For Mr. Obama, the lion’s-den strategy of addressing a Republican audience reinforced his effort in the State of the Union address this week to reclaim a more bipartisan image and reach out to disaffected independents. Although he and other presidents have addressed opposition caucuses before, they usually close the doors for questions, but this time the White House insisted on letting the news media record the give and take.

That worked to his benefit as he took advantage of the staging that comes with being president. He commanded the lectern with the presidential seal and the camera was trained mainly on him, while his interlocutors were forced to look up to him from the audience. Moreover, Mr. Obama gave long, confident and informed answers and felt free to interrupt questioners, while it is typically harder for others to interrupt a president.

But Republicans said they believed they had achieved a victory as well, demonstrating that while Democrats might not like some of their policy ideas, they had advanced some proposals as evidenced by the president’s acknowledgment that he had read them and even incorporated some of them into his initiatives.

“For him to say, ‘I have read your proposals and they are substantive proposals,’ that is a huge thing for Republicans,” Representative Jeff Flake, Republican of Arizona, said afterward.

Just to make the point that they have been more than the party of no, Representative John A. Boehner of Ohio, the House Republican leader, as he introduced the president handed him a booklet called “Better Solutions” compiling a variety of Republican ideas that they said the president had ignored or resisted over the last year.

“We don’t expect you to agree with us on every one of our solutions,” Mr. Boehner said, “but we do hope that you and your administration will consider them.”