OCR GCSE Mathematics: Congruence and congruence proofs (J560)

This study guide covers congruence for the OCR GCSE (9-1) Mathematics specification (J560), Version 2.1 (August 2026). It follows content references 9.02a and 9.02b, which sit in content area 9, Congruence and similarity. The content applies to both Foundation tier and Higher tier, and nothing in this unit is marked Higher tier only. The specification expects you to support answers with appropriate working, and some questions ask for an extended line of reasoning, so proofs are written out in full.

Use this page with the revision notes and the practice questions. The course hub is at OCR GCSE Mathematics, the printable list is the checklist, and the free diagnostics show where to start. For wider exam habits see OCR GCSE Mathematics exam preparation.

What this unit covers

Ref	What you must be able to do	Tier
9.02a	Identify congruent triangles	Both
9.02a	Prove two triangles are congruent using SSS, ASA, SAS or RHS	Both
9.02b	Apply congruent triangles in calculations	Both
9.02b	Use congruent triangles in simple proofs, for example that the base angles of an isosceles triangle are equal	Both

Congruent triangles (9.02a)

Two shapes are congruent if they are exactly the same shape and size. One can be moved onto the other by a combination of translation, rotation and reflection, so a congruent triangle may be turned round or flipped over. Enlargement is not allowed, because it changes the size.

If two triangles are congruent, every side and angle in one has a partner in the other. Partners are called corresponding sides and corresponding angles, and they are equal.

Write the congruence with the vertices in matching order. The statement ABC ≅ ZXY means A matches Z, B matches X and C matches Y. So AB = ZX, BC = XY, angle A = angle Z, and so on. Always read the matching from the order of the letters, never from the picture alone.

Spot the congruent pair

Four triangles are described below.

Triangle	Information
1	sides 7 cm, 9 cm, 11 cm
2	sides 11 cm, 7 cm, 9 cm
3	angles 40°, 60°, 80°, one side 7 cm
4	angles 40°, 60°, 80°, one side 12 cm

Triangles 1 and 2 have the same three side lengths, so they are congruent (SSS). Triangles 3 and 4 have the same angles, so they have the same shape, but a 7 cm side and a 12 cm side cannot be equal. They are not congruent. One is an enlargement of the other, so they are similar but not congruent.

The four conditions (9.02a)

You do not need to check all six measurements. Any one of these four sets of facts is enough to prove two triangles congruent.

Condition	What you need	Notes
SSS	all three sides equal	no angles needed
SAS	two sides equal and the angle between them equal	the angle must be included
ASA	two angles equal and one side equal	the side must match in position
RHS	both triangles right-angled, hypotenuses equal, one other side equal	right angle is essential

Each condition is named by the order of facts around the triangle. In SAS the angle sits between the two sides. In ASA the side sits between the two angles, although a side in any matching position works, because the third angle follows from the angle sum.

Two sets of facts that are not enough

AAA (three equal angles) does not work. Triangles 3 and 4 above show why: same angles, different size.

SSA, two sides and an angle that is not between them, does not work either. Take sides of 8 cm and 6 cm with a 40° angle opposite the 6 cm side. Two different triangles can be drawn, because the 6 cm side can swing to two positions. RHS is a safe special case because the right angle removes that choice. If you reject a condition in an answer, give the reason, as above.

Example: using the angle sum with ASA

Triangle ABC has angle A = 38°, angle B = 74° and AB = 6.5 cm. Triangle PQR has angle P = 38°, angle R = 68° and PQ = 6.5 cm. Are they congruent?

Angle Q = 180° − 38° − 68° = 74°
Angle B = 74° = angle Q
Angle A = 38° = angle P
AB = PQ = 6.5 cm, and AB lies between A and B just as PQ lies between P and Q

Two angles and the matching side are equal, so the triangles are congruent (ASA). The statement is ABC ≅ PQR.

Writing a congruence proof (9.02a)

A proof is a chain of reasons. Use this layout every time.

1. Name the two triangles, using letters in matching order.
2. List three equal pairs of facts, each with a reason.
3. State the condition (SSS, SAS, ASA or RHS) and say the triangles are congruent.
4. Use “corresponding” to state what follows from the congruence.

Common reasons you can quote:

Fact	Reason to write
Two lines cross	vertically opposite angles are equal
A side shared by both triangles	common side
Parallel lines	alternate angles are equal
Corners of a square or rectangle	angle of 90°
Marked equal lengths	given in the question
A point is the midpoint	it divides the line into two equal parts

Proof 1: SAS

Lines AB and CD cross at M. M is the midpoint of AB and the midpoint of CD. Prove that AC = BD.

Statement	Reason
AM = BM	M is the midpoint of AB
angle AMC = angle BMD	vertically opposite angles
CM = DM	M is the midpoint of CD

The triangles AMC and BMD are congruent (SAS). The angle is between the two equal sides in both triangles. Corresponding sides are equal, so AC = BD.

Proof 2: ASA

ABCD is a parallelogram. Prove that triangle ABC is congruent to triangle CDA, and deduce that AB = CD.

Statement	Reason
angle BAC = angle DCA	alternate angles, AB parallel to DC
AC = CA	common side
angle BCA = angle DAC	alternate angles, BC parallel to AD

The triangles are congruent (ASA), with AC lying between the two equal angles in each. The matching is A to C, B to D, C to A. So AB = CD, and BC = DA as well.

Proof 3: RHS

Triangle ABC has AB = AC. M is the point on BC where the perpendicular from A meets it. Prove that M is the midpoint of BC.

Statement	Reason
angle AMB = angle AMC = 90°	perpendicular from A
AB = AC	given (hypotenuses)
AM = AM	common side

The triangles AMB and AMC are right-angled with equal hypotenuses and one equal side, so they are congruent (RHS). Corresponding sides are equal, so BM = MC. M is the midpoint of BC.

Applying congruent triangles (9.02b)

Calculations

Congruence turns known measurements in one triangle into unknown ones in the other.

Example. Triangle ABC has AB = 9.6 cm, BC = 7.2 cm, angle A = 41° and angle B = 77°. Triangle PQR is congruent to triangle ABC with A matching R, B matching P and C matching Q. Find PQ, RP and angle Q.

PQ matches BC, so PQ = 7.2 cm
RP matches AB, so RP = 9.6 cm
Angle C = 180° − 41° − 77° = 62°
Angle Q matches C, so angle Q = 62°

The letters P, Q, R appear in a different order from A, B, C, so match them by position in the congruence statement, not by where they sit on the page.

The isosceles triangle result

An isosceles triangle has two equal sides. The base angles, which sit opposite those sides, are equal. Congruence proves this.

Proof (base angles equal). In triangle ABC, AB = AC. Draw AD, the bisector of angle A, meeting BC at D. Prove angle B = angle C.

Statement	Reason
AB = AC	given
angle BAD = angle CAD	AD bisects angle A
AD = AD	common side

The triangles ABD and ACD are congruent (SAS). The equal angle lies between the two equal sides in each triangle. So angle ABD = angle ACD, which means the base angles are equal.

You can now use the result in questions without proving it again unless the question says “prove”. If a question asks you to prove it, write out the table above.

Example. In triangle ABC, AB = AC and angle A = 48°. Find angle B.

Angle B = angle C (base angles of an isosceles triangle)
2 × angle B = 180° − 48° = 132°
Angle B = 66°

A second example turns this round. If a base angle is 71°, the angle at the top is 180° − 2 × 71° = 38°.

Combining the results

The RHS proof above shows that the perpendicular from the apex of an isosceles triangle bisects the base. A numerical use follows.

Example. An isosceles triangle ABC has AB = AC = 17 cm. The perpendicular from A meets BC at M, and AM = 15 cm. Find BC.

M is the midpoint of BC (RHS congruence)
BM² = 17² − 15² = 289 − 225 = 64
BM = 8 cm
BC = 2 × 8 = 16 cm

Longer proofs

In a longer proof, the first congruence often leads to a second one. Prove one pair of triangles congruent, state which sides or angles are equal as a result, and use those new facts in the next congruence. Write each stage separately. If a diagram has a shared side, mark it “common” so you do not forget it.

Common errors

• Writing the letters in the wrong order, so that the corresponding sides are wrong.
• Using the pairs “two sides and any angle”. The angle must be between the sides for SAS.
• Treating AAA or SSA as a proof. Neither is on the list of four conditions.
• Missing the reason beside a statement. “AB = CD” on its own is weaker than “AB = CD (opposite sides of a parallelogram)”, because the reason shows why it is true.
• Assuming something from the diagram, such as a right angle or equal lengths that are not marked or stated.
• Forgetting to finish the proof. Say “congruent (SAS)” and then state what you deduce.
• Using RHS when the triangles are not right-angled.

Where to go next

Work through the practice questions, then use the revision notes before a mock. Tick off 9.02a and 9.02b on the printable checklist.

Official syllabus

This page is aligned to the OCR GCSE (9-1) in Mathematics (J560) specification, Version 2.1, August 2026, published by OCR for first assessment in 2017. The relevant content is 9.02a (Congruent triangles) and 9.02b (Applying congruent triangles).