Question

If the diagonals of a quadrilateral bisect each other, prove that the quadrilateral is a parallelogram.

Answer 1

$$\begin{gathered} \mathrm{Given}:ABCD\mathrm{is}\mathrm{a}\mathrm{quadrilateral}\text{whose d}\mathrm{iagonals}\mathrm{bisect}\mathrm{each}\mathrm{other}. \\ \mathrm{To}\mathrm{prove}:ABCD\mathrm{is}\mathrm{a}\mathrm{paralleogram}. \\ \mathrm{Proof}:\mathrm{In}△AOD\mathrm{and}△BOC,\mathit{} \\ AO=CO\mathit{}\mathit{}\left(\mathrm{Given}\right) \\ DO=OB\left(\mathrm{Given}\right) \\ \angle AOD=\angle COB(\text{V}\mathrm{ertically}-\mathrm{opposite}\mathrm{angle}s) \\ \mathrm{So},△AOD\cong △COB\mathit{}(\mathrm{By}\mathrm{SAS}\mathrm{congruency}) \\ AD=BC(\mathrm{c}.\mathrm{p}.\mathrm{c}.\mathrm{t}) \end{gathered}$$
$$\begin{gathered} \mathrm{In}△AOB\mathrm{and}△DOC\mathit{} \\ AO=CO\mathit{}\mathit{}\left(\mathrm{Given}\right) \\ BO=OD\left(\mathrm{Given}\right) \\ \angle AOB=\angle COD(\mathrm{vertically}\mathrm{opposite}\mathrm{angle}) \\ \mathrm{So},△AOB\cong △COD\mathit{}(\mathrm{By}\mathrm{SAS}\mathrm{congruency}) \\ AB=DC(\mathrm{c}.\mathrm{p}.\mathrm{c}.\mathrm{t}) \end{gathered}$$
Since, the opposite sides of the quadrilateral are equal, $\square$ABCD is a parallelogram.