Final answer:
The Ptolemaic rulers legitimized their authority by adopting Egyptian religious practices, creating a hybrid Greco-Egyptian culture, and declaring themselves as pharaohs to align with Egypt's ancient traditions of kingship.
Step-by-step explanation:
The Ptolemaic rulers of Egypt sought to legitimate their rule primarily by adopting Egyptian religious practices and styling themselves as pharaohs. To convincingly adopt the mantle of pharaoh, the Ptolemies engaged in a series of religious and cultural integrations. For example, one of Ptolemy's first acts was to take possession of the body of Alexander the Great and entomb it in a grand mausoleum in Alexandria, declaring Alexander a god.
They also established the cult of Serapis, a deity combining elements of Greek and Egyptian religion, which helped to bridge the cultural gap between the two peoples. The Ptolemaic dynasty, being of Macedonian Greek origin, found it imperative to connect themselves to the ancient and revered Egyptian institution of the pharaoh through these religious and cultural syntheses.