Congruent triangle proofs GCSE questions ask you to show two triangles are identical in shape and size using one of four criteria: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and a side), or RHS (right angle, hypotenuse, side). State the criterion and give matching evidence for each part.
What does congruent mean?
Two triangles are congruent if they are exactly the same size and shape — every side matches a corresponding side, and every angle matches a corresponding angle, even if the triangle has been rotated, reflected, or translated. Congruent triangles are similar triangles with a scale factor of exactly 1.
Proving congruence matters because it lets you transfer known facts (equal sides, equal angles, equal areas) from one triangle to another without measuring — a technique used throughout GCSE geometry, from isosceles triangle proofs to circle theorem proofs.
What are the four congruence criteria?
You only need three matching pieces of information to prove two triangles are congruent — but they must be the right three pieces, in the correct combination:
| Criterion | Meaning | What must match |
|---|---|---|
| SSS | Side-Side-Side | All three sides |
| SAS | Side-Angle-Side | Two sides and the angle between them |
| ASA | Angle-Side-Angle | Two angles and the side between them (or connecting them) |
| RHS | Right angle-Hypotenuse-Side | A right angle, the hypotenuse, and one other side |
Note that "SSA" (two sides and a non-included angle) is not a valid criterion — it does not guarantee congruence, because two different triangles can share those measurements. Always check the angle is genuinely between the two named sides before writing SAS.
How do you prove triangles congruent using SSS?
Use SSS when you can show all three sides of one triangle equal the three corresponding sides of the other.
- Identify and label the three pairs of corresponding sides.
- State each equality, giving a reason (given information, shared side, or a calculated length).
- Conclude with "Triangles [name] and [name] are congruent (SSS)."
Worked example: Triangle ABC has AB = 5 cm, BC = 7 cm, AC = 9 cm. Triangle DEF has DE = 5 cm, EF = 7 cm, DF = 9 cm. Prove the triangles are congruent.
- AB = DE = 5 cm (given)
- BC = EF = 7 cm (given)
- AC = DF = 9 cm (given)
- All three pairs of corresponding sides are equal, so triangle ABC ≡ triangle DEF (SSS).
How do you prove triangles congruent using SAS?
Use SAS when two sides and the angle between them match.
- Identify the two pairs of corresponding sides and confirm the angle between them is the one given.
- State each equality with a reason.
- Conclude with the SAS statement.
Worked example: In triangles PQR and STU, PQ = ST = 6 cm, QR = TU = 8 cm, and angle PQR = angle STU = 40°. Prove the triangles are congruent.
- PQ = ST = 6 cm (given)
- QR = TU = 8 cm (given)
- Angle PQR = angle STU = 40°, and this angle lies between the two given sides in both triangles
- Therefore triangle PQR ≡ triangle STU (SAS).
How do you prove triangles congruent using ASA and RHS?
ASA requires two angles and the side that connects them (or lies between them). Because the two angles are fixed, the third angle is automatically fixed too (angles in a triangle sum to 180°), so this criterion pins down the whole shape once one side length is matched.
RHS applies specifically to right-angled triangles: if the right angle, the hypotenuse, and one other side all match, the triangles are congruent — even though this is really a special case of SAS (Pythagoras' theorem forces the third side to match too).
Worked example (RHS): Two right-angled triangles both have a right angle, a hypotenuse of 13 cm, and one shorter side of 5 cm. Prove they are congruent.
- Both triangles have a right angle (given)
- Both hypotenuses are 13 cm (given)
- Both triangles have a matching shorter side of 5 cm (given)
- Therefore the triangles are congruent (RHS). By Pythagoras, the remaining side in both triangles is $\sqrt{13^2 - 5^2} = 12$ cm, confirming all three sides match.
How do you set out a full congruence proof?
Exam mark schemes reward a clear, structured proof over a correct-looking sketch. Follow this layout every time:
- Draw or label the two triangles clearly, marking equal sides and angles with tick marks or arcs if working from a diagram.
- List each piece of matching information on its own line, giving a reason for each (given, common/shared side, vertically opposite angles, alternate angles on parallel lines, and so on).
- Name the criterion you are using (SSS, SAS, ASA, or RHS) only after you have justified all three required facts.
- Write the conclusion using the congruence symbol, matching vertices in the same order: triangle ABC ≡ triangle DEF.
Matching vertex order matters — writing "ABC ≡ DEF" implies A corresponds to D, B to E, and C to F, so examiners check the letters are in the correct correspondence.
What common mistakes lose marks in congruence proofs?
The most frequent error is quoting "SSA" as if it were a valid criterion — always double-check the angle you have is genuinely between the two known sides before using SAS. Another common mistake is forgetting to justify a "shared side" or "shared angle" explicitly — a side common to both triangles (like a diagonal of a quadrilateral) still needs the reason "common side" stated, not just assumed. A third mistake is stating the conclusion with vertices in the wrong order, which can cost a mark even when the working is otherwise correct.
Frequently asked questions
What is the difference between congruent and similar triangles?
Congruent triangles are identical in both shape and size — every side and angle matches exactly. Similar triangles have the same shape (equal angles, sides in the same ratio) but can be different sizes. Congruent triangles are a special case of similar triangles where the scale factor equals 1.
Why isn't SSA a valid congruence criterion?
SSA (two sides and a non-included angle) can produce two different triangles from the same measurements, because the third side has two possible positions that both satisfy the given lengths and angle. Because congruence proofs must guarantee a unique triangle, only SSS, SAS, ASA and RHS are accepted at GCSE.
How do you find a "common side" or "shared angle" in a proof?
Look for a side or angle that belongs to both triangles in the diagram — often a shared diagonal in a quadrilateral, or vertically opposite angles where two lines cross. State it explicitly with the reason "common side" or "vertically opposite angles" rather than leaving it unlabelled, since mark schemes require every piece of evidence to be justified.
Do I need to prove congruence before using circle theorems?
Not usually — most circle theorems (like the angle at the centre) are proved once and then quoted as facts. However, some harder GCSE and A-level geometry questions ask you to prove a related fact, such as two chords being equal, by first proving two triangles congruent (often using RHS with the circle's radius as the common hypotenuse or side).
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