Question

State true or false:

In the given figure, the diagonals $AC$ and $BD$ intersect at point $O$. If $OB=OD$ and $AB\parallel DC$, then $ABCD$ is a parallelogram.

• A. True
• B. False

Answer 1

The correct option is A True

To: ABCD a parallelogram
Construction :Draw $OX\parallel DC$
In $△BCD$
O is the mid point of BD and $OX\parallel DC$
$\therefore Byconverseofmidpointtheorem$
X is the mid point of BC and $OX=\frac{1}{2}DC$
$\therefore OX=\frac{1}{2}AB$
Now in $△ABC$,
X is the mid point of BC and $OX=\frac{1}{2}AB$
$\therefore Byconverseofmidpointtheorem$
O is the mid point of AC and $OX\parallel AB$
As $OX\parallel ABandOX\parallel DC$
$=AB\parallel DC$
Now $AB\parallel DCandAB=DC$
$\therefore ABCDisaparallelogram$