Any two angles are called Supplementary angles when they add up to 180°. When two angles add to 180°, we say that the angles are “Supplement” to each other. In the pair of supplementary angles if one angle is x°, then its supplement is an angle of (180 - x)° .
Supplementary Angles
From the above figures, we have $\angle$ABD and $\angle$DBC are supplementary because 55°+125°=180°. $\angle$EFG and $\angle$HIK are supplementary because 74°+106°=180°. Also, $\angle$ABD and $\angle$DBC together form straight angle $\angle$ABC. Thus, if two angles are together and are supplement to each other then they have a common vertex and share one side. The other two non-shared sides of the angles form a straight line and have a straight angle. Such kind of angles are called as linear pair. The angles need not be together, but together they add up to 180°. For example, in geometrical figures, the adjacent angles of a rectangle are supplementary, and also the opposite angles of a cyclic quadrilateral are supplementary.

Two angles are supplementary, only when one will be an acute angle and the other will be an obtuse angle or both of them will be right angles.

Supplementary angles are seen in real life in many places. Few examples are shown here.

Supplementary Angles in Real Life

In the figure (i) we notice that the two streets meet at angle of A° and B° at the corners. These two angles sum to 180°. So, these kinds of intersections are examples of supplementary angles.

In the figure (ii) we see a tree which has two branches. We can see supplementary angles U° and D° which are the angle of a branch on a tree from the ground and the angle of that same branch from the sky. These angles add up to 180°, thus forming supplementary angles.

Consecutive angles of a four cornered room, consecutive angles of a four cornered window, consecutive angles of a piece of A4 sheet all are supplement angles.
Theorem 1:
If two straight lines meet, then the adjacent angles so formed are supplementary.
Given: A straight line AC meets straight line BD at B.

Supplementary Angles Theorem

To Prove: $\angle$ ABD + $\angle$ CBD = 180°

StatementReason
1. ∠ ABD + ∠ CBD = ∠ ABC ABC is a straight line
2. ∠ ABC = 180°Straight angle
3. ∠ ABD + ∠ CBD = 180°From (1) and (2)

Theorem 2:
If two adjacent angles are supplementary, then their exterior arms lie in a straight line.
Given: $\angle$ ABD and $\angle$ CBD are adjacent angles.
$\angle$ ABD + $\angle$ CBD = 180°

Supplementary Angle Theorem

To Prove: ABC is a straight line.
Proof:
StatementReason
1.Assume a straight line ABKAssumption
2. ∠ ABD + ∠ DBK = 180°From (1)
3. ∠ ABD + ∠ CBD = 180°Given
4. ∠ ABD + ∠ DBK = ∠ ABD + ∠ CBD From (1) and (2)
5. ∠ DBK = ∠ CBD
Which is not possible
From (4)
A part cannot be equal to whole
6. Assumption is wrongFrom (1)
7. Hence ABC is a straight lineFrom (5) and (6)

Theorem 3:
If two angles are each supplementary to a third angle, then they’re congruent to each other. In short, Supplements of the same angle are congruent.
Proof: Let $\angle$A and $\angle$B be supplementary angles. Thus, $\angle$B is a supplement of $\angle$A.
Then $\angle$A + $\angle$B = 180° ……..(1)
Let $\angle$A and $\angle$C be supplementary angles. Thus, $\angle$C is a supplement of $\angle$A.
Then $\angle$A + $\angle$C = 180° ……..(2)
From (1) and (2) we have,
$\angle$A + $\angle$B = $\angle$A + $\angle$C
$\angle$B = $\angle$C.
Thus the supplements of same angle are congruent.

Theorem 4:
If two angles are supplementary to two other congruent angles, then they’re congruent. In short, Supplements of congruent angles are congruent.
Proof: Let $\angle$A and $\angle$B be congruent angles.
Then $\angle$A = $\angle$B ……..(1)
Let $\angle$A and $\angle$C be supplementary angles. Thus, $\angle$C is a supplement of $\angle$A.
Then $\angle$A + $\angle$C = 180° ……..(2)
Let $\angle$B and $\angle$D be supplementary angles. Thus, $\angle$B is a supplement of $\angle$D.
Then $\angle$B + $\angle$D = 180° ……..(3)
From (2) and (3) we have,
$\angle$A + $\angle$C = $\angle$B + $\angle$ D
From (1),
$\angle$A + $\angle$C = $\angle$A + $\angle$ D
$\angle$C = $\angle$D.
Thus, the Supplements of congruent angles are congruent.

Solved Examples

Question 1: Find x from the figure if ABC is a straight line.

 Supplementary Angle Example
Solution:
 
Given that ABC is a straight line. So, $\angle$ABK and $\angle$CBK are linear pair and thus supplementary.
                    Thus they add up to 180°.
$\angle$ABK + $\angle$CBK = 180°
x + 24° = 180°
x = 180° - 24°  
x = 156°.

 

Question 2: Find two supplementary angles such that the measure of the first angle is 25° less than four times the measure of the second.
Solution:
 
Let x be the first angle and let y be the second angle such that they are supplementary.
Since the angles are supplementary, we have x + y = 180 …… (i)
From the given data we have, x= 4y – 25……. (ii)
Now there are 2 equations with 2 unknowns which can be solved for x and y.
Substituting the expression for x from (2) into (1) we get,
(4y – 25) + y = 180
Solving for y, 5y – 25 = 180.
5y = 180 + 25 = 205
y = 41°
Substitute this value for y into (1) and solving for x, we get,
x + 41 = 180
x = 180 – 41 = 139
x = 139°.

Thus 139° and 41° are two supplementary angles such that the measure of the first angle is 25° less than four times the measure of the second.