Hilbert's Foundations: Proving the Vertical Angle Theorem

Here's how to prove the vertical angle theorem. Suppose we have vertical angles.

Let's call $\angle BAE = x$ and $\angle DAC = y$.

Due to the definition of supplementary angles, $\angle BAD = 180^\circ-x$. But also due to the definition of supplementary angles, $\angle BAD = 180^\circ - y$ If $180^\circ-x=180^\circ-y$, then $-x=-y$, and $x=y$, therefore proving that $\angle BAE \cong \angle DAC$.