How many triangles can be formed from two given angle measures and the length of their included side?

Since the two angles are already given, therefore the value of the third angle is already fixed. That is, the third angle is the difference between 180 and the sum of the two angles. If the angles are only given, then we can form any number of angles as long as the ratio of angles are consistent.

However since the measure of the length of one of the sides is already given, hence this constrict the number of triangles we can form into 1. We cannot simply dictate the measure of the two sides or else we will disobey the Angle Side Angle postulate. The measure of the other two sides are calculated using triangle formulas.

Answer:

1 triangle