OCR GCSE Mathematics: Congruence and congruence proofs (J560) – Revision Notes

These are condensed revision notes for OCR GCSE (9-1) Mathematics (J560), Version 2.1 (August 2026), content references 9.02a and 9.02b. Both tiers are covered and no content here is Higher tier only. For full explanations and longer worked proofs, read the study guide. Test yourself afterwards with the practice questions. Course pages: OCR GCSE Mathematics hub, checklist and the free diagnostics.

Key definitions

Term	Meaning
Congruent	exactly the same shape and size; can be moved onto each other by translation, rotation or reflection
Corresponding	matching sides or angles in congruent shapes; they are equal
Included angle	the angle between two named sides
Common side	a side shared by both triangles in a diagram
Hypotenuse	the longest side of a right-angled triangle, opposite the right angle
Isosceles triangle	two equal sides; the two angles opposite them are equal

Congruent is not the same as similar. Similar shapes have the same shape but may differ in size. Congruent shapes are identical in both.

The four conditions (9.02a)

Name	Facts needed	Picture in words
SSS	three pairs of equal sides	all three lengths match
SAS	two equal side pairs, with the equal angle lying between those sides	angle sits inside the two sides
ASA	two pairs of equal angles and one pair of equal sides	side matches in the same position
RHS	right angle, equal hypotenuses, one more equal side	both triangles right-angled

Not accepted as proof: AAA (same shape, size may differ) and SSA, meaning two sides and a non-included angle (can give two different triangles).

Method in steps

1. Name the triangles with matching letters, for example ABC ≅ ZXY.
2. Write three facts about equal pairs, each with a reason.
3. Say which condition applies and that the triangles are congruent.
4. State what follows from corresponding sides or angles.

Reasons to quote:

Situation	Reason
Crossing lines	vertically opposite angles are equal
Shared side	common side
Parallel lines	alternate angles are equal
Corner of a square or rectangle	angle of 90°
Midpoint	divides the line into two equal lengths
Bisector of an angle	splits the angle into two equal angles
Equal sides in an isosceles triangle	base angles are equal

Worked reminders

Identify congruent triangles

Triangle 1 has sides 6 cm, 7 cm, 9 cm. Triangle 2 has sides 9 cm, 6 cm, 7 cm. All three sides match, so they are congruent (SSS). Triangle 3 has angles 50°, 60°, 70° and a 4 cm side, while triangle 4 has the same angles and a 9 cm side. They are not congruent, because AAA does not prove congruence.

Isosceles triangles (9.02b)

If AB = AC, the angles at B and C are equal. To prove it, draw the bisector of angle A to meet BC at D. Triangles ABD and ACD have AB = AC, equal angles at A and the common side AD, so they are congruent (SAS). Then angle B = angle C.

Two uses:

Apex angle 56°  →  each base angle = (180° − 56°) ÷ 2 = 62°
Base angle 35°  →  apex angle = 180° − 2 × 35° = 110°

In an equilateral triangle all three sides are equal, so each pair of base angles is equal and every angle is 60°.

Perpendicular from the apex (RHS)

In an isosceles triangle ABC with AB = AC, let M be the foot of the perpendicular from A to BC. The triangles AMB and AMC share AM, have equal hypotenuses AB and AC, and each contains a right angle. They are congruent by RHS, so BM = MC. This result is often used with Pythagoras.

Right-angled triangle with hypotenuse 10 cm and one side 6 cm
Other side² = 10² − 6² = 64, so the other side = 8 cm
Any other right-angled triangle with hypotenuse 10 cm and a side of 6 cm
is congruent to it (RHS).

Algebra with congruence

If corresponding sides are 5x − 4 and 3x + 10, set them equal.

5x − 4 = 3x + 10
2x = 14
x = 7, so each side = 5(7) − 4 = 31

Always substitute back into both expressions to check they agree.

Proof of equal diagonal segments

Lines PQ and RS cross at O. O is the midpoint of both lines. Triangles POR and QOS have PO = QO, equal vertically opposite angles at O, and RO = SO, so they are congruent (SAS). Hence PR = QS.

Proof templates

Most proofs follow one of these patterns. Learn the pattern, then fit the facts to it.

Pattern	Typical facts
SAS from crossing lines	two pairs of equal lengths from midpoints, plus vertically opposite angles
ASA in a parallelogram	two pairs of alternate angles from the parallel sides, plus a common diagonal
SSS in a kite or rhombus	pairs of equal sides, plus the diagonal as a common side
SAS in an isosceles triangle	equal sides, base angles equal, common base
RHS from perpendiculars	two right angles, equal hypotenuses such as radii, one equal side

A proof using a shared angle

Triangle ABC has AB = AC. D lies on AB and E lies on AC with AD = AE. Prove BE = CD.

Triangles ABE and ACD
AB = AC (given)
Angle A is common
AE = AD (given)
So ABE ≅ ACD (SAS), and BE = CD

A shared angle at the apex counts as a common angle, so write “angle A is common” as the reason.

Must-know distinctions

• SAS needs the angle between the two sides. Two sides and a different angle is SSA, which fails.
• ASA needs matching positions for the equal side. Two angles fix the third, so a side in the same position on both triangles works.
• RHS needs a right angle in both triangles. It is not a general SSA rule.
• Congruent has equal sizes. Similar does not.
• Corresponding means in the same position, not in the same place on the page.
• To prove congruence you must give three facts. State a condition without the three facts and the proof is incomplete.

Quick self-test

1. What does it mean to say two triangles are congruent?
2. Triangle 1 has sides 5 cm, 6 cm, 7 cm. Triangle 2 has sides 6 cm, 7 cm, 5 cm. Which condition shows they are congruent?
3. Why do three equal angles not prove two triangles are congruent?
4. In triangle ABC, AB = AC and angle A = 56°. Find angle B.
5. An isosceles triangle has a base angle of 35°. Find the angle at the apex.
6. Two right-angled triangles each have a hypotenuse of 10 cm and one other side of 6 cm. State the condition that shows they are congruent and find the third side.
7. Corresponding sides of two congruent triangles are 5x − 4 and 3x + 10. Find x and the side length.
8. In the statement ABC ≅ ZXY, which side of triangle ZXY is equal to BC?
9. An equilateral triangle has side 8 cm. Give the size of each angle and explain why.
10. Two triangles have sides 8 cm and 6 cm and a 40° angle, but in one triangle the 40° angle is between the sides and in the other it is not. Can you say they are congruent?

Answers

1. They are exactly the same shape and size, so all corresponding sides and angles are equal.
2. SSS, because all three sides match.
3. They have the same shape but can be different sizes. A small and a large triangle can have the same angles.
4. Base angles are equal, so angle B = (180° − 56°) ÷ 2 = 62°.
5. Angle at apex = 180° − 2 × 35° = 110°.
6. RHS (right angle, hypotenuse, one side). The third side is √(10² − 6²) = 8 cm.
7. 5x − 4 = 3x + 10 gives 2x = 14, so x = 7 and each side = 31.
8. B matches X and C matches Y, so BC = XY.
9. The three sides are equal, so by the base angles result all angles are equal. Each is 180° ÷ 3 = 60°.
10. No. The angle must be between the sides for SAS. Two sides and a non-included angle does not prove congruence.

Where marks are usually lost

• Writing the congruence statement with letters in the wrong order.
• Quoting a condition with no supporting facts, or giving a fact with no reason.
• Using SSA or AAA as if they were accepted conditions.
• Calling a side “common” when it is not shared by both triangles.
• Assuming a right angle or a pair of parallel lines from the picture alone.
• Forgetting the final deduction, such as “so AC = BD” or “so angle B = angle C”.
• Applying the isosceles angle result to the wrong pair of angles (the equal angles are opposite the equal sides).
• Setting corresponding sides equal in the wrong pairs when forming an equation in x.
• Giving a fully worked proof of a result the question only asked you to use.

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).