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0580/13 May/Jun 2026, Question 9: Tangent Properties of a Circle

Cambridge IGCSE Mathematics 0580 · Paper 13 (Core) · Non-calculator · 5 marks

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The question, in short

Circle with centre O, tangent DAE at A, DE parallel to BC, BOE straight, angle AEO = 35°. Find reasons for angle AOE = 55°, then x and y.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 2 marks

Give two geometrical reasons explaining why angle AOE equals 55°.

  1. 1Angle OAE = 90° because the tangent is perpendicular to the radius
  2. 2Angles in a triangle add to 180°
  3. 3Angle AOE = 180 − 90 − 35 = 55°

AnswerAngle between tangent and radius = 90°; angles in a triangle add to 180°

Part (b) · 1 mark

Find the value of x, the angle at B between BO and BC.

  1. 1DE is parallel to BC
  2. 2Angle OBC = angle BEA (alternate angles)
  3. 3x = 35

Answer35

Part (c) · 2 marks

Find the value of y, the angle at O between OB and OC.

  1. 1OB = OC (radii), so triangle OBC is isosceles
  2. 2Angle OCB = x = 35°
  3. 3y = 180 − 2 × 35 = 110

Answer110

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