Question

In the given figure, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that $\square$PQRS is a parallelogram.

Answer 1

ABCD is a parallelogram
So, opposite pair of sides will be congruent and parallel.
$\Rightarrow \mathrm{AD}\cong \mathrm{BC}\mathrm{and}\mathrm{AB}\cong \mathrm{CD}$
Given, AP = BQ = CR = DS
$$\begin{gathered} \mathrm{AD}-\mathrm{SD}=\mathrm{BC}-\mathrm{BQ} \\ \Rightarrow \mathrm{AS}=\mathrm{CQ}.....\left(1\right) \end{gathered}$$
In $△$APS and $△$RCQ
AS = CQ (From (1))
AP = CR (Given)
$$\begin{gathered} \angle \mathrm{PAS}=\angle \mathrm{RCQ}\left(\mathrm{Opposite}\mathrm{angles}\mathrm{of}\mathrm{a}\mathrm{parallelogram}\mathrm{are}\mathrm{equal}\right) \\ \mathrm{Thus},△\mathrm{APS}\cong △\mathrm{RCQ}\left(\mathrm{SAS}\mathrm{congruency}\right) \\ \Rightarrow \mathrm{SP}=\mathrm{RQ}\left(\mathrm{CPCT}\right) \end{gathered}$$
Similarly, PQ = SR
Thus, opposite pair of sides are congruent.
Hence, PQRS is a parallelogram.