Quick questions on Circle properties and angle theorems explained: O-Level E-Maths

An angle subtended at the circumference by a diameter is a right angle. This is the special case of the centre-circumference theorem where the central angle is the straight angle $180^\circ$, half of which is $90^\circ$. Spotting a diameter often instantly gives a $90^\circ$ angle.

Angles subtended by the same arc, on the same side, at the circumference are equal. So two angles standing on the same chord, both above it, must match. This lets you transfer a known angle to another point on the circle.

A cyclic quadrilateral has all four vertices on a circle. Its opposite angles are supplementary, adding to $180^\circ$. An exterior angle of a cyclic quadrilateral equals the interior opposite angle.

A tangent touches a circle at exactly one point. The radius drawn to the point of contact is perpendicular to the tangent, giving a $90^\circ$ angle. Two tangents drawn from the same external point are equal in length, and they make equal angles with the line to the centre.

A correct value without the named theorem loses the reasoning marks.

The angle at the centre of a circle on arc $PQ$ is $84^\circ$. Find the angle at the circumference on the same arc. [1 mark]

$PQ$ is a diameter and $R$ lies on the circle. State the size of angle $PRQ$. [1 mark]

In a cyclic quadrilateral one angle is $108^\circ$. Find the angle opposite it. [1 mark]