Congruent triangles - Edexcel GCSE Maths (Non-calculator) Exam Questions

Practise 28 exam-style questions on Congruent triangles for Edexcel GCSE Maths (Non-calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.

28 questions
1–5 marks each
Instant marking
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20 questions, marked as you go

Sample questions with model answers

Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.

Question 1

2 marks

The diagram shows two triangles, ABC and PQR.

Explain why triangles ABC and PQR are congruent.

Geometry diagram, marked "Diagram NOT accurately drawn". Triangle ABC. Triangle PQR. Side AB is labelled 5 cm. Side BC is labelled 7 cm. Angle ABC is marked 40°. Side PQ is labelled 5 cm. Side QR is labelled 7 cm. Angle PQR is marked 40°.

Model answer

AB = PQ = 5 cm
BC = QR = 7 cm
Angle ABC = angle PQR = 40°, and in each triangle this angle is between the two given sides.
So the triangles are congruent (SAS).

Explanation

For SAS the angle must be the included angle, between the two equal sides. Here the 40° angle is at B, between AB and BC, and at Q, between PQ and QR, so SAS applies. The triangles face different ways, but that does not matter for congruence.

Mark scheme

1 mark
C1 for identifying the equal parts: AB = PQ = 5 cm, BC = QR = 7 cm and angle ABC = angle PQR = 40°
1 mark
C1 for the angle is between the two sides in both triangles, so they are congruent by SAS
must state SAS (or side, angle, side)

Question 2

3 marks

In triangle ABC, angle A = 50°, angle B = 60° and AB = 8 cm.
In triangle PQR, angle P = 50°, angle R = 70° and PQ = 8 cm.

Are triangles ABC and PQR congruent? You must give a reason for your answer.

Model answer

Angle Q = 180° − 50° − 70° = 60°
So angle A = angle P = 50° and angle B = angle Q = 60°.
AB = PQ = 8 cm, and in each triangle this side is between the two angles.
Yes, the triangles are congruent (ASA).

Explanation

Find the third angle of triangle PQR so you can compare the angles at the ends of the 8 cm sides. AB joins the 50° and 60° angles, and so does PQ, so ASA applies.

Mark scheme

1 mark
M1 for finding a missing angle, e.g. angle Q = 180 − 50 − 70 = 60° (or angle C = 70°)
1 mark
C1 for angle A = angle P, angle B = angle Q and the equal sides AB and PQ are between these angles
1 mark
C1 for Yes, congruent by ASA

Question 3

3 marks

ABCD is a rectangle.

Use congruent triangles to prove that the diagonals AC and BD are equal in length.

Model answer

Consider triangles ABC and BAD.
AB is a common side.
BC = AD (opposite sides of a rectangle are equal)
angle ABC = angle BAD = 90° (angles of a rectangle)
The right angle is between the two sides in each triangle.
So triangles ABC and BAD are congruent (SAS).
Hence AC = BD (corresponding sides).

Explanation

Pick the two triangles that share side AB and each contain one diagonal. They have two equal sides with a right angle between them, so they are congruent by SAS, and the diagonals are corresponding sides.

Mark scheme

1 mark
M1 for choosing two suitable triangles, e.g. ABC and BAD, with BC = AD (opposite sides of a rectangle)
1 mark
M1 for angle ABC = angle BAD = 90° and AB common
1 mark
C1 for triangles ABC and BAD are congruent (SAS), so AC = BD

Question 4

4 marks

The diagram shows a square ABCD.
E is a point on BC and F is a point on CD such that BE = CF.

Prove that triangles ABE and BCF are congruent.
Hence prove that AE = BF.

Geometry diagram, marked "Diagram NOT accurately drawn". Square ABCD. Line segments AE and BF are drawn. E lies on side BC. F lies on side CD. BE and CF are each marked with one short dash (equal lengths).

Model answer

In triangles ABE and BCF:
AB = BC (sides of a square are equal)
angle ABE = angle BCF = 90° (angles of a square)
BE = CF (given)
The right angle is between the two equal sides in each triangle.
So triangles ABE and BCF are congruent (SAS).
Hence AE = BF (corresponding sides of congruent triangles).

Explanation

Match the triangles carefully: AB with BC, the right angle at B with the right angle at C, and BE with CF. In each triangle the right angle lies between the two sides, so SAS applies. Then AE and BF are corresponding sides, opposite the right angles.

Mark scheme

1 mark
M1 for AB = BC (sides of a square)
1 mark
M1 for angle ABE = angle BCF = 90° (angles of a square)
1 mark
M1 for BE = CF (given)
1 mark
C1 for triangles ABE and BCF are congruent (SAS), so AE = BF (corresponding sides)
the right angles must be between the pairs of equal sides

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