The first theorem used to prove that two angles are complementary states that if two angles are complementary to a third angle, then they are typically congruent to each other. The second theorem states that complements of congruent angles are congruent, which means that if two angles are complements of congruent angles, they are congruent to each other. Complementary angles are those that have a sum of 90 degrees.

Complementary angles are typically adjacent angles, which means that they share an arm and have a sum of 90 degrees. The proof of the complementary angle theorem involves two steps. The first step is to show that two angles, A and B, are complements of each other. The first step is accomplished by adding angle A to angle B and equating them to 90 degrees. The second step is to show that angle A is a complement of another angle, C, by adding angle A to angle C and equating them to 90 degrees. The two steps yield two equations that add up to 90 degrees. Equating the two equations shows that they have a common term, angle A, which means that both angle B and angle C are congruent, and this proves the theorem.