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0580/22 Feb/Mar 2023, Question 17: Combining Circle Theorems

Cambridge IGCSE Mathematics 0580 · Paper 22 (Extended) · Calculator · 6 marks

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The question, in short

In a circle with cyclic quadrilateral ABCD, AB extended to X, tangent PQ at A, given angle CBX = 85°, angle BAQ = 55°, angle CAD = 42°, find five angles.

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Part (a) · 1 mark

Find angle CBD.

  1. 1Angle CBD = angle CAD = 42° (angles in the same segment)

Answer42

Part (b) · 1 mark

Find angle ACB.

  1. 1Angle ACB = angle BAQ = 55° (alternate segment theorem)

Answer55

Part (c) · 1 mark

Find angle ADC.

  1. 1Angle ADC = angle CBX = 85° (exterior angle of a cyclic quadrilateral equals the interior opposite angle)

Answer85

Part (d) · 2 marks

Find angle BCD.

  1. 1Angle ADC = angle CBX = 85° (exterior angle of a cyclic quadrilateral equals the interior opposite angle)
  2. 2Angle ACD = 180 − 85 − 42 = 53° (angle sum of triangle ACD)
  3. 3Angle ACB = angle BAQ = 55° (alternate segment theorem)
  4. 4Angle BCD = angle ACB + angle ACD = 55 + 53 = 108°

Answer108

Part (e) · 1 mark

Find angle PAD.

  1. 1Angle PAD = angle ACD = 53° (alternate segment theorem)

Answer53

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