How can you tell how many solutions there are for a triangle given two sides and a non included angle?

Answer 1

Draw a figure! Then you have to look at cases.

Case I

The angle is a right angle.

If the 2 given sides are equal, there are no triangles.

As long as one side is longer than the other there is one triangle (or 2 congruent ones if you are really pedantic!)

Case II

The angle is acute

There may be one triangle...a right triangle.

There may be no triangles if the 2nd leg is too short to form a triangle.

There may be 2 triangles if the 2nd leg is longer than would be required to make a right triangle and one of these will be an obtuse triangle.

Case III

The angle is obtuse

There may be one triangle if the second leg is long enough, no triangles if the 2nd leg is too short.

Looking at the figures for each case will make clear what is happening. You can, of course, do this analytically, but it is much easier to understand if you draw a figure will the numbers you are given.