In the figure at left, PQ is a diameter of a circle with centre O. If ∠PQR = 55°, ∠SPR = 25°and ∠PQM = 50°.

Find (i) ∠QPR, (ii) ∠QPM and (iii) ∠PRS.

According to theorem the angle subtended by a diameter on the circumference of a circle is 90°

∠PRQ = 90°

As the sum of all angles of a triangle is 180°

In triangle PRQ,

∠QPR = 180 – 55 – 90 = 35°

∠PMQ = 90°

As sum of all angles of triangle = 180°

In triangle PQM,

∠QPM = 180– 90– 50 = 40°

Triangle formed by POS is an isosceles triangle as two of sides are radius OP and OS.

Hence ∠OPS = ∠OSP = 35 + 25 = 60°

As sum of all angles of a triangle is 180°.

∠POS = 180 –60–60 = 60°

According to theorem that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference.

Angle subtended by chord SP on centre = 60°

Hence angle subtended by chord SP on circumference is

Hence ∠PRS = 30°