Question

Given two angles that measure 50 degrees and 80 degrees and side that measures 4 feet, how many triangles, if any, can be constructed?

Answer 1

The angles are the only constraint here that counts.  If one of the three interior angles of a supposed triangle is 50 degrees and another is 80 degrees, then the third angle must be 50 degrees.  Thus, we have a 50-50-80 triangle, which is isosceles though not a right triangle.  If 4 feet is a measure of one of the equal sides of a supposed triangle, then obviously the adjacent side also has measure 4 ft.

The set of angles remains the same (50-50-80), but subject to the constraint mentioned above, the measure of any one of the sides has infinitely many possible values, so long as those values are positive.