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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