House managers began laying out their case for convicting and removing Mr. Trump from office, accusing him of a corrupt scheme to enlist the help of a foreign government to tarnish his domestic political rivals and pave his way to a second term.

In a series of sober, methodical speeches, the Democratic managers, or prosecutors, presented extracts of testimony that they were able to gather during the House inquiry despite the president’s refusal to cooperate to argue that Mr. Trump’s pressure on Ukraine to incriminate Democrats amounted to an assault on America’s constitutional democracy.

“President Trump solicited foreign interference in our democratic elections, abusing the power of his office to seek help from abroad to improve his re-election prospects at home,” Mr. Schiff said from the well of the Senate. “President Trump,” he added, “withheld hundreds of millions of dollars in military aid to a strategic partner at war with Russia to secure foreign help with his re-election. In other words, to cheat.”

While the Senate has so far refused to allow witnesses, Mr. Schiff and his fellow managers in effect brought a few to the floor anyway by playing video clips from current and former officials like Fiona Hill, Gordon D. Sondland, William B. Taylor Jr. and David Holmes, who testified before Mr. Schiff’s House Intelligence Committee last year. And they played several from Mr. Trump himself, showing the president in 2016 publicly calling on Russia to hack Hillary Clinton’s email and last year publicly calling on Ukraine and even China to investigate former Vice President Joseph R. Biden Jr.

“Shocking video,” Representative Sylvia Garcia, Democrat of Texas, said after showing Mr. Trump telling George Stephanopoulos of ABC News last spring that he would still be willing to accept damaging information about a campaign opponent from Russia or another foreign government.

With three days to make their case, the seven House impeachment managers started outlining the events in narrative form, how the president and his associates sought to pressure Ukraine to announce investigations of Mr. Biden and Democrats while withholding $391 million in American aid. The managers will later explore the constitutional ramifications of the case.

Mr. Schiff was followed by Representative Jerrold Nadler of New York. Next came Ms. Garcia, then Representatives Jason Crow of Colorado, Val Demings of Florida and Hakeem Jeffries of New York. Mr. Schiff returned to the microphone after dinner, before handing off to Representative Zoe Lofgren of California, then returned to finish the day. Mr. Schiff was by far the main face of the effort, speaking for more than four hours compared with three hours for all six of his colleagues combined.