Abstract: This paper primarily addresses the discrepancies appeared in the existing literature for characterizing the theorem of angular moment of a rigid body, especially those in planar motion, with respect to the instantaneous center of velocity of the body. Starting from the fundamental concepts of a moving moment center, we introduce the general forms of the angular momentum theorem for a system of particles about a moving point including the instantaneous center of velocity of the moving rigid body, which were obtained in literature. After the general conditions are discussed for the cases when the theorems can be degenerated to the form of the angular momentum theorem about a fixed moment center, in particular, a weak condition is proposed for those widely applied in the planar motion of rigid bodies. That is, when "the distance from the instantaneous center of velocity to the mass center remains constant in the moving progress", we give a strict proof of the conclusion that general theorem with absolute movement, when the moving moment center is taken as the instantaneous center of velocity, can be degenerated to the form of angular momentum theorem about a fixed moment center. It is obvious that this weak condition provides us an intuitive and simple judgement for those planer moving bodies when the form of angular momentum theorem about a fixed moment center can be used in the case of instantaneous center of velocity. Finally, several examples are presented to illustrate cases where the degenerated form of angular momentum theorem can be used when the weak condition is satisfied, and where the degenerated form can not be applied when the weak condition is not remained. In the case of the weak condition being unsatisfied, the solution operation should be taken in the general form of the theorem about the instantaneous center of velocity rather than the degenerated form such that the correct result is obtained. From them, one can know that those discrepancies on the form obtained in the literature for the angular momentum theorem of a planer moving body about its instantaneous center of velocity expressed in the degeneration form with its fixed moment center without any condition are obviously appeared. In conclusion, the degeneration form obtained in literature has no university in application.