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If two lines intersect, prove that the vertically opposite angles are equal.

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It is given that if two lines intersect, the vertically opposite angles are equal.

Now let AB and CD be two lines intersecting at point O.

From the figure, we have two pairs of vertically opposite angles namely:

(i) \angle AOC and \angle BOD

(ii) \angle AOD and \angle BOC

Now we have to prove that \angle AOC = \angle BOD

And \angle AOD = \angle BOC

\Rightarrow Now ray OA stands on line CD

\therefore \angle AOC + \angle AOD = 180^{\circ} … (i) (linear pair angles)

Similarly, can we write

\angle AOD + \angle BOD = 180^{\circ} … (ii) (linear pair angles)

From equation (i) and (ii) comparing

\angle AOC + \angle AOD = \angle AOD + \angle BOD

\Rightarrow \angle AOC = \angle BOD

Similarly, we can prove that \angle AOD = \angle BOC

Hence Proved.

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