Congruence Tests: SSS, SAS, AAS and RHS

Congruent triangles have six equal parts — three sides and three angles — but you do not have to check all six. Only three of them, chosen correctly, are enough to guarantee congruence, and the other three then follow automatically. There are exactly four valid tests. SSS means three sides equal. SAS means two sides and the included angle. AAS means two angles and one matching side. And RHS means a right angle, the hypotenuse, and one other side. Every congruence proof you write will use exactly one of these four, so it is worth knowing them cold.

The two side-based tests come first. SSS means all three sides are equal, and three sides fix a triangle completely — there is only one triangle you can build from three given lengths. SAS means two sides and the included angle, and the word included is doing all the work. The angle must sit between the two sides you are using. Picture it: fix the angle, fix the two arms coming off it, and the third side has only one place it can go. Move that angle somewhere else and the triangle is no longer pinned down, so two sides and an angle in the wrong position is not a valid test at all.

Now the angle-based tests. AAS means two angles and one matching side. It works because two angles automatically give you the third, since the angle sum is 180 degrees — so you really know all three angles, which fixes the shape, and the one side then fixes the size. RHS is for right-angled triangles only: a right angle, the hypotenuse, and one other side. It is interesting because it is really side-side-angle, which is normally not a valid test, but the right angle removes the ambiguity since Pythagoras forces the third side to a single value. In RHS the hypotenuse must be one of the equal sides, so check the diagram carefully.

Two combinations look plausible but prove nothing. AAA — all three angles equal — fails because angles fix the shape but never the size, so you get similar triangles rather than congruent ones. SSA fails for a different reason: because the angle is not included between the two sides, two genuinely different triangles can fit the same data. That is known as the ambiguous case. There is one exception, which is that when the angle is 90 degrees, SSA becomes RHS and does work, because Pythagoras removes the ambiguity. So the valid list is exactly four: SSS, SAS, AAS and RHS. Naming AAA or SSA in a proof scores zero however neat the rest is.

In practice, choosing the test means reading the marks on the diagram. Tick marks show equal sides, arcs show equal angles, and a small square shows a right angle. Three sides marked equal gives SSS. Two sides with the angle between them gives SAS, and you must check that the angle really is included. Two angles with a matching side gives AAS, and the third angle comes free. A right angle with the hypotenuse and one other side gives RHS. Three angles alone, or two sides with an angle not between them, gives no test at all. Count what you are given, match it to the list, and do not guess.