The "Donkey Theorem", making an ASS of you
- Author:
- Herholdt Bezuidenhout
- Topic:
- Triangles
This method will NOT always prove triangles congruent!
The SSA (or ASS) combination deals with two sides and the non-included angle. This combination is humorously referred to as the "Donkey Theorem". SSA (or ASS) is NOT a universal method to prove triangles congruent since it cannot guarantee that the shapes of the triangles formed will always be the same.
Using SSA to prove congruency, can make an ASS of you!
Let's look at the problems associated with SSA:
When using SSA (or ASS), the lengths of the sides being used, and their position in relation to the angle, will determine if congruent triangles will always be created. Let's see which special conditions will create congruent triangles, and which ones will not.
ASS is not an universal method for proving triangles congruent
If BC CD AB
There exists no triangles as no triangle can be drawn. Since no triangles are possible, no congruent triangles are possible.
If BC = CD AB
The minimum (shortest) distance from point C to the ray from A through B, is the perpendicular distance. In this case there exist one right-angled triangle and we can use RHS to prove congruence. This specific case of ASS is the basis for the acceptable method RHS which applies only in right triangles.
If BC > CD AB and BC < AC
The "swinging" nature of BC, creating possibly two different triangles, is the problem with this method. Since this situation is open to two interpretations, it is referred to as the Ambiguous Case. (This is a reference we will be examining further in trigonometry.)
If BC > CD AB and BC > AC
In this case only one possible triangle can be drawn and you are safe and the two triangles will be of the same shape and size (congruent).
Since the SSA (or ASS) method allows for the possibility of creating triangles of various shapes (or even no triangles at all), this method is not an universal method for proving triangles congruent.