Question

PQRS is a Trapezium and Digonal meets at M if PQ || RS show that PM/MR =QM/MS
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Answer 1

Here, PQ || RS and PR and QS are transversals.

∴∠RPQ=∠PRS ............(1)

and ∠SQP = ∠RSQ ............(2)

Now, in ΔPMQ and ΔRMS, we have:

(i) ∠PMQ = ∠RMS [vertically opposite angles]

(ii) ∠MPQ = ∠MRS [using (1)]

(iii) ∠MQP = ∠MSR [using (2)]

∴ΔPMQ  is similar to ΔRMS

∴$\frac{PM}{RM} =\frac{QM}{SM}$ [ratio of corresponding sides of similar triangles is the same]

Hence proved.