Final answer:
The House Committee on Rules is responsible for establishing the rules under which bills are presented to the House of Representatives, including the terms for debate and amendment.
Step-by-step explanation:
The committee responsible for the rules under which bills will be presented to the House of Representatives is the House Committee on Rules. When the bill reaches the House floor, it must first go through this committee which establishes the rules of debate, including time limits and the types and numbers of amendments that may be considered.
The House Committee on Rules plays a crucial role because it determines the procedure by which the full House will consider the bill. This includes setting the terms for debate and indicating which, if any, amendments to the bill can be proposed from the floor. Such procedural rules can significantly influence the outcome of the legislative process and are essential in managing the flow of legislation.