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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

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