E and F are respectively the midpoints of nonparallel sides AD and BC of a trapezium ABCD. Prove that EF||AB.

Explanation

Let ABCD be a trapezium in which AB||DC. Let E and F be midpoints of AD and BC respectively. E and F are joined.
We have to show that EF || AB.
If possible. Let EF be not parallel to AB then draw EG||AB, meeting BC at G.
Now, AB||EG||DC and the transversal AD cuts them at A, E, D respectively. such that AE = ED.
Also, BGC is the other transversal cutting AB, EG and DC at B, G and C respectively.
`therefore ` BG = GC (by intercept theorem).
This shows that G is the midpoint of BC.
Hence, G must coincide with F [`because` F is the midpoint of BC].
Thus, our supposition is wrong.
Hence, EF || AB.