[CliffordK]

Whew,
That was much simpler than what I had ended up with using the SOH, CAH, TOA & building right triangles.

As with problems in many math books, the solutions often work out to be whole numbers.  While the left most triangle is an isosceles triangle, the bottom one, and the right most ones are not. 

In my solution, I could have stopped once I discovered that it was an isosceles triangle, but proving that without some advanced trig functions might have been illusive. 

And, of course, a general solution would need to be independent from such constructions.  Perhaps the next step would be to scramble the triangles so that the given angles are whole numbers, and the calculated angles are not.

[damocles]

Update: I have tried to generalize the problem to find out if there might be a geometric method.
The result is that the angles a & c are not in whole number of degrees for any of the other angle inputs (which are in whole numbers of degrees) for this particular generalization.
The conclusion is that there is no particular geometric construction that could help to solve this problem except possibly for the particular case.
Finding the appropriate geometric construction may not be possible, and would certainly involve more time and labour than my trigonometric solution, especially because the trigonometric solution
• can be applied to more general modifications of the problem
• is well adapted to Excel spreadsheet