Question

D and E are the mid-points of the sides AB and AC respectively of ∆ABC. DE is produced to F. To prove that CF is equal and parallel to DA, we need an additional information which is ______.

Options

• ∠DAE = ∠EFC

• AE = EF

• DE = EF

• ∠ADE = ∠ECF

Answer 1

D and E are the mid-points of the sides AB and AC respectively of ∆ABC. DE is produced to F. To prove that CF is equal and parallel to DA, we need an additional information which is DE = EF.

Explanation:

We have produced DE to F such that

DE = EF  ...(1)

In ΔADE and ΔCFE,

AE = CE  ...[Since, E is the mid-point of AC]

∠AED = ∠CEF  ...[Vertically opposite angles]

DE = FE    ...[By (1)]

∴ ΔADE ≅ ΔCFE   ...[By SAS congruency]

∴ AD = CF and ∠ADE = ∠CFE   ...[By C.P.C.T.]

This shows that alternate interior angles are equal.

Thus, AD || CF

Therefore, the additional information which we need is DE = EF