Final answer:
Roosevelt secured territorial rights in Panama by supporting Panamanian independence, recognizing Panama swiftly, and negotiating the Hay-Bunau-Varilla Treaty, which allowed the U.S. control over the Panama Canal zone.
Step-by-step explanation:
Theodore Roosevelt gained territorial rights in Panama through a series of strategic actions aimed at supporting Panama's independence from Colombia, ultimately to facilitate the construction of the Panama Canal. In an act of what Roosevelt dubbed the use of the "big stick" diplomacy, American battleships were sent to the Panamanian coast, preventing Colombian troops from suppressing a Panamanian revolt. Roosevelt's decision was motivated by the Colombian Senate's rejection of a treaty that would have allowed the U.S. to lease land for the canal.
By swiftly recognizing Panama's independence and negotiating the Hay-Bunau-Varilla Treaty, the United States was granted "use, occupation, and control" of the canal zone, paying Panama $10 million plus an annual fee of $250,000. This strategic move was intertwined with Roosevelt's corollary to the Monroe Doctrine, preventing foreign establishments in the Caribbean and ensuring U.S. intervention rights in Latin America.