Answer :


Given, ABCD is a trapezium in which AB||CD. Also, E and F are mid-points of sides AD and BC.


Now, joining BE and produce it to meet CD produced at G.


In ΔGCB, by mid-point theorem


EF||GC


But GC = CD, and CD||AB


EF||AB


Now, draw BD which intersects EF at O.


In ΔADB, AB||EO and E is mid-point of AD.


By converse of mid-point theorem, O is mid-point of BD.


Also, EO = AB …(i)


In ΔBDC, OF||CD and O is mid-point of BD.


OF = CD …(ii)


Adding (i) and (ii),


EO + OF = AB + CD


EF = (AB + CD)


Hence, proved.


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