Question

1. Two angles of a triangle measure 50 degrees and 30 degrees. If the longest side is 9, find the length of the shortest side

2. Why is there no solution for a triangle with side lengths 22, 13, and 8?

Answer 1

If the 2 angles are 50 and 30 degrees, then the third angle must be
180-(50+30) or 100°

so the shortest side must be opposite the smallest angle, and 9 must be opposite 100°.
By the sine law:
x/sin30 = 9/sin100, where x is the shortest side
x = 9sin30/sin100 = appr 4.57

As to you second question:
the sum of any two sides of a triangle must
be greater than the third side to form a physical triangle.

Since 13+8 < 22, the lines cannot meet to form a triangle.