An optic fiber is made of clear plastic with index of refraction of 1.50. for what angle of incidence will light remain within the plastic "guide"?

Well what you need to do in order to know that is doing the following procedure:
 n sin (theta) = 1 where n = 1.5 
theta = arcsin(1/n) = 41.8.
so the angle will be 41.8 degrees