# In figure 5.25, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that 􀀍PQRS is a parallelogram. Given ABCD is a parallelogram so

and DC = AB and DC AB

also AP = BQ = CR = DS

AS = CQ and PB = DR

in ΔAPS and Δ CRQ

A = C (opposite s of a parallelogram are congruent)

AS = CQ

AP = CR

ΔAPS Δ CRQ( SAS congruence rule)

PS = RQ (c.p.c.t.)

Similarly PQ= SR

Since both the pair of opposite sides are equal

PQRS is gram.

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In figure 5.23, G is the point of concurrence of medians of ΔDEF. Take point H on ray DG such that D-G-H and DG = GH, then prove that 􀀍GEHF is a parallelogram. MHB - Math Part-II