jmc

The sum of the ratios is $10$. Since the sum of the angles of a triangle is $180^{\circ }$, the ratio can be scaled up to $54:54:72$ $(3\cdot 18:3\cdot 18:4\cdot 18).$ The numbers in the ratio $54:54:72$ represent the angles of the triangle. The question asks for the largest, so the answer is $\boxed{72}$.