Two triangles are congruent (identical in shape and size) if they satisfy one of 4 tests: SSS (all 3 sides equal), SAS (2 sides and the included angle equal), AAS (2 angles and a corresponding side equal), or RHS (right angle, hypotenuse, and one other side equal — right-angled triangles only). A full-marks proof states which test is used and lists each matching pair of sides or angles explicitly, not just the conclusion.
Congruent triangles are identical — same shape and same size, so every corresponding side and angle matches exactly. Similar triangles only need the same shape; sizes can differ, connected by a scale factor. Congruence is really just a special case of similarity where the scale factor is 1. Mixing up which one a question is asking for is an easy way to apply the wrong test entirely.
SSS — Side, Side, Side
All 3 sides of one triangle equal the 3 corresponding sides of the other.
SAS — Side, Angle, Side
2 sides and the angle between them (the included angle) are equal.
AAS — Angle, Angle, Side
2 angles and any one corresponding side are equal (the side does not need to be between the two angles).
RHS — Right angle, Hypotenuse, Side
Both triangles are right-angled, with equal hypotenuses and one other equal side. Only applies to right-angled triangles.
✗ Not a valid test: AAA. Matching all 3 angles only proves the triangles are similar, not congruent — the sizes could still differ by any scale factor. AAA never proves congruence on its own.
In the diagram, AB = AD and AC bisects angle BAD. Prove that triangle ABC is congruent to triangle ADC.
Solution
AB = AD (given), and AC bisects angle BAD, so ∠BAC = ∠DAC (given — bisector splits the angle equally).AC = AC (common side to both triangles).△ABC ≡ △ADC (SAS).The 3-line proof structure examiners want: (1) state each matching pair with its reason — given, common side, or a calculated angle fact; (2) name the test being used; (3) write the congruence statement with vertices in matching order, e.g. △ABC ≡ △ADC — not △ABC ≡ △CDA. The order of the letters tells the reader exactly which vertices correspond.
In the diagram, ABCD is a kite with AB = AD, and BD is a diagonal. AC is perpendicular to BD at point M, the midpoint of BD. Prove that triangle ABM is congruent to triangle ADM.
Solution
AB = AD (given — kite property), BM = DM (given — M is the midpoint of BD).∠AMB = ∠AMD = 90° (given — perpendicular).AM = AM (common side).△ABM ≡ △ADM (RHS).Common mistake
Students sometimes use BM = DM as the "S" for an SAS argument, forgetting that RHS is the correct (and required) test whenever a right angle and the hypotenuse are both confirmed equal. If a right angle is present, always check RHS before defaulting to SAS — RHS is the test specifically built for right-angled triangles.
O-Level questions rarely ask you to prove congruency as an end goal by itself. More often, congruency is a stepping stone: once you've shown two triangles are congruent, you can then state that their corresponding sides or angles are equal — and use that fact in a later part of the question (e.g. to find a missing length, or to prove a shape is a parallelogram or rhombus, drawing on the property tables for triangles, quadrilaterals and symmetry).
After proving congruency, state what it unlocks: "Since △ABC ≡ △ADC, BC = DC (corresponding sides of congruent triangles)." This follow-up sentence is often exactly what the next part of the question needs, and forgetting to state it explicitly can cost marks even when your congruency proof itself was correct.
Why isn't SSA (or ASS) a valid congruence test?
When the angle is not between the two given sides, the same two side lengths and angle can sometimes form two different possible triangles — so the triangle isn't uniquely determined. This is why the angle in SAS must specifically be the included angle, sitting between the two known sides.
Does the order of letters in the congruence statement actually matter?
Yes — writing △ABC ≡ △PQR specifically means A corresponds to P, B corresponds to Q, and C corresponds to R. Getting the order wrong (e.g. writing △ABC ≡ △QPR when it should be △PQR) misstates which sides and angles actually match, and can cost marks even if the triangles genuinely are congruent.
Can I use AAA to prove two triangles are congruent?
No — AAA only proves the triangles are similar (same shape), since matching all three angles says nothing about size. Two triangles with the same angles could be different sizes entirely, connected only by a scale factor. Congruency requires size to match too, which AAA cannot confirm.
What's the difference between "included angle" and just "an angle"?
The included angle is specifically the angle formed at the vertex where your two known sides meet — it sits between them. If the angle you know is not at that shared vertex, it isn't the included angle, and SAS cannot be applied directly; check whether AAS fits instead.
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