Complements of the same angle are equal

Dependencies: Protractor axiom (angles can be treated as real numbers).

Statement

Let $\alpha$ and $\beta$ both be complementary to the same angle $\gamma$:

Then $\alpha = \beta$.

Proof

Step 1: translate the figure into two equations. The right angle on the left is split into $\alpha$ and $\gamma$; the right angle on the right is split into $\beta$ and the same $\gamma$. The two $\gamma$'s are the same angle — that is precisely the "same angle" hypothesis of the theorem.

Step 2: subtract the two equations; the $\gamma$ terms cancel. Protractor axiom turns each angle into a real number, so this is just ordinary arithmetic with reals. Subtracting the two equations cancels $\gamma$:

i.e. $\alpha - \beta = 0$, hence $\alpha = \beta$. $\blacksquare$

Immediate consequences

• Substitutability of co-complements: in any figure containing $\alpha + \gamma = 90^{\circ}$, as long as you also verify $\beta + \gamma = 90^{\circ}$ you may swap $\alpha$ for $\beta$ without changing any other angle relations. This is the cleanest version of the kind of substitution used later in right triangles (the two acute angles are complementary $\Rightarrow$ acute angles are equal).
• Supplements of the same angle are equal (the $180^{\circ}$ version) has a parallel proof: replace $90^{\circ}$ by $180^{\circ}$ and "complementary" by "supplementary"; the argument is word-for-word the same.

Remarks

The entire content of this theorem lives inside Protractor axiom: once an angle can be assigned a real number, the equality of complements is just one subtraction step in a simple linear system. The geometry is reduced to "admit that angles are real numbers"; everything else is algebra.