Prove the circle properties – the angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc – an angle in a semicircle is a right angle – angles at the circumference of a circle subtended by the same arc are equal – the alternate segment theorem – the opposite angles of a cyclic quadrilateral are supplementary and its converse – a tangent drawn to a circle is perpendicular to the radius at the point of contact and its converse.

A circle has centre \(O\). Points \(P\), \(Q\), and \(R\) lie on the circumference such that \(\angle POQ = 3x + 20^\circ\) and \(\angle PRQ = x + 15^\circ\), where both angles are subtended by the same arc \(PQ\). Prove that the angle at the centre is twice the angle at the circumference, and hence determine the value of \(x\).

A circle has centre O and a chord AB. Point P lies on the major arc AB. Which statement correctly relates the angles in this configuration?

A circle has centre \(O\). Points \(A\), \(B\), and \(C\) lie on the circumference of the circle such that \(\angle AOB = 110^\circ\). Point \(D\) is a point on the major arc \(AB\) (the arc not containing \(C\)). Prove that \(\angle ACB + \angle ADB = 180^\circ\).