Congruent Complements Theorem

The case where two angles are complements of congruent angles can be reduced to the case where two angles are complements to the same angle using reflections, translations, and rotations. Consider the case where two angles are complements of the same angle. The sum of measures of complementary angles is equal to $90^{\circ }.$ In this case, $\angle 1$ and $\angle 2,$ and $\angle 2$ and $\angle 3$ are pairs of complementary angles. Therefore, $m\angle 1+m\angle 2$ and $m\angle 2+m\angle 3$ are equal by the Transitive Property of Equality. Using the Subtraction Property of Equality, this equation can be simplified by subtracting $m\angle 2$ from both sides. Two angles have equal measures when they are congruent.