Repairing A Broken Derailleur Hanger
Derailleur Hangers are small pieces of metal that hold the derailleur to the frame. This is my broken one:
Derailleur Hangers cost around $30 to replace depending on the type of bike you have. On your bike broken hangers look like this with the Derailleur hanging loose on the cable which can get caught up in your wheel and cause all sorts of damage:
To replace the broken hanger with the new one take off your rear wheel, undo the bolt on the frame and the one on the Derailleur to remove both broken pieces. The new Derailleur Hanger will look similar to this when on the bike only:
Now screw on the Derailleur to the hanger and she'll be good as new.
Put the back wheel on and the bikes ready to roll.
Helpful tips
Check the Derailleur alignment and check if its bent if it got caught in spokes or anything like that.