Question

In the given figure, $AB$ and $CD$ are intersecting at ${}^{\prime }{O}^{\prime },OA=OB$ and $OD=OC$. Show that
$AD\parallel BC$

Answer 1

From the figure,

$OA=OB$ and $OD=OC$

Line $CD$ and $AB$ intersect

$\therefore \angle AOD=\angle BOC$ [Vertically opposite angles]

In $△AOD$ and $△BOC$,

$OA=OB$

$\angle AOD=\angle BOC$

$\therefore △AOD\cong △BOC$ [SAS Congruence Rule]

$OD=OD$

$\Rightarrow \angle OAD=\angle OBC$ [by CPCT]

But $\angle OAD$ and $\angle OBC$ form a pair of alternate angles.

If a transversal intersects two lines such that pair of alternate interior angles is equal, then lines are parallel.

$\Rightarrow AD||BC$