Vertical Angle Theorem

Theorem: Vertically opposite angles formed by a pair of intersecting lines are equal

Prerequisites:
Angle on a straight line (proof)

Proof:

In the given figure, $\angle AOD$ and $\angle BOC$ are vertically opposite. Similarly, $\angle AOC$ and $\angle BOD$ are vertically opposite.

Since angle at a point on a straight line is $180^o$, hence due to line CD,

$\quad\qquad\angle AOD + \angle AOC = 180^o\\
\therefore\quad\;\;  \angle AOD = 180^o - \angle AOC\qquad\qquad\qquad\qquad\text{$...(1)$}$

Also, due to line AB,

$\quad\qquad \angle AOC + \angle BOC = 180^o\\
\therefore\quad\;\;  \angle BOC = 180^o - \angle AOC\qquad\qquad\qquad\qquad\text{$...(2)$}$

Hence, from $(1)$ and $(2)$,

$\qquad\quad\angle AOD = \angle BOC$

Similarly,

$\qquad\quad\angle AOC = \angle BOD$

Hence the result.


Recommended:
Corresponding angles theorem
Alternate angles theorem
Interior angles theorem