There are four valid combinations a pair of triangles' sides and angles must meet if those two triangles are to be said to be congruent. They are often referred to by their three-letter abbreviatations, and are as follows:
- SAS (side, angle, side): Two pairs of common sides must be present, each with an identical angle between them.
- RHS (right angle, hypoteneuse, side): a right angle, the hypoteneuse and another side must all be common to both triangles.
- SSS (side, side, side): All three sides must be the same.
- AAS (angle, angle, side): Two angles and the side joining them must be common.
Any other combination of sides or angles common to both triangles is, sadly, not enough to consider the triangles congruent to one another. Below are two which are commonly thought to be valid conditions, but are not.
- AAA (angle, angle, angle): Not congruent, because this would make an equilateral triangle of unknown side length. There is no limit to the number of equilateral triangles one could make.
- ASS (angle, side, side): Not congruent, because there are two possible ways of building a triangle like this (see below).*
Example of ASS triangles:
Imagine a pair of triangles. Both have two known sides (lengths 9cm and 5cm) and one known angle (30 degrees). Assuming that the angle is placed at the left vertex, and the 9cm side is used as the base, we could place the 5cm side either adjacant to or opposite the angle. This gives us two possible triangles, whereas to be congruent both triangles must be and can only be identical to each other.
* Yes, an RHS triangle is, technically, an ASS triangle. However, in this combination, only one triangle is ever possible, as opposed to the usual two.