The ambiguous case is one of the most difficult concepts to get across to Geometry students. When I teach congruent triangles, I think that some of the students are convinced that they can't use the ASS Theorem to prove triangles congruent simply because I don't want them to write bad words on their paper! (No matter how many times I try to explain it!!!)
Then when you actually try to explain to Algebra 2 or Pre Calculus students the SSA case (have you noticed that it is never written as the ASS case? LOL) their heads start to explode!
I started my search online and came across a couple of things...
1) a really great explanation of the ambiguous case can be found at the regents prep center
Ambiguous Case Explanation
2) A really great activity that even geometry students can use to explore the ambiguous case (see this link
Ambiguous Case with Pipe Cleaners ) The basic idea is that you give the students two pipe cleaners to start with. They must also start with a given angle - they use 30 degrees in this example. Then students try shortening one of the pipe cleaners by 1 inch intervals. After the students shorten the pipe cleaner a few times, they will get to a point where two triangles can be formed - in other words, being given an angle, a side, and a side does not guarantee that two triangles will be congruent. Although not all geometry students are ready to see the trig involved, (especially during the beginning of the school year when we study congruent triangles), I think that they can see by experimentation that two triangles can be formed in certain situations. I am definitely trying this out this year!
Here are a couple of other links that I found for other possible activities.
3) Ambiguous case activity with paper fasteners and construction paper.
Ambiguous Case Activity
4) Here is a link to a calculator activity that you can use with your TI-Nspire - you could even just use it as a visual representation if you don't want to take the time for all the students to do it on their own.
Calculator Ambiguous Case
5) Finally, here is a real world example from the
CK-12 Book that would be a good starting point for your lesson.