Triangle congruence criteria
Edexcel Higher tier tests congruence formally. Two triangles are congruent if all corresponding sides and angles are equal — but you don't need to check all six; one of four shortcuts suffices.
The four congruence criteria
1. SSS (side-side-side)
All three pairs of sides equal.
2. SAS (side-angle-side)
Two pairs of sides equal AND the included angle (the angle between them) is equal.
3. ASA (angle-side-angle) — also AAS
Two pairs of angles equal AND a corresponding side equal. (If two angles are equal, the third automatically is.)
4. RHS (right-angle-hypotenuse-side)
Both triangles have a right angle, equal hypotenuses, and one other equal side.
Note: SSA does NOT prove congruence
Two sides and a non-included angle is not enough — there can be two distinct triangles satisfying these (the "ambiguous case" of the sine rule).
Stating a congruence proof (Edexcel exam style)
A typical Edexcel "prove that △ABC ≡ △PQR" question requires:
- State the equal sides and angles (each as B1 or M1).
- Quote the congruence criterion (SSS / SAS / ASA / RHS).
- Conclude "therefore △ABC ≡ △PQR".
The conclusion line is mandatory — the C1 communication mark depends on it.
✦Worked example— Worked example (Higher)
In a kite ABCD where AB = AD and CB = CD, prove that △ABC ≡ △ADC.
| Statement | Reason |
|---|---|
| AB = AD | given |
| CB = CD | given |
| AC = AC | common side |
| ∴ △ABC ≡ △ADC | by SSS |
Edexcel exam tip
When two triangles share a side, always state "common side" as a reason — not just write the equality.
⚠Common mistakes— Common errors
- Citing SSA — invalid.
- Forgetting to write the criterion (SSS, SAS, etc.) at the end.
- Mixing up the order of vertices in △ABC ≡ △PQR — the correspondence A↔P, B↔Q, C↔R is implicit.
- Including a "fourth" piece of evidence — only state what's needed.
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