Question

$\square$ABCD is a parallelogram. Points P, Q, R and S are points on the side such that seg AP $\cong$ seg CR $\cong$ seg DS
Prove that $\square$PQRS is a parallelogram (see fig. 5.12)

Answer 1

Given: seg AP $\cong$ seg BQ $\cong$ seg CR $\cong$ seg DS BQ = CR = DS

We have:
seg AB $\cong$ seg CD (Opposite sides of parallelogram $\square$ABCD are equal.)
Subtracting seg AP from both sides, we get:
seg AB $-$ seg AP $\cong$ seg CD $-$ seg AP
or,
seg AB $-$ seg AP $\cong$ seg CD $-$seg CR ($\because$seg AP$\cong$ seg CR)
or,
seg BP $\cong$seg DR ...(1)

Again, we have:
seg AD $\cong$ seg BC (Opposite sides of parallelogram $\square$ABCD are equal.)
Subtracting seg BQ from both sides, we get:
seg AD $-$seg BQ $\cong$seg BC $-$ seg BQ
or,
seg AD $-$ seg DS $\cong$seg BC $-$ seg BQ ($\because$ seg DS $\cong$seg BQ)
or,
seg AS $\cong$ seg CQ ...(2)

Consider $△$DSR and $△$BQP,
seg DS $\cong$seg BQ (Given)
$\angle$SDR = $\angle$QBP (Opposite angles of parallelogram ABCD are equal.)
seg DR$\cong$ seg BP [Using (1)]


By SAS congruency, $△$DSR $\cong$ $△$BQP
seg SR$\cong$seg PQ (by c.s.c.t) ....(3)

Consider $△$APS and $△$RCQ
seg AP $\cong$seg RC (Given)
$\angle$SAP = $\angle$QCR (Opposite angles of parallelogram ABCD are equal.)
seg AS $\cong$ seg CQ [Using (2)]


By SAS congruency, $△$SAP $\cong$ $△$QCR.
seg SP$\cong$ seg RQ (by c.s.c.t) ...(4)
From eq (1) , (2), (3) & (4) we can say that
Since, opposite sides of quadrilateral PQRS are equal,
Hence, $\square$PQRS is a parallelogram.