|Representing Great Britain|
|1924 Paris||4×100 m relay|
|1928 Amsterdam||200 m|
|1928 Amsterdam||4×100 m relay|
|British Empire Games|
|1934 London||4×110 yd relay|
|1934 London||220 yd|
Rangeley competed for Great Britain in the 1924 Summer Olympics held in Paris, France in the 4×100 metre relay where he won the silver medal with his teammates Harold Abrahams, Wilfred Nichol and Lancelot Royle. In the 100 metres event he was eliminated in the quarter-finals.
At the following Games in 1928 held in Amsterdam Rangeley won a silver medal in the 200 metres. He also joined with a new relay team consisting of teammates Cyril Gill, Edward Smouha and Jack London which won the bronze medal in the 4×100 metres competition. In the 100 metres event he was eliminated in the quarter-finals again.
After not participating in the 1932 Games his final Olympic appearance was in 1936 when he was a member of the British relay team which was eliminated in the first round of the 4×100 metre relay contest.
At the 1934 British Empire Games he won the bronze medal in the 220 yards event. He also won the gold medal with the English relay team in the 4×110 yards competition.
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