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0580/43 Oct/Nov 2023, Question 4: Combining Circle Theorems

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 7 marks

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The question, in short

Calculate a regular decagon's interior angle, then use a circle diagram with tangent FG at D and parallel chords EB and DC to determine the four marked angles w, x, y and z.

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

Part (a) · 2 marks

Work out the size of a single interior angle in a regular polygon with 10 sides.

  1. 1Exterior angle = 360 ÷ 10 = 36°
  2. 2Interior angle = 180 − 36 = 144°

Answer144

Part (b) · 5 marks

Using the circle with tangent FG at D, EB parallel to DC, angle ACD = 75°, angle FDE = 20° and angle GDC = 25°, determine w, x, y and z.

  1. 1w = angle EDF = 20 (alternate segment theorem)
  2. 2x = angle EBD = 20 (angles in the same segment)
  3. 3Angle DAC = angle GDC = 25 (alternate segment), so angle ADC = 180 − 75 − 25 = 80; angle BDC = w = 20 (alternate angles, EB ∥ DC)
  4. 4y = 80 − 20 = 60
  5. 5Angle EDC = 180 − 20 − 25 = 135, so z = 180 − 135 = 45 (co-interior angles, EB ∥ DC)
  6. 6w = 20, x = 20, y = 60, z = 45

Answerw = 20, x = 20, y = 60, z = 45

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