Breaking thin pieces of glass can be tricky, but there are a few things you can do to help direct the break the way you want it to go.
Relieving scores made alongside curved and tapering pieces make the breaking more certain. A relieving score is one that is in addition to the primary score. This additional score will allow you to break the thin or tapering piece from the larger sheet safely, and then go on to break out the delicate piece.
The object is to always be breaking away less glass than is retained. The use of two breaking/grozing pliers, one on each side of the narrow pieces gives more even pressure than fingers or cut running pliers with wide jaws.
When breaking tapering pieces of glass you should normally grasp the thin end in fingers or pliers and run the score toward the thick end and ease the run of the score. When the score opens an initial distance, turn the glass end for end and run the score back to the opened one.