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If you want to act like a local, do as the locals do and see a soccer match (just remember to call it "football"). Estadio do Maracanã is one of the biggest soccer stadiums in the world, so unless you have a problem with crowds, going to a game is a must.
One of the can't-miss tourist attractions is a large lagoon, Lagoa, in the middle of the city. It's a great place to walk or ride a bicycle, or just sit back and watch paddle-boaters and take in the view of the iconic Christ the Redeemer statue and Ipanema beach while sipping on fresh coconut juice.