In the diagram, A, B, C and D are four points on a circle, centre O. Angle $ABO = x^\circ$ and angle $ADO = y^\circ$. Find angle $BCD$ in terms of $x$ and $y$. Give reasons for each step of your working.

1. Find the angle between $BA$ and $AO$.

Since $OA = OB$ radii, $\triangle AOB$ is isosceles.

Therefore
$\angle BAO = \angle ABO = x^\circ.$

2. Find the angle between $AO$ and $AD$, then form $\angle BAD$.

Since $OA = OD$ radii, $\triangle AOD$ is isosceles.

Therefore
$\angle OAD = \angle ADO = y^\circ.$

Hence
$\angle BAD = \angle BAO + \angle OAD = x^\circ + y^\circ.$

3. Use the cyclic quadrilateral angle property to find $\angle BCD$.

Since $A$, $B$, $C$ and $D$ lie on a circle, $ABCD$ is a cyclic quadrilateral.

Opposite angles of a cyclic quadrilateral are supplementary, so
$\angle BAD + \angle BCD = 180^\circ.$

Thus
$x + y + \angle BCD = 180,$
$\angle BCD = (180 - x - y)^\circ.$