Question

Question 1
If two lines intersect, prove that the vertically opposite angles are equal.

Answer 1

Given, two lines AB and CD intersect at point O.
To prove
(i) $\angle AOC=\angle BOD$
(ii) $\angle AOD=\angle BOC$
Proof:
(i) Ray OA stands on line CD.
$\therefore \angle AOC+\angle AOD={180}^{\circ }$ [linear pair axiom]…(i)
Ray OD stands on line AB.
$\therefore \angle AOD+\angle BOD={180}^{\circ }$ [linear pair exiom]…(ii)

From the equations (i) and (ii),
$\Rightarrow \angle AOC+\angle AOD=\angle AOD+\angle BOD$
$\angle AOC=\angle BOD$
(ii) Ray OD stands on line AB.
$\therefore \angle AOD+\angle BOD={180}^{\circ }$ [linear pair axiom]…(iii)
Ray OB stands on line CD.
$\therefore \angle DOB+\angle BOC={180}^{\circ }$ …(iv)
From equations (iii) and (iv)
$\Rightarrow \angle AOD+\angle BOD=\angle DOB+\angle BOC$
$\angle AOD=\angle BOC$
Hence proved.