Final answer:
The United Kingdom's legislature consists of the popularly elected House of Commons and the partly hereditary House of Lords, fulfilling the criteria described in the question.
Step-by-step explanation:
The country where the legislature is divided into two houses with one house elected by popular vote and the other based on hereditary membership is the United Kingdom (UK).
The UK's Parliament is comprised of the House of Commons, which is elected via popular vote, and the House of Lords, where some of its members are hereditary peers, though many are now appointed. This bicameral legislative system reflects a longstanding tradition within UK governance, where historical class distinctions played a significant role in the makeup of its legislature.