Let $ABC$ be an isosceles triangle, and point $D$ in its interior such that $$D \hat{B} C=30^\circ, D \hat{B}A=50^\circ, D \hat{C}B=55^\circ$$ 

(a) Prove that $\hat B=\hat C=80^\circ$.
(b) Find the measure of the angle $D \hat{A} C$.