On Tuesday, British lawmakers are scheduled to vote again on May’s withdrawal agreement, along with the supplemental statements. A possible series of votes this week will help determine whether Britain will depart on schedule on March 29 — or whether there will be more uncertainty, division and delay.

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“Today we have secured legal changes. Now is the time to come together, to back this improved Brexit deal and to deliver on the instruction of the British people,” May said on Monday night.

But whether the additional language would be enough to win over a divided House of Commons was unclear.

Keir Starmer, the opposition Labour Party’s point person on Brexit, responded that the changes announced were not a breakthrough. Waving a letter in the air that was sent from E.U. chiefs to May in January, he said, “It’s not new. That’s not today, that was in the letter.”

Steve Baker, a prominent Brexiteer, told the BBC that it was “very good gloss” on something that “falls short” of expectations.

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May’s Brexit deal, negotiated over two years with the E.U., was rejected by Parliament in January by a vote of 432 to 202. Since then, she has pressed for adjustments that might induce support from hard-line Brexiteers in her Conservative Party.

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This deal is only about the terms of departure and does not include what the future relationship between Britain and the E.U. will look like. It sets out the $50 billion divorce settlement that Britain will pay; it allows for a two-year transition period, when things will remain essentially as they are now in terms of trade, migration, security and travel; and it seeks a guarantee to preserve the free and open border on the island of Ireland.

It is that guarantee — the Irish backstop — that is at the heart of the impasse. Many British lawmakers fear that it would limit their country’s sovereignty, requiring them to continue to abide by European rules and regulations on customs and trade forever, without having any say. Some had hoped for a sunset clause, or a provision allowing Britain to unilaterally terminate the backstop.

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E.U. leaders on Monday offered fresh pledges that they would seek all possible ways to avoid invoking the politically toxic plan. They wanted to give May a fig leaf, allowing her to say she had received concessions and win over wavering British lawmakers ahead of Tuesday’s vote. But the assurances were framed as the E.U.’s final offer.

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“The backstop is an insurance policy. Nothing more, nothing less. The intention is for it never to be used, like every insurance policy,” European Commission President Jean-Claude Juncker told reporters after meeting with May in Strasbourg. “There will be no new negotiations. It is this.”

If Parliament accepts May’s deal Tuesday, then Britain will leave the European Union on March 29.

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If Parliament rejects her deal, lawmakers are expected to vote Wednesday on whether to allow Britain to leave the E.U. without a deal, a scenario that many hard-line Brexiteers embrace, but one that others warn could send shock waves through the British and European economies.

If the lawmakers decide they still want to try for a managed withdrawal, they’re expected to vote Thursday on whether to postpone Brexit. Delaying Britain’s departure would require permission from E.U. leaders.

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May said earlier that if such a delay were necessary, it would be granted only once, and that it shouldn’t go beyond the end of June.

Anti-Brexit lawmakers hope that if Britain’s departure is delayed, momentum will build for a second referendum — a do-over — to ask voters whether they really want to leave.

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Although Labour has endorsed a second referendum, many lawmakers in all parties are wary, and there does not appear to be majority support for it in Parliament.

As May scrambled on Monday, her opponents circled.

“The prime minister cannot keep dodging scrutiny after failing to get changes to her overwhelmingly rejected deal,” Labour leader Jeremy Corbyn said. “Theresa May must come to Parliament this afternoon and face up to the mess her government has made of the Brexit negotiations.”

Charles Grant, director of the Centre for European Reform, said the outcome of this week’s votes could play into May’s political future. “Can she remain as prime minister if Parliament is taking control and guiding the Brexit process?” he said. “It’s not entirely clear.”

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May survived a Conservative Party vote of no-confidence in her leadership in December and cannot be challenged again that way for a year. But if her government as a whole lost a no-confidence vote, there would be a 14-day window in which any party could try to form a new government.

Lawmakers also have been speculating about how long May would remain at 10 Downing Street if the votes this week don’t go her way.

George Freeman, a Conservative lawmaker, told the BBC that his party shouldn’t change leaders — at the moment. “A panicked change of leader now, I think, will solve nothing. . . . Vote for the deal, and then we can change,” he said.