Give the proof of Theorem of vertically opposite angles.

Theorem:

In a pair of intersecting lines the vertically opposite angles are equal.

Proof:

Let us consider two lines $AB$ and $CD$ which intersect each other at $O$. 

$\angle AOD$ and $\angle AOC$ form a linear pair.

Therefore,

$\angle AOD + \angle AOC = 180^o$........(i)

$\angle AOC$ and $\angle BOC$ form a linear pair.

Therefore,

$\angle AOC + \angle BOC = 180^o$........(ii)

From (i) and (ii),

$\angle AOD + \angle AOC = \angle AOC + \angle BOC$

$\Rightarrow \angle AOD = \angle BOC$.........(iii)

$\angle AOD$ and $\angle BOD$ form a linear pair.

Therefore,

$\angle AOD + \angle BOD = 180^o$........(iv)

From (i) and (iv),

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

$\Rightarrow \angle AOC = \angle BOD$

Therefore, the pair of opposite angles are equal.

Hence proved.

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Updated on: 10-Oct-2022

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