Vertically Opposite Angles

From a point $O$ in a line $AB,$ ray $OC$ and $OD$ are drawn on opposite sides of $AB$ such that $\angle BOC=\angle AOD$. Prove that $OC$ and $OD$ are opposite rays.

If two straight lines intersect each other, prove that the ray opposite to the bisector of one angles thus formed bisects the vertically opposite angle.

Prove that the bisectors of a vertically opposite angles are in the same straight line.

In the adjoining figure, $\angle AOC=\left(2y-13\right)°$ and $\angle BOC=\left(3y-12\right)°$, find all the four angles.

In the adjoining figure, find the value of $x$.