BRASÍLIA—Eduardo Cunha, leader of the lower house of Brazil’s congress, had a hectic day on Dec. 15. Federal police raided his home around 6 a.m. seeking evidence he had received kickbacks in a wide-ranging embezzlement scandal relating to the state oil company Petrobras.

Hours later he was in congress, impeccably dressed in a blue suit, vowing to move forward with his top legislative issue, a motion to impeach Brazilian President Dilma Rousseff for allegedly using accounting tricks to mask deficits.

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