Why is the Normal force in the picture not perpendicular to the surface? 
My teacher said that the first is true, but the normal force should be perpendicular to the surface?
 A: One should remember that a Normal force is actually only one component of the contact force. It is often convenient to resolve the contact force into Normal and Friction components, particularly in cases of either negligible friction or limiting friction. 
Certainly one can argue that the normal force of the plank on the fulcrum is perpendicular to the plank, but then one should not forget the friction force, meaning that the net reactive force of the fulcrum on the Plank is not normal to the Plank. Moreover this reactive contact force is not naturally split into normal and friction components precisely because the fulcrum is a corner.
In a case like this it is almost certainly easier to resolve the contact force of the fulcrum on the Plank into $X$ and $Y$ horizontal and vertical components. Then since there are no other horizontal forces, and any horizontal components of acceleration are likely to be negligible (if the question should involve acceleration) we can easily see that $X=0$
