Lemmara logoLemmaraWorked solution

0580/21 May/Jun 2023, Question 12: Congruent Triangles: Proof Conditions

Cambridge IGCSE Mathematics 0580 · Paper 21 (Extended) · Calculator · 4 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Circle geometry: choose the congruence criterion for two triangles formed by radii and equal chords, then use a tangent, parallel chords and angle RQU = 42° to calculate three angles.

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

Part (a) · 1 mark

Select which congruence test (SAS, RHS, SSS or ASA) justifies triangle AOB being congruent to triangle COB.

  1. 1OA = OC (radii), OB common, AB = BC: three pairs of equal sides
  2. 2SSS

AnswerSSS

Part (bi) · 1 mark

Calculate angle QSR using the tangent at Q and angle RQU = 42°.

  1. 1Alternate segment theorem: angle between tangent QU and chord QR = angle QSR
  2. 2Angle QSR = 42°

Answer42

Part (bii) · 1 mark

Calculate angle PQS using the circle geometry and parallel lines given.

  1. 1PQ is parallel to SR, so angle PQS = angle QSR (alternate angles)
  2. 2Angle PQS = 42°

Answer42

Part (biii) · 1 mark

Calculate angle POS at the centre using the earlier angle results.

  1. 1Angle at the centre is twice the angle at the circumference
  2. 2Angle POS = 2 × angle PQS = 2 × 42 = 84°

Answer84

Practise more congruent triangles: proof conditions questions

Real 0580 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on congruent triangles: proof conditions

Also searched as: 0580/21/M/J/23 · 0580_s23_qp_21 · 0580 s23 21 · question 12.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.