Due to limited transit network coverage and infrequent service, suburban commuters often face the transit first mile/last mile (FMLM) problem. To deal with this, they either drive to a park-and-ride location to take transit, use carpooling, or drive directly to their destination to avoid inconvenience. Ridesharing, an emerging mode of transportation, can solve the transit first mile/last mile problem. In this setup, a driver can drive a ride-seeker to a transit station, from where the rider can take transit to her respective destination. The problem requires solving a ridesharing matching problem with the routing of riders in a multimodal transportation network. We develop a transit-based ridesharing matching algorithm to solve this problem. The method leverages the schedule-based transit shortest path to generate feasible matches and then solves a matching optimization program to find an optimal match between riders and drivers. The proposed method not only assigns an optimal driver to the rider but also assigns an optimal transit stop and a transit vehicle trip departing from that stop for the rest of the rider's itinerary. We also introduce the application of space-time prism (STP) (the geographical area which can be reached by a traveler given the time constraints) in the context of ridesharing to reduce the computational time by reducing the network search. An algorithm to solve this problem dynamically using a rolling horizon approach is also presented. We use simulated data obtained from the activity-based travel demand model of Twin Cities, MN to show that the transit-based ridesharing can solve the FMLM problem and save a significant number of vehicle-hours spent in the system.