Final answer:
Burma, now known as Myanmar, was the country in Southeast Asia that was under British rule.
Step-by-step explanation:
The country in Southeast Asia that fell under British rule was Burma. This region was part of the British Empire's extensive South Asian territories. British rule in Burma lasted until the Japanese invasion during World War II. Although Burma (now known as Myanmar) was under Japanese control briefly, it was previously one of Britain's most prosperous colonies. Following the war, Burma attained independence and established a democratic government in 1948, which lasted until the military dictatorship took over in 1962.