Angle of projection is theta to the horizontal, when vertical velocity is half the max velocity?

You need to learn the words. "Maximum height" is the altitude at the peak of the trajectory, but "maximum velocity" can be anything so it has to be defined, and you have not defined it. I might assume that you meant "muzzle velocity" or "launch velocity", but that is still meaningless because velocity includes a direction, and the launch direction is not the same as the vertical component.

Anyway, solving this involves drawing a picture, and I can't draw pictures here. So get back with your teacher to straighten this out.