Question

Diagonals of parallelogram $$ABCD$$ intersect at $$O$$ as shown in Fig. $$XY$$ contains $$O$$, and $$X, Y$$ are points on opposite sides of the parallelogram. Give reasons for each of the following:
$$\triangle BOY\cong \triangle ODX$$.
Now, state if $$XY$$ is bisected at $$O$$.

Answer 1

$$\textbf{Step-1: Apply the concept of congruency.}$$

$$\text{In}$$ $$\triangle BOY $$ $$\text{and}$$ $$\triangle DOX,$$

$$(i) BO = OD$$

$$(ii) \angle OBY = \angle ODX$$ $$\textbf{[Alternate angles]}$$

$$(iii) \angle BOY = \angle DOX $$ $$\textbf{[Vertically opposite angles]}$$

$$\triangle BOY \cong \triangle DOX$$ $$\textbf{[ASA congruency]}$$

$$\therefore OX = OY$$ $$\textbf{[C.P.C.T.C.]}$$

$$XY$$ $$\text{is bisected at O}$$

$$\textbf{Hence, XY is bisected at O.}$$