Question

if two sides intersect each other then the vertical opposite angles are equal
​

Answer 1

Given:

Two sides (lines)are interesting each other.

To Prove:

The vertically opposite angles are equal.

Proof:

Let us say two lines AB and CD intersect each other at point O .

(For figure refer to attachment:)

Now ,

⇒ $\angle$ AOC + $\angle$ BOC = 180° ( linear pair) ................(i)

Similarly ,

⇒ $\angle$AOC + $\angle$ AOD = 180°. .................(ii)

From (i) and (ii) we have ,

⇒$\angle$ AOD = $\angle$ BOC

Again ,

⇒$\angle$BOC+$\angle$BOD=180°(linear pair) ...............(iii)

And

⇒$\angle$ BOD + $\angle$ AOD = 180° (linear pair) ..................(iv)

So from (iii) and (iv) we have ,

⇒$\angle$ AOD = $\angle$BOC

But $\angle$ AOD and $\angle$ BOC are Vertically opposite angles .

Similarly $\angle$AOD and $\angle$BOC are Vertically opposite angles ,and they are found to be equal .

Hence proved.