Question 4: a) Find the size of angles x, y and z. b) The diagram shows a circle with centre O. RS is a diameter. What is the size of angle x?

The angles $x$ and $68^\circ$ are opposite angles in the cyclic quadrilateral.

$$x + 68^\circ = 180^\circ$$ $$x = 180^\circ - 68^\circ$$ $$x = 112^\circ$$

The size of angle $x$ is $\boxed{112^\circ}$.

The angles $y$ and $38^\circ$ are opposite angles in the cyclic quadrilateral.

$$y + 38^\circ = 180^\circ$$ $$y = 180^\circ - 38^\circ$$ $$y = 142^\circ$$

The size of angle $y$ is $\boxed{142^\circ}$.

The angles $z$ and $55^\circ$ are opposite angles in the cyclic quadrilateral.

$$z + 55^\circ = 180^\circ$$ $$z = 180^\circ - 55^\circ$$ $$z = 125^\circ$$

The size of angle $z$ is $\boxed{125^\circ}$.

The sum of angles in a triangle is $180^\circ$.

$$\angle RUS + 50^\circ + 45^\circ = 180^\circ$$ $$90^\circ + 50^\circ + 45^\circ = 180^\circ$$

Since $\angle RUS = 90^\circ$, the angle $z$ is the remaining angle in the triangle.

$$z = 180^\circ - 90^\circ - 50^\circ$$ $$z = 40^\circ$$

The size of angle $z$ is $\boxed{40^\circ}$.