In the adjoining figure, ABCD is a parallelogram whose diagonals inter

1.	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.
Answer» 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.

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.

Answer»

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.

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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.

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