# Show that the lin

We can draw the trapezium as

Here, let EF be the line segment joining the oblique sides of the trapezium at midpoints E and F (say) correspondingly.

Construction: Extend AD and BC such that it meets at P.

To Prove: EF DC and EF AB

Proof: Given that, ABCD is trapezium which means DC AB. …(statement (i))

In ∆PAB,

DC AB (by statement (i))

So, this means we can apply Thale’s theorem in ∆PAB. We get

…(ii)

E and F are midpoints of AD and BC respectively, we can write

DA = DE + EA

Or DA = 2DE …(iii)

CB = CF + FB

Or CB = 2CF …(iv)

Substituting equation (iii) and (iv) in equation (ii), we get

By applying converse of Thale’s theorem, we can write DC EF.

Now if DC EF, and we already know that DC AB.

EF is also parallel to AB, that is, EF AB.

This means, DC EF AB.

Hence, proved.

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