Question

In the given figure, $AB\parallel CD$ and $O$ is the midpoint of $AD$.
Show that (i) $△AOB\cong △DOC$
(ii) $O$ is the midpoint of $BC$.

Answer 1

(i) From the figure, $△AOB$ and $△DOC$

We know that $AB\parallel CD$ and $\angle BAO$ and $\angle CDO$ are alternate angles

So we get

$\angle BAO=\angle CDO$

From the figure, we also know that $O$ is the midpoint of the line $AD$

We can write it as $AO=DO$

According to the figure we know that $\angle AOB$ and $\angle DOC$ are vertically opposite angles.

So we get $\angle AOB=\angle DOC$

Therefore, by $ASA$ congruence criterion we get

$△AOB\cong △DOC$.

(ii) We know that $△AOB\cong △DOC$

So we can write it as

$BO=CO(c.p.c.t)$

Therefore, it is proved that $O$ is the midpoint of $BC$.