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0580/33 May/Jun 2022, Question 8: Angles in a Triangle

Cambridge IGCSE Mathematics 0580 · Paper 33 (Core) · Calculator · 13 marks

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The question, in short

Triangle PQR is isosceles with PQ = PR and QRS straight, angle PRS = 114°; circle with tangent AB at E parallel to CD, angle BED = 68°; sector radius 7.5 cm, angle 60°.

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

Part (ai) · 1 mark

State the mathematical name of triangle PQR, given PQ = PR.

  1. 1PQ = PR, so the triangle is isosceles

AnswerIsosceles

Part (aii) · 3 marks

Calculate angle QPR using the isosceles triangle and the straight line QRS with angle PRS = 114°.

  1. 1Angle PRQ = 180 − 114 = 66° (angles on a straight line)
  2. 2Angle PQR = 66° (isosceles triangle)
  3. 3Angle QPR = 180 − 66 − 66 = 48°

Answer48

Part (bi) · 2 marks

Find angle CDE and give a reason, using tangent AB parallel to CD and angle BED = 68°.

  1. 1Angle CDE = angle BED = 68°
  2. 2Reason: alternate angles (AB parallel to CD)

Answer68, alternate angles

Part (bii) · 2 marks

Find angle DEF and give a reason, using the circle with centre O and tangent at E.

  1. 1FE is a diameter, so angle BEF = 90° (angle between tangent and radius = 90°)
  2. 2Angle DEF = 90 − 68 = 22°

Answer22, angle between tangent and radius = 90°

Part (biii) · 3 marks

Calculate angle EFD and state two further geometrical properties needed to find it.

  1. 1Angle EDF = 90° (angle in a semicircle = 90°)
  2. 2Angle EFD = 180 − 90 − 22 = 68° (angles in a triangle add to 180°)

Answer68, with reasons: angle in a semicircle = 90° and angles in a triangle add to 180°

Part (c) · 2 marks

Calculate the arc length PQ of the sector with radius 7.5 cm and sector angle 60°.

  1. 1Arc = 60/360 × 2 × π × 7.5
  2. 2= 2.5π = 7.853… ≈ 7.85 cm

Answer7.85 or 7.86 or 7.853 to 7.855

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Also searched as: 0580/33/M/J/22 · 0580_s22_qp_33 · 0580 s22 33 · question 8.
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