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NBC and Fox News both projected Tuesday night that the Texas senator would take the state.
With about 44% of the state's precincts reporting, Cruz had a 41.7% to 29% lead over Trump. Sen. Marco Rubio of Florida was in third with 18.4% of the vote.
Earlier in the evening, GOP frontrunner Donald Trump won primaries in Michigan and Mississippi, further paving the real-estate mogul's path to the nomination.
Hawaii's Republican caucuses are also on Tuesday, but won't report the results until at least early Wednesday morning.