But with a victory in Ohio, his home state, Gov. John Kasich denied Mr. Trump one of the night's biggest prizes and made it harder for him to clinch the nomination outright before primary voting ends in June.

Senator Ted Cruz of Texas finished second in Illinois and North Carolina and was locked in a tight race with Mr. Trump in Missouri, ensuring that he, too, would earn a share of delegates.

Mr. Trump has faced mounting criticism from Republicans for the vitriolic tone of his candidacy, but he struck a defiant note Tuesday night, describing himself proudly as a candidate of the angry and disaffected. "There is great anger," he said. "Believe me, there is great anger."