Question

A line segment $AB$ is parallel to another line segment $CD$. $O$ is the midpoint of $AD$. Show that $O$ is also the midpoint of $BC$.

Answer 1

Step 1: State the given data and draw a diagram

Let, us draw a suitable diagram.

It is given that $AB\left|\right|CD$.

And, $O$ is the mid-point of $AD$.

i.e., $AO=DO$ $...\left(\mathrm{i}\right)$

Now, since $AB\left|\right|CD$ and $AD$ is a transversal.

So, $\angle BAO=\angle CDO$ [alternate interior angles]

Also, since $AB\left|\right|CD$ and $CB$ is a transversal.

So, $\angle OBA=\angle OCD$ [alternate interior angles]

Step 2: Show that $△AOB$ is congruent to $△DOC$

In the triangles $△AOB$ and $△DOC$,

$\angle OBA=\angle OCD$ [alternate interior angles]

$\angle BAO=\angle CDO$ [alternate interior angles]

$AO=DO$ [from equation $\left(\mathrm{i}\right)$]

Then, by the AAS (Angle-Angle-Side) congruency criterion,

$△AOB\cong △DOC$

Step 3: Equate $BO$ and $CO$

Now, as we know, the corresponding parts of the congruent triangles are congruent (CPCTC). So,

$\because$ $△AOB\cong △DOC$

$\Rightarrow$ $BO=CO$

i.e., the point $O$ is the mid-point of $BC$.

Hence, it is proved that $O$ is the mid-point of $BC$.