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0580/32 May/Jun 2025, Question 10: Combining Circle Theorems

Cambridge IGCSE Mathematics 0580 · Paper 32 (Core) · Calculator · 5 marks

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The question, in short

Using a circle with centre O and diameter AC, find angles x and y and give geometrical reasons for x, y and for angle ABC being 90°.

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

Part (a) · 2 marks

Find the value of angle x (angle ABO) and state the geometrical reason.

  1. 1OA = OB (radii), so triangle OAB is isosceles
  2. 2x = angle BAO = 65 because base angles of an isosceles triangle are equal

Answer65 base angles of an isosceles triangle are equal

Part (b) · 1 mark

State the geometrical reason why angle ABC equals 90°.

  1. 1AC is a diameter, so angle ABC = 90° because the angle in a semicircle equals 90°

AnswerAngle in a semicircle equals 90

Part (c) · 2 marks

Find the value of angle y (angle BCO) and state the geometrical reason.

  1. 1AC is a diameter, so angle ABC = 90° (angle in a semicircle).
  2. 2In triangle ABC: y = 180 − 90 − 65
  3. 3y = 25 because the angle sum of a triangle equals 180°

Answer25 Angle sum of a triangle equals 180

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