Final answer:
The United States supported Panama's independence to gain control over the Panama Canal, which was crucial for military and trade reasons that fit President Theodore Roosevelt's progressive agenda of strategically expanding American influence.
Step-by-step explanation:
In the early 1900s, the United States played a significant role in the Panama revolution and the subsequent construction of the Panama Canal. President Theodore Roosevelt supported Panama's independence from Colombian rule largely because of the strategic importance of building a canal across Panama. This was pivotal to Roosevelt's progressive agenda, and the most accurate answer to why this was important to President Roosevelt's progressive agenda is Option 2: To gain control over the Panama Canal. The canal would serve military and trade purposes, expanding U.S. influence and control in Latin America while also facilitating faster and more efficient maritime trade routes between the Atlantic and Pacific Oceans.
Roosevelt asserted influence in Panama by backing the Panamanian revolution, which was partially motivated by the Colombian government's hesitance to ratify a treaty allowing the construction of the canal. Once Panama gained its independence with American support, the United States quickly recognized the new country and secured the rights to build the canal through the Hay-Bunau-Varilla Treaty. This action reflected Roosevelt's belief in the importance of American intervention in strategic endeavors and his assertion that the United States had the right to exercise influence in this region under his corollary to the Monroe Doctrine which prevented the establishment of foreign bases in the Caribbean.