Vertical Angle Theorem

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The Vertical Angle Theorem is a theorem that states that all vertical angles are congruent.

Proof

Assume all angle measures to be in radians. A pair of vertical angles are formed by lines $\overline A \overline B$ and $\overline C \overline D$ and the intersection of these lines is P. Angles $\angle APC$ and $\angle BPD$ are vertical angles. Let $m \angle APC = x$. From this, $m\angle CPB = \pi-x$, so $m \angle BPD = \pi-(\pi-x)=x=m\angle APC.$

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