Congruent Complements Theorem

Based on this diagram, the following conditional statement holds true. The sum of angles $1$ and $2$ is $90^{\circ }.$ The sum of angles $\angle 2$ and $\angle 3$ is also $90^{\circ }.$ Since both angles $\angle 1$ and $\angle 3$ are each complementary to angle $\angle 2,$ they are congruent.

The Congruent Complements Theorem also applies when two angles are complementary to congruent angles.

Based on this pair of diagrams, the following conditional statement holds true. The sum of angles $\angle 1$ and $\angle 2$ is $90^{\circ }.$ The sum of angles $\angle 3$ and $\angle 4$ are also complentary, so their sum is also $90^{\circ }.$ If angle $\angle 1$ is congruent to angle $\angle 4,$ then by the Congruent Complements Theorem, it follows that angle $\angle 2$ is congruent to angle $\angle 3.$

Proof

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.

$$\begin{array}{r}m\angle 1+m\angle 2=90^{\circ }\\ m\angle 2+m\angle 3=90^{\circ }\end{array}$$

Therefore, $m\angle 1+m\angle 2$ and $m\angle 2+m\angle 3$ are equal by the Transitive Property of Equality.

$$\begin{array}{c}m\angle 1+m\angle 2=m\angle 2+m\angle 3\end{array}$$

Using the Subtraction Property of Equality, this equation can be simplified by subtracting $m\angle 2$ from both sides.

$$\begin{array}{c}m\angle 1+m\angle 2=m\angle 2+m\angle 3\\ \Downarrow \\ m\angle 1=m\angle 3\end{array}$$

Two angles have equal measures when they are congruent.

$\angle 1\cong \angle 3$