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.