Answer :

Figure according to given information



SK = KL = QL … given … (i)


Consider ΔPKQ and ΔRLS


PQ = SR … opposite sides of parallelogram PQRS


PQK = RSL … alternate pair of interior angles for parallel lines PQ and SR with transversal as SQ


SL = SK + KL and KQ = KL + LQ so using (i) we can say that


SL = KQ


Therefore, ΔPQK ΔRSL


PKQ = RLS … corresponding angles of congruent triangles


Thus PM || NR because PKQ and RLS are pair of alternate interior angles with transversal as KL


PM || NR … (ii)


Consider ΔPMS and ΔRNQ



PMS = NRM … corresponding pair of angles for parallel lines PM and NR with transversal as SR … (a)


NRM = RNQ … alternate interior angles for parallel lines PQ and SR with transversal as NR … (b)


PMS = RNQ … using equation (a) and (b)


PSM = RQN … opposite pair of angles for parallelogram PQRS


PS = QR … opposite pair of sides for parallelogram PQRS


Therefore, ΔPMS ΔRNQ … AAS test for congruency


PM = NR … corresponding sides of congruent triangles … (iii)


SM = NQ … corresponding sides of congruent triangles … (c)


As PQ = SR … opposite sides of parallelogram PQRS … (d)


From figure PN = PQ – NQ and MR = SR – SM


Using (c) and (d)


PN = SR – SM


PN = MR … (iv)


As PQ || SR and PN and MR lie on the lines PQ and SR respectively hence we can conclude that


PN || MR … (v)


Using equations (ii), (iii), (iv) and (v) we can conclude that for quadrilateral PMRN the opposite sides are congruent and parallel therefore, PMRN is a parallelogram


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