The interior angle of a regular polygon is $4$ times its exterior angle. How many sides does the polygon have?

The diagonals of a parallelogram $ABCD$intersect at $O$. A line through $O$ intersects $AB$ at $X$ and $DC$ at $Y$. Prove that $OX=OY$.

$\mathrm{ABC}$ is a right-angled triangle and $\mathrm{O}$ is the mid-point of the side opposite to the right angle. Explain why $\mathrm{O}$ is equidistant from $\mathrm{A},\mathrm{B}$ and $\mathrm{C}$.

The angles of a quadrilateral are in the ratio $1:2:3:4$. Find the measure of each angle.