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 = 1/2AB …(i)
In ΔBDC, OF||CD and O is mid-point of BD.
OF = 1/2CD …(ii)
Adding (i) and (ii),
EO + OF = 1/2AB + 1/2CD
EF = 1/2(AB + CD)
Hence, proved.