Question

ABCD is a parallelogram,AD is produced to E so that DE=DC and EC produced meets AB produced in F. prove that BF=BC

Answer 1

Refer to the attached figure.

Since, ABCD is a parallelogram, therefore AB is parallel to CD.

Therefore, AF is parallel to CD.

And we can observe that EF is a traversal.

$\angle 2 = \angle 4$    (Corresponding angles)    (Equation 1)

Therefore, AE is parallel to BC.

So, $\angle 1 = \angle 3$    (Corresponding angles)   (Equation 2)

Now, in triangle DEC,

DE = DC

Therefore,  

So,  $\angle 1 = \angle 2$    (Equation 3)

(Angles opposite to the equal opposite sides are also equal)

Therefore,  $\angle 3 = \angle 4$    (By equations 1,2 and 3)

Therefore, BC = BF

Hence, proved.

Answer 2

Pic have the solution