Triangle congruence: SSS, SAS, ASA, RHS
Two triangles are congruent if they are the same shape AND size — exactly one can be superimposed on the other (allowing flips). This is stronger than similarity (which is just same shape).
The four congruence rules
You only need ONE of these to prove two triangles congruent:
SSS (Side-Side-Side)
All three corresponding sides are equal.
SAS (Side-Angle-Side)
Two sides and the included angle (the angle between them) are equal.
ASA (Angle-Side-Angle)
Two angles and the included side (between them) are equal.
(Equivalent: AAS — two angles and a non-included side. AAS works because the third angle is determined.)
RHS (Right-angle, Hypotenuse, Side)
Right-angled triangles only. Hypotenuse and one other side equal.
What does NOT prove congruence
- AAA (three equal angles) — only proves SIMILARITY, not congruence (one could be larger).
- SSA (Side-Side-non-included Angle) — ambiguous case; doesn't always determine the triangle.
Setting out a proof
A neat proof has THREE statements + a conclusion:
- State which sides/angles are equal and why (use given information or facts like "common side").
- Identify the congruence rule (SSS / SAS / ASA / RHS).
- Conclude: "Therefore △ABC ≅ △DEF (rule)".
✦Worked example
In rectangle ABCD, prove △ABC ≅ △CDA.
Statements:
- AB = CD (opposite sides of rectangle equal).
- BC = DA (opposite sides of rectangle equal).
- AC = AC (common side).
- All three pairs of sides equal → SSS congruence.
- Therefore △ABC ≅ △CDA. ✓
⚠Common mistakes
- Using SSA — not a valid congruence rule.
- Using AAA as congruence — only similarity.
- Forgetting "common side" — when two triangles share an edge.
- Wrong angle position — for SAS, the angle MUST be between the two sides quoted.
- Listing properties without naming the rule — examiners want the abbreviation explicitly stated.
➜Try this— Quick check
Two right-angled triangles each have hypotenuse 13 cm and one leg 5 cm. Are they congruent? By which rule?
- Yes, by RHS.
AI-generated · claude-opus-4-7 · v3-deep-geometry