Lemmara logoLemmaraWorked solution

0580/43 May/Jun 2023, Question 4: Congruent Triangles: Proof Conditions

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 13 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Use bearings and circle theorems to find angles: bearing of B from C from triangle ABC, angles in a cyclic quadrilateral with a tangent, and complete a congruence proof that CB is parallel to OT.

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

Part (a) · 3 marks

Given bearing of C from A is 114°, B due south of A, AC = BC, find the bearing of B from C.

  1. 1Angle BAC = 180° − 114° = 66°
  2. 2AC = BC, so angle ABC = 66° and angle ACB = 180 − 66 − 66 = 48°
  3. 3Bearing of A from C = 114 + 180 = 294°
  4. 4Bearing of B from C = 294 − 48 = 246°

Answer246°

Part (bi) · 1 mark

With tangent MPN and angles MPS = 58°, PSR = 74°, QPN = 27°, find angle PRS.

  1. 1Alternate segment theorem: angle PRS = angle MPS
  2. 2Angle PRS = 58°

Answer58°

Part (bii) · 1 mark

Using the same circle and tangent data, find angle PQR.

  1. 1PQRS is a cyclic quadrilateral, opposite angles sum to 180°
  2. 2Angle PQR = 180 − 74 = 106°

Answer106°

Part (biii) · 2 marks

Using the same circle and tangent data, find angle RPQ.

  1. 1Alternate segment theorem: angle PRQ = angle QPN = 27°
  2. 2Angles in triangle PQR: angle RPQ = 180 − 106 − 27
  3. 3= 47°

Answer47°

Part (c) · 6 marks

Complete the statements, giving values and reasons, to prove CB is parallel to OT using congruent triangles AOT and BOT and angle ATO = 34°.

  1. 1Angle OAT = angle OBT = 90° because a radius is perpendicular to a tangent
  2. 2AT = BT because tangents from the same point to a circle are equal
  3. 3Congruent by RHS
  4. 4Angle BOC = 180 − 56 − 56 = 68° because angles on a straight line AOC add up to 180°
  5. 5OB = OC (radii), so triangle OBC is isosceles: angle OBC = (180 − 68)/2 = 56°
  6. 6Angle OBC = angle BOT = 56° are alternate angles, so CB is parallel to OT

AnswerRadius is perpendicular to tangent; tangents from the same point are equal; RHS; angle BOC = 68° because angles on a straight line add up to 180°; angle OBC = 56° because triangle OBC is isosceles (base angles equal); angle OBC = angle BOT = 56° so they are alternate angles

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: 0580/43/M/J/23 · 0580_s23_qp_43 · 0580 s23 43 · question 4.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.