2007 Toyota Corolla car won't start

My 2007 Corolla has started without any sign of a problem for the two years I've owned it. After sitting a few days it won't turn over. I assumed some light had been left on and drained the battery. I push started it (it fired right up) and made my trip to the store. I parked on a hill and started it by rolling to get back home. Upon returning home and checking the battery, it shows 12.9 volts. I know that is good enough. The positive battery terminal was quite corroded, but even after cleaning it thoroughly, all I get is a clicking sound when the key is turned to start the car. Any ideas? Thanks.