Question

How to solve for each angle measure.

Answer 1

Have you considered applying the Law of Cosines?

For angle B:      b^2 = a^2 + c^2 - 2ac*cos B

a, b and c are all known, leaving cos B as the only unknown.  Find cos B, and then find angle B itself.  From the diagram we know that angles A, B and C are all acute.