Old glass never cuts or breaks well, its a liquid and the "grain" is much stronger in old glass that has settled with time.
there are two ways I know to break glass safely ... both involve scoring the glass first with hard steel or a diamond tip, right up to an edge. Just a score is sufficient, but the more even, the better. New glass is easier.
1. place the glass on a clean flat surface with a matchstick under the score at the edge close to you, then push down on the shorter piece of glass next to the matchstick, it should snap off neatly, even if the piece of glass is quite large.
2. hold the glass each side of the score with the thumbs on top facing away from you, fingers underneath. then twist your hands *apart* as if you are breaking a biscuit. the glass should "snic" and break leaving you with a neat break. I prefer this second method but its not as safe as the first. You can cut very long thin curvy slices this way.
If your score is only a "start" from the edge, the break will tear away from the score as the glass breaks, often a curve.
Or you can score the entire curve first. Even with a scored line, the bottom part of the break is very sharp.
cutting glass is quite fun if the glass isn't too old. (If it is, it must be a bit like bending the ribs on the Pappas bass ...)