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

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

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Solution.

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

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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