You could say simply that if you have prepared a (half) spin state |z+> the angular momentum along the x axis is zero but if you measure the spin along the x axis you will find a non-zero value.
I like your example because it clearly shows the subtlety that the original comment by rssoconnor also misses. Energy, momentum, and angular momentum absolutely are conserved quantities. But if you prepare your initial state such that it does not have a definite value of these quantities then you cannot with certainty predict the measured value, either.