EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Exam Practice: Geometric Proof and Congruence
Geometric proof and congruence · Similarity and congruence · Lesson 2 of 5 · 18 marks · 30 minutes
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Instructions
• Answer all the questions.
• Write your answers in the spaces provided.
• The marks for each question are shown in brackets - use this as a guide to how much to write.
• The answers are on separate pages at the back. Attempt every question before you look at them.
• Answer all questions. Give reasons for every step.
Question 1 PROVE (4 marks)
ABCD is a parallelogram. The diagonals AC and BD meet at M. Prove that triangles AMB and CMD are congruent.
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(Total for Question 1 = 4 marks)
Question 2 PROVE (2 marks)
Using the congruent triangles from the last question, prove that AM = MC.
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(Total for Question 2 = 2 marks)
Question 3 PROVE (4 marks)
Triangle ABC is isosceles with AB = AC. D is the midpoint of BC. Prove that angle ABC = angle ACB.
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(Total for Question 3 = 4 marks)
Question 4 PROVE (3 marks)
PQRS is a kite with PQ = PS and RQ = RS. Prove that angle PQR = angle PSR.
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(Total for Question 4 = 3 marks)
Question 5 EXPLAIN (2 marks)
Explain why "the triangles look the same" is not a good reason in a proof.
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(Total for Question 5 = 2 marks)
Question 6 PROVE (3 marks)
ABCD is a rectangle. The diagonals AC and BD are drawn. Prove that AC = BD.
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(Total for Question 6 = 3 marks)
TOTAL FOR PAPER = 18 MARKS
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Answers and mark scheme
Check your answer only once you have written one.
Question 1 (4 marks)
AB = DC (opposite sides of a parallelogram). Angle BAM = angle DCM (alternate angles, AB parallel to DC). Angle ABM = angle CDM (alternate angles). So the triangles are congruent by ASA.
• AB = DC with a reason B1
• One pair of equal angles with a reason M1
• The second pair of equal angles with a reason M1
• Congruent by ASA A1
Question 2 (2 marks)
Triangles AMB and CMD are congruent, so corresponding sides are equal. AM corresponds to CM, so AM = MC.
• Corresponding sides of congruent triangles M1
• AM = MC A1
Question 3 (4 marks)
AB = AC (given). BD = DC (D is the midpoint). AD is common. So triangles ABD and ACD are congruent by SSS, and angle ABC = angle ACB (corresponding angles).
• AB = AC and BD = DC with reasons M1
• AD common M1
• SSS A1
• Conclusion about angles C1
Question 4 (3 marks)
PQ = PS and RQ = RS (given). PR is common. The triangles PQR and PSR are congruent by SSS, so angle PQR = angle PSR.
• Equal sides and common side M1
• SSS M1
• Conclusion A1
Question 5 (2 marks)
A proof must use exact facts, not appearances or measurements. Each step needs a mathematical reason such as SSS or a parallel-line rule.
• Not appearances or measurements M1
• Each step needs a mathematical reason C1
Question 6 (3 marks)
AB = DC (opposite sides of a rectangle). BC is common to triangles ABC and DCB. Angle ABC = angle DCB = 90°. So triangles ABC and DCB are congruent by SAS, and AC = BD.
• Equal sides and right angles M1
• SAS M1
• AC = BD A1