OCR GCSE Mathematics: Congruence and congruence proofs (J560) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover OCR GCSE (9-1) Mathematics (J560), Version 2.1 (August 2026), content references 9.02a and 9.02b. Both tiers are covered and none of it is Higher tier only. The mark allocations are set by Marlbridge for practice. Read the study guide first, then use the revision notes as a quick reminder. Course pages: OCR GCSE Mathematics hub, checklist and the free diagnostics.

Questions

1. Four triangles are described. Triangle P: sides 8 cm, 10 cm, 12 cm. Triangle Q: sides 12 cm, 8 cm, 10 cm. Triangle R: sides 8 cm and 6 cm with a 40° angle between them. Triangle S: sides 8 cm and 6 cm with a 40° angle opposite the 6 cm side.

(a) Show that P and Q are congruent, naming the condition. [2] (b) Explain whether R and S can be proved congruent. [2]

2. Name the condition (SSS, SAS, ASA or RHS) that proves each pair congruent.

(a) Both triangles have sides 5 cm, 12 cm and 13 cm. [1] (b) Both have sides 6 cm and 9 cm with a 70° angle between them. [1] (c) Both are right-angled, with hypotenuse 10 cm and another side 6 cm. [1] (d) Both have angles 50° and 60°, and the 8 cm side between those two angles. [1]

3. Triangle ABC is congruent to triangle ZXY, with A matching Z, B matching X and C matching Y. AB = 12.4 cm, BC = 9.1 cm, angle A = 33° and angle C = 95°.

(a) Write down the lengths of ZX and XY. [2] (b) Find the sizes of angle Z and angle X. [2]

4. ABCD is a square. E lies on AB and F lies on BC, with AE = BF. Prove that DE = AF. [4]

5. PQRS is a parallelogram. The diagonals PR and QS meet at O. Prove that OP = OR. [5]

6. O is the centre of a circle. AB and CD are chords. M is the foot of the perpendicular from O to AB, and N is the foot of the perpendicular from O to CD. OM = ON. Prove that AM = CN. [4]

7. In triangle ABC, AB = AC. D lies on AB and E lies on AC with BD = CE. Prove that BE = CD. [5]

8. (a) Rani says two triangles are congruent because both have angles of 50°, 60° and 70°. Explain why she is wrong. [2] (b) Tom says two triangles are congruent because both have sides of 5 cm and 7 cm and a 30° angle. What extra information about the 30° angle would make his argument valid? [1]

9. Triangle ABC is congruent to triangle DEF, with A matching D, B matching E and C matching F. AB = (4x − 3) cm, DE = (2x + 9) cm, BC = (y + 7) cm, EF = (3y − 5) cm, angle C = (2z + 14)° and angle F = (3z − 16)°.

(a) Find x and the length of AB. [2] (b) Find y and the length of BC. [2] (c) Find z and the size of angle C. [2]

10. ABCD is a rhombus, so all four sides are equal. The diagonal AC is drawn.

(a) Prove that triangle ABC is congruent to triangle ADC. [3] (b) Hence prove that AC bisects angle DAB. [2] (c) Angle ABC = 70°. Find angle BAC, giving a reason. [3]

11. Triangle PQR has PQ = PR. X lies on PQ and Y lies on PR with QX = RY. The lines QY and RX meet at Z.

(a) Prove that triangle XQR is congruent to triangle YRQ. [4] (b) Angle XRQ = 35°. Explain why angle YQR = 35°. [1] (c) Find angle QZR. [2]

Answers

1. (a) P has sides 8, 10, 12 and Q has sides 12, 8, 10, so all three pairs of sides are equal [1]. Condition SSS, so P and Q are congruent [2] [1] (b) In R the 40° angle is between the 8 cm and 6 cm sides, but in S it is opposite the 6 cm side [1]. The equal angle is in a different position, so the SAS condition is not met and congruence cannot be proved [2] [1] Examiner insight: in part (b) say where the angle is in each triangle. A bare “no” gives the reader nothing to credit.

2. (a) SSS [1] (b) SAS [1] (c) RHS [1] (d) ASA [1] Examiner insight: name the condition with all its letters. For (c), the right angle is part of RHS, so say it is present rather than just writing “H”.

3. (a) ZX matches AB, so ZX = 12.4 cm [1]. XY matches BC, so XY = 9.1 cm [1] (b) Angle Z matches angle A, so angle Z = 33° [1]. Angle B = 180° − 33° − 95° = 52°, and angle X matches B, so angle X = 52° [1] Examiner insight: the order of the letters in ABC ≅ ZXY decides the matching. Write the pairs before substituting numbers.

4. Triangles DAE and ABF. DA = AB, because the sides of a square are equal [1] Angle DAE = angle ABF = 90°, the angles of a square [1] AE = BF, given [1] The triangles are congruent (SAS), with the right angle between the equal sides. Corresponding sides are equal, so DE = AF [4] [1] Examiner insight: each fact needs its reason, and you must state the condition before the conclusion. A correct conclusion on its own does not show the proof.

5. Triangles OPQ and ORS. Angle OPQ = angle ORS, alternate angles, PQ parallel to SR [1] Angle OQP = angle OSR, alternate angles, PQ parallel to SR [1] PQ = SR, opposite sides of a parallelogram [1] The triangles are congruent (ASA) [1] Corresponding sides are equal, so OP = OR [5] [1] Examiner insight: the equal side PQ = SR lies between the two equal angles in each triangle, so say that when you quote ASA.

6. Triangles OMA and ONC. Angle OMA = angle ONC = 90°, perpendiculars [1] OA = OC, both are radii, so the hypotenuses are equal [1] OM = ON, given [1] The triangles are congruent (RHS), so AM = CN [4] [1] Examiner insight: in RHS, label the equal sides as hypotenuses. A right angle alone is not a full reason.

7. Triangles DBC and ECB. BD = CE, given [1] Angle DBC = angle ECB, base angles of the isosceles triangle ABC [1] BC = CB, common side [1] The triangles are congruent (SAS) [1] Corresponding sides are equal, so BE = CD [5] [1] Examiner insight: the base angles result may be quoted as a reason here. Make sure the angles you name are the ones between the equal sides.

8. (a) Three equal angles mean the triangles have the same shape [1]. They can still be different sizes, for example one with a 4 cm side and one with a 9 cm side, so AAA does not prove congruence [2] [1] (b) The 30° angle would have to be between the 5 cm and 7 cm sides (SAS) [1] [1] Examiner insight: give a concrete case in (a), such as two sizes of triangle. “They might not be the same” is vague.

9. (a) 4x − 3 = 2x + 9 [1], so x = 6 and AB = 4(6) − 3 = 21 cm [2] [1] (b) y + 7 = 3y − 5 [1], so y = 6 and BC = 6 + 7 = 13 cm [2] [1] (c) 2z + 14 = 3z − 16 [1], so z = 30 and angle C = 2(30) + 14 = 74° [2] [1] Examiner insight: substitute back into both expressions. If the two values differ, you have set up the wrong equation.

10. (a) Triangles ABC and ADC. AB = AD, sides of a rhombus [1] BC = DC, sides of a rhombus [1] AC is common, and the triangles are congruent (SSS) [1] (b) Corresponding angles are equal, so angle BAC = angle DAC [1]. Hence AC bisects angle DAB [2] [1] (c) Triangle ABC has AB = BC, so it is isosceles [1]. Base angles are equal, so angle BAC = (180° − 70°) ÷ 2 [1] = 55° [3] [1] Examiner insight: in (b) point to the pair of corresponding angles. For (c), say which two sides are equal so the base angles can be justified.

11. (a) Triangles XQR and YRQ. XQ = YR, given [1] Angle XQR = angle YRQ, base angles of isosceles triangle PQR [1] QR is common [1] The triangles are congruent (SAS) [4] [1] (b) Corresponding angles in congruent triangles are equal, so angle YQR = angle XRQ = 35° [1] [1] (c) Triangle ZQR has angles YQR = 35° and XRQ = 35° [1]. Angle QZR = 180° − 35° − 35° = 110° [2] [1] Examiner insight: in (c) use the result from (b) rather than starting again.

Where marks are usually lost

• Writing a claim without a reason, such as “AB = CD” with no reference to the question or a property.
• Naming the wrong condition, or using SSA or AAA.
• Quoting SAS when the angle is not between the two sides.
• Writing the vertices in an order that does not match, so the final deduction names the wrong sides.
• Leaving out “common side” or “vertically opposite angles” when a fact comes from the diagram.
• Ending with “congruent” and not stating the length or angle that follows.
• Setting corresponding quantities equal in the wrong pairs when solving for x, y or z.
• Using an isosceles result without saying which sides are equal.

Next steps

• Review the revision notes and the study guide.
• Go to the OCR GCSE Mathematics hub and the printable checklist.
• Try the free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

Aligned to the OCR GCSE (9-1) in Mathematics (J560) specification, Version 2.1, August 2026, published by OCR, for first assessment in 2017. Content references 9.02a (Congruent triangles) and 9.02b (Applying congruent triangles).