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4024/12 May/Jun 2022, Question 17: Congruent Triangles: Proof Conditions

Cambridge O Level Mathematics 4024 · Paper 12 (Non-calculator) · Non-calculator · 4 marks

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The question, in short

Identify which listed quadrilaterals always have both diagonals splitting them into two congruent isosceles triangles, then prove two triangles congruent using midpoint and parallel lines.

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

Part (a) · 1 mark

From square, rectangle, rhombus, parallelogram, trapezium and kite, select every quadrilateral whose diagonals always create two congruent isosceles triangles.

  1. 1Each diagonal must split the shape into two congruent isosceles triangles, so all four sides must be equal
  2. 2Square, Rhombus

AnswerSquare, Rhombus

Part (b) · 3 marks

Using the diagram with X the midpoint of AB and AC parallel to DB, prove triangle AXC congruent to triangle BXD, justifying each step.

  1. 1AX = BX because X is the midpoint of AB (given)
  2. 2Angle AXC = angle BXD (vertically opposite angles)
  3. 3Angle CAX = angle DBX (alternate angles, AC parallel to DB)
  4. 4So triangle AXC is congruent to triangle BXD (ASA)

AnswerTwo correct pairs of angles from ∠AXC = ∠BXD, ∠CAX = ∠DBX and ∠ACX = ∠BDX with correct reasons and AX = BX with correct reason and ASA stated e.g. AX = BX given; ∠AXC = ∠BXD vertically opposite; ∠CAX = ∠DBX alternate angles; congruent ASA

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Also searched as: 4024/12/M/J/22 · 4024_s22_qp_12 · 4024 s22 12 · question 17.
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