Final answer:
The 'republican virtues' referred to a) Personal freedom and self-sufficiency, which were essential civic virtues needed for the success of the American republic, as the nation's citizens were expected to prioritize the common good and civic responsibilities, especially property holders who were seen as having a greater stake in society.
Step-by-step explanation:
While crossing the Appalachian mountains, the "republican virtues" referred to were a) Personal freedom and self-sufficiency. This answer emphasizes that, within the political theory of republicanism at the time of the American Revolution, citizens were expected to exhibit virtuous behaviors for the republic to succeed. Virtues included personal freedoms and self-sufficiency, indicating that property holders, who had a greater stake in society due to their investments, were seen as more capable of exercising civic responsibility and thus were more inclined to be entrusted with political rights.
Republican virtues favored a form of government that placed the common good above individual desires and petty concerns—a system thought to be more sustainable in smaller republics where commonality of interests was more apparent. Nonetheless, Americans believed that these virtues were necessary even in the larger republic they were creating the ran along the Eastern Seaboard to the Appalachian Mountains.
By casting off British control and forming independent states, American revolutionaries were also casting off loyalty to the British monarchy and rejecting centralized government control and religious conformity in favor of a government that emphasized personal liberty, self-reliance, and a quest to secure the blessings of liberty through virtuous behavior as outlined by the likes of George Mason in the Virginia Declaration of Rights.