We should get rid of our cat, but my mom wont listen!?

Well , if the cat gets attacked from the dogs try to find a bruise on him/her maybe that would convince her. Or show her how the dogs are eating all her food. Maybe you could try to explain to her all the drama it causes, how your allergic and what can happen when your allergic reaction catches up with you also , tell her how the cat would have a way better life with a family that can really handle it.