Triangle congruence
Two triangles are congruent if one can be moved (rotation, reflection, translation) to coincide exactly with the other. Same shape AND same size. Examined explicitly on Intermediate and Higher.
The four congruence conditions
| Condition | What it means |
|---|---|
| SSS | All three sides equal |
| SAS | Two sides and the included angle equal |
| ASA | Two angles and the included side equal (also AAS — two angles and a non-included side, since the third angle is determined) |
| RHS | Right angle, hypotenuse and one other side equal (right-angled triangles only) |
SSA / ASS is NOT a congruence condition — two sides and a non-included angle can produce two different triangles (the "ambiguous case"). Don't quote it.
Method for proving congruence (WJEC standard)
- State which sides/angles are equal and why (given, common side, vertically opposite, etc.).
- List exactly three matching pieces of information.
- State the congruence condition you are using.
- Conclude: "Therefore △ABC ≡ △DEF (SAS)".
The mark scheme awards:
- M1 for naming three matching pairs with reasons.
- M1 for explicitly naming the congruence condition (SSS / SAS / ASA / RHS).
- A1 for the conclusion.
Why congruence matters
If two triangles are congruent, every corresponding part is equal — sides, angles, areas, perimeters. WJEC exploits this: prove congruence first, then deduce a length or angle as a consequence.
Example: in a kite ABCD with AB = AD and CB = CD and shared diagonal AC, prove △ABC ≡ △ADC.
- AB = AD (given)
- CB = CD (given)
- AC = AC (common side)
- Therefore △ABC ≡ △ADC (SSS).
- So ∠BAC = ∠DAC, i.e. AC bisects ∠BAD. (Bonus consequence.)
WJEC exam tip
ALWAYS write the congruence condition as four letters at the end of your reasoning. A correct argument that leaves out "(SAS)" or "(SSS)" loses the dedicated condition mark.
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