What power allows the U.S. president to reject a bill passed by the legislative branch that does not have a two-thirds majority?

The power that allows the U.S. president to reject a bill passed by the legislative branch, particularly when it does not have a two-thirds majority backing, is known as a veto. This constitutional authority enables the president to refuse to sign a bill into law, effectively preventing its enactment unless Congress can subsequently muster enough support (a two-thirds majority in both the House of Representatives and the Senate) to override that veto.

Understanding the function of a veto is essential, as it serves as a fundamental check and balance within the U.S. government. It ensures that no single branch becomes too powerful by allowing the executive branch to intercede on legislative decisions. The ability to veto laws protects against legislation that may be considered unwise or not in the public's best interest.

The other options, while relevant in legislative processes, do not describe this specific power. A filibuster refers to a tactic employed in the Senate to delay or block legislative action, an override is the process by which Congress can counter a presidential veto if enough votes are gathered, and amend refers to altering a bill or constitution. Each of these serves different functions within the legislative process but does not provide the specific power of rejecting a bill that is held by the president.