Which of the following is used to cut different shaped edges on a wooden cabinet door?

The tool most appropriate for cutting different shaped edges on a wooden cabinet door is a router. A router is specifically designed to hollow out an area in the face of a piece of wood, allowing for the creation of various decorative edges and profiles. Its versatility enables woodworkers to cut intricate designs and shapes, making it the ideal choice for shaping cabinet door edges with precision and detail.

While a saw is primarily used for making straight cuts or cutting through thick materials, it does not effectively create the curved or decorative edges that routers can manage. Other tools like a file are used for smoothing surfaces rather than cutting shapes, and a lathe is typically employed for shaping cylindrical objects rather than edges. The router stands out for its ability to enhance the aesthetics of woodworking projects through its shaping capabilities.