Safety is an important topic in autonomous driving since any collision may cause serious damage to people and the environment. Hamilton-Jacobi (HJ) Reachability is a formal method that verifies safety in multi-agent interaction and provides a safety controller for collision avoidance. However, due to the worst-case assumption on the car's future actions, reachability might result in too much conservatism such that the normal operation of the vehicle is largely hindered. In this paper, we leverage the power of trajectory prediction, and propose a prediction-based reachability framework for the safety controller. Instead of always assuming for the worst-case, we first cluster the car's behaviors into multiple driving modes, e.g. left turn or right turn. Under each mode, a reachability-based safety controller is designed based on a less conservative action set. For online purpose, we first utilize the trajectory prediction and our proposed mode classifier to predict the possible modes, and then deploy the corresponding safety controller. Through simulations in a T-intersection and an 8-way roundabout, we demonstrate that our prediction-based reachability method largely avoids collision between two interacting cars and reduces the conservatism that the safety controller brings to the car's original operations.