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Fager's postulate for the congruence of two triangles.

Two triangles are congruent if it can be shown that any three member permutation of corresponding sides and angles are congruent except for AAA and jackASS, or if it can be shown that the hypotenuse and a leg correspond.

Discussion. Congruent triangles are those with the same size and shape. Two triangles are congruent only if three pairs of corresponding angles and three pairs of corresponding sides are congruent. In other words six corresponding things must be congruent--all three corresponding angles and all three corresponding sides. Fortunately, the mathematics of Geometry allows for the proof of congruence of triangles by showing that only three very specific pairs of corresponding members are congruent. For example, it can be shown by rigorous mathematical proof that two triangles are congruent if just three sides of one triangle are congruent to three sides of the other. In other words if three pairs of sides are equal in measure, then the three corresponding angles must also be of equal measure. Geometric shorthand for this example is "side-side-side" or just SSS. Valid Geometric permutations to prove the congruence of two triangles are in Figure 1 below.

What will work

AAS	angle-angle-side (two angles and the side opposite to one of the angles). Notice that AAS and SAA are equivalent.
ASA	angle-side-angle (two angles and the included side).
SAS	side-angle-side (two sides and the included angle).
SSS	side-side-side
Hy-Leg	In the special case of a right triangle, two right triangles are congruent if it can be shown that the hypotenuse and either leg of one triangle are congruent to the corresponding parts of the other right triangle.

Figure 1.

Now that we know what will work, we need to ask ourselves what won't work. That is in Figure 2.

What won't work

AAA

Angle-Angle-Angle (AAA) does not satisfy congruency. You can have two 60-60-60 triangles meaning all angles are 60 degrees with one triangle having sides of 6" by 6" by 6" and the other 10' by 10' by 10'. In other words AAA proves that two triangles are similar, that is their legs will be in proportion, but it doesn't prove that corresponding sides are congruent.

jackASS

Angle-side-side (ASS or SSA) does not prove congruence either. Ask Dr. Math why this is so at: http://mathforum.org/library/drmath/view/54659.html
Here is an easy way to remember that ASS can not be used:

There is no jackASS in Geometry!

Figure 2.

There are eight possible permutations of two items taken three at a time. They are shown in Figure 3 below.

But to show congruence of triangles AAA and ASS (SSA) are eliminated for the reasons stated in Figure 2. Any other permutation is valid (notice that AAS and SAA are equivalent).

Conclusion. In other words, any good student of Geometry has proved that two triangles are congruent if AAS, ASA, SAS or SSS is true. But his dilema months later when trying to solve congruency problems is that it is often confusing trying to recall just which triplets work and which don't. The unified answer is "don't bother." Don't remember which permutations are valid, rather remember this:

ANYTHING (INCLUDING HY-LEG) IS VALID EXCEPT AAA AND ASS,
or there is no jackASS in Geometry!

If, for example, the problem solver can find SAS then he simply says that the two triangles are congruent by SAS (just by remembering that SAS is neither AAA or ASS).

Epilog. Due to pressing research in other areas, the author has not had time to make a rigorous search to see whether others have come to this same conclusion. The idea is so intuitively obvious that the author assumes that a host of others have already come to the same conclusion, including perhaps Pythagoras himself. If such is the case then the reasoning process contained herein is simply the author's independent discovery and validation of the idea.