Question

19. In the adjoining figure, ABCD is a parallelogram
whose diagonals intersect each other at O. A line
segment EOF is drawn to meet AB at E and DC at
F. Prove that OE = OF.
o
A
E​

Answer 1

We know that ABCD is a parallelogram whose diagonals intersect each other at O.

Consider △ AOE and △ COF.

We know that ∠ CAE and ∠ DCA are alternate angles.

From the figure we know that the diagonals are equal and bisect each other AO = CO.

We know that ∠ AOE and ∠ COF are vertically opposite angles.

∠ AOE =∠ COF

By ASA congruence criterion

△ AOE ≅ △ COF

OE = OF (c. p. c. t).

Therefore, it is proved that OE = OF.