ABCD is a rhombus and its diagonals intersect at O.

(i) Yes

In $\Delta \mathrm{BCO}$ and $\Delta \mathrm{DCO}:$

$\mathrm{OC}=\mathrm{OC}$ (common)

$\mathrm{BC}=\mathrm{DC}$ (all sides of a rhombus are equal)

$\mathrm{BO}=\mathrm{OD}$ (diagonal $s$ of a rhomus bisect each other)

$\mathrm{By} \mathrm{SSS}$ congruence :

$\Delta \mathrm{BCO} \cong \Delta \mathrm{DCO}$

(ii) Yes

By c.p.c.t:

$\angle B C O=\angle D C O$