Definition: Any two angles are called Complementary angles when they add up to 90°. When two angles add to 90°, we say that the angles are “Complement” to each other. In the pair of Complementary angles if one angle is x°, then its Complement is an angle of (90 - x)° .
Complementary Angles
From the above figures, we have $\angle$ABD and $\angle$DBC are complementary because 30° + 60° = 90°. $\angle$FGH and $\angle$IJK are complementary because 40.2° + 49.8° = 90°. Also, $\angle$ABD and $\angle$DBC together form right angle $\angle$ABC. Thus, if two angles are together and are complement to each other then they have a common vertex and share one side and are called Adjacent complementary angles. The other two non-shared sides of the angles form right angle. The angles need not be together, but together they add up to 90°. This can be seen from $\angle$FGH and $\angle$IJK. They are called non-adjacent complementary angles.

Two angles are complementary, only when both are acute angles.

Theorem 1:
If two angles are each complementary to a third angle, then they are congruent to each other. In short, Complements of the same angle are congruent.

Proof: Let $\angle$A and $\angle$B be complementary angles. Thus, $\angle$B is a complement of $\angle$A.
Then $\angle$A + $\angle$B = 90° ……..(1)
Let $\angle$A and $\angle$C be complementary angles. Thus, $\angle$C is a complement of $\angle$A.
Then $\angle$A + $\angle$C = 90° ……..(2)
From (1) and (2) we have,
$\angle$A + $\angle$B = $\angle$A + $\angle$C
$\angle$B = $\angle$C.
Thus the complements of same angle are congruent.

Theorem 2:
If two angles are complementary to two other congruent angles, then they are congruent. In short, complements 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 complementary angles. Thus, $\angle$C is a complement of $\angle$A.
Then $\angle$A + $\angle$C = 90° ……..(2)
Let $\angle$B and $\angle$D be complementary angles. Thus, $\angle$B is a complement of $\angle$D.
Then $\angle$B + $\angle$D = 90° ……..(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 complements of congruent angles are congruent.
Complementary angles are seen in real life in many places. Few examples are shown here.

Complementary Angles in Real Life

In the figure (i) we notice that the three streets meet at angle of A° and B° at the corners. These two angles sum to 90°. So, these kinds of intersections are examples of complementary angles.

In the figure (ii) we see an envelope. The angles of a flap of an envelope from the corner (each side measured from the corner of the fold) are Complementary angles ( A° and B° ). These angles add up to 90°, thus forming complementary angles.

In the figure (iii) the angles made at the pointer head C° and D° are complementary.
In any pair of complementary angles if one angle is x°, then its complement is an angle of (90 - x)° .

Solved Examples

Question 1: Find x from the figure if $\angle$ABC is a right angle.

 Complementary Angles Example
Solution:
 
Given that $\angle$ABC is a right angle. So, $\angle$ABD and $\angle$CBD are complementary.
                    Thus they add up to 90°.
$\angle$ABD + $\angle$CBD = 90°
50° + x = 90°
x = 90° - 50°  
x = 40°.
 

Question 2: If one angle is 75°, find the second angle if the two angles are complementary to each other.
Solution:
 
Given  $\angle$1 = 75° ……. (i)
$\angle$1 and $\angle$2 are complementary angles.
Since they are complementary angles they must add up to 90° .
So, $\angle$1 + $\angle$2 = 90°  ……… (ii)
From (i) and (ii),
75° + $\angle$2 = 90°.
$\angle$2 = 90° - 75°
$\angle$2 = 15°.
 

Question 3: Find two complementary 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 complementary.
Since the angles are complementary, we have x + y = 90 …… (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 = 90
Solving for y, 5y – 25 = 90.
5y = 90 + 25 = 115
Y = $\frac{115}{5}$
y = 23°
Substitute this value for y into (1) and solving for x, we get,
x + 23 = 90
x = 90 – 23 = 67
x = 67°.

Thus 67° and 23° are two complementary angles such that the measure of the first angle is 25° less than four times the measure of the second.
 

Question 4: If two equal angles are complements to each other, then find the measure of each angle.
Solution:
 
Let the two equal angle be x° and y°.  
Thus we have x = y ……..  ( i )
Also given that x° and y° are complementary.
Thus x + y = 90° …….      (ii)
From (i) and (ii) we have,
x + x = 90°
2x = 90°
x = $\frac{90^o}{2}$
x = 45 °