The angle on the cube answer is really bugging me. I understand the isoceles triangle answer but if it's a cube made of squares, why isn't the angle two bisections ie, 90 degrees?
I think it's because the projection is confusing you. The angle in question is not being measured on the surface of the cube, but is "inside" it. Try finding a hand cube, draw those two lines, and you'll see that 90 degrees isn't the angle between just those two lines, but between one line and the edge added to the angle between that edge and the other line. The 60 degree angle is defined without reference to the edge of the cube, which is just there to try and confuse the issue.