Question

Determine the number of possible triangles, ABC, that can be formed given B = 45°, b = 4, and c = 5.

Answer 1

There are 2 triangles here.  You have to drop an altitude and determine the length of it.  You can use the fact that sin45 = h/5 (h is the height) and get that the height is 3.5.  The rule then is if h is smaller than b which is in turn smaller than c, you have 2 triangles.  This is the case of the ambiguous triangle, that SSA that you cannot use to prove congruency.  The rules are very particular for whether or not 0, 1, or 2 triangles exist.  Learn them.