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4024/13 May/Jun 2026, Question 15: Combining Circle Theorems

Cambridge O Level Mathematics 4024 · Paper 13 (Non-calculator) · Non-calculator · 5 marks

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The question, in short

Using the circle with tangent PT at T and straight line PCA, given angle APT = 27° and angle TAP = 30°, find the four marked angles.

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Part (a) · 1 mark

Find angle TBC.

  1. 1Angles in the same segment are equal
  2. 2angle TBC = angle TAC = 30°

Answer30

Part (b) · 1 mark

Find angle PTC.

  1. 1Alternate segment theorem
  2. 2angle PTC = angle TAC = 30°

Answer30

Part (c) · 2 marks

Find angle ATC.

  1. 1In triangle APT: angle PTA = 180 − 27 − 30 = 123°
  2. 2angle ATC = 123° − angle PTC = 123 − 30
  3. 3= 93°

Answer93

Part (d) · 1 mark

Find angle ABC.

  1. 1ABCT is a cyclic quadrilateral: opposite angles add to 180°
  2. 2angle ABC = 180 − 93 = 87°

Answer87

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