Answer: E

Solution:

The angles $\angle AMD$ and $\angle MDC$ are equal since $AB \parallel DC.$

As such, $\angle MDC = \angle DMC ,$ making $MDC$ isosceles and $MC = DC = 6.$

As we can see, $\sin (\angle CMB) = \frac 12,$ making $\angle CMB = 30^\circ .$

Therefore, $\angle AMC = 150^\circ .$ Since $\angle AMD$ is half of that, $\angle AMD = 75^\circ .$

Thus, the correct answer is E .