Diagonals of parallelogram ABCD intersect at O as shown in Figure.

1.	Diagonals of parallelogram ABCD intersect at O as shown in Figure. XY contains O, and X, Y are points on opposite sides of the parallelogram. Give reasons for each of the following:(i) OB = OD(ii) ∠OBY = ∠ODX(iii) ∠BOY = ∠DOX(iv) ΔBOY = ΔDOXNow, state if XY is bisected at O.
Answer» (i) OB = OD OB = OD. Since diagonals bisect each other in a parallelogram. (ii) ∠OBY =∠ODX ∠OBY =∠ODX. Since alternate interior angles are equal in a parallelogram. (iii) ∠BOY= ∠DOX ∠BOY= ∠DOX. Since vertical opposite angles are equal in a parallelogram. (iv) ΔBOY ≅ ΔDOX ΔBOY and ΔDOX. Since OB = OD, where diagonals bisect each other in a parallelogram. ∠OBY =∠ODX [Alternate interior angles are equal] ∠BOY= ∠DOX [Vertically opposite angles are equal] ΔBOY ≅ΔDOX [by ASA congruence rule] OX = OY [Corresponding parts of congruent triangles] ∴ XY is bisected at O.

Diagonals of parallelogram ABCD intersect at O as shown in Figure. XY contains O, and X, Y are points on opposite sides of the parallelogram. Give reasons for each of the following:(i) OB = OD(ii) ∠OBY = ∠ODX(iii) ∠BOY = ∠DOX(iv) ΔBOY = ΔDOXNow, state if XY is bisected at O.

Answer»

(i) OB = OD

OB = OD. Since diagonals bisect each other in a parallelogram.

(ii) ∠OBY =∠ODX

∠OBY =∠ODX. Since alternate interior angles are equal in a parallelogram.

(iii) ∠BOY= ∠DOX

∠BOY= ∠DOX. Since vertical opposite angles are equal in a parallelogram.

(iv) ΔBOY ≅ ΔDOX

ΔBOY and ΔDOX. Since OB = OD, where diagonals bisect each other in a parallelogram.

∠OBY =∠ODX [Alternate interior angles are equal]

∠BOY= ∠DOX [Vertically opposite angles are equal]

ΔBOY ≅ΔDOX [by ASA congruence rule]

OX = OY [Corresponding parts of congruent triangles]

∴ XY is bisected at O.