary 1:

It two lines intersect, then the vertically opposite angles so formed are equal.

$\angle$ AOC = $\angle$ BOD and $\angle$ BOC = $\angle$ AOD .

Corollary 2:

When a number of lines meet at a point, the sum of the angles so formed is four right angles.

$\angle$a + $\angle$ b + $\angle$ c + $\angle$ d + $\angle$ e = 3600

## Supplementary Angles in Real Life

In real life, supplementary angles are seen in many places. Few examples for it are shown below:

## Examples of Supplementary Angles

Below you could see how to solve supplementary angles.

### Solved Examples

Question 1: Evaluate the supplementary angles for the angle 29 degree.
Solution:

Step 1: The given angle is 29 degrees.

Step 2: The supplementary angle of 29 degree is 180 - 29 = 151 degree.

Step 3: Therefore, 29 degree is the supplementary angle of 151 degree.

Question 2: Evaluate the supplementary angles for the angle 36 degree.
Solution:

Step 1: The given angle is 36 degree.

Step 2: The supplementary angle of 36 degree is 180 - 36 = 144 degree.

Step 3: Therefore 36 degree is the supplementary angle of 144 degree.

Question 3:

Find x from the following figure:

Solution:

$\angle$ABD + $\angle$CBD = x + 65$^o$ = 180$^o$

x = 180$^o$ - 65$^o$ = 130$^o$.

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