AUGUSTA, Maine — Liberal consensus-builder Emily Cain and conservative firebrand Bruce Poliquin won their parties’ primaries on Tuesday for the open congressional seat in Maine’s massive rural district, which the GOP hopes to wrest away from Democrats for the first time in two decades.
Cain easily defeated Troy Jackson, a logger from Allagash, to win the Democratic nomination for the Second Congressional District. On the GOP ticket, Poliquin beat establishment Republican and former chief of staff to US Senator Olympia Snowe, Kevin Raye.
While the Democrats’ pick says she will work across the aisle in Washington, Republicans are putting their hope in a conservative who has pledged to fight for GOP values, including smaller government and lower taxes.
Cain, a state senator who also works for the honors college at the University of Maine, campaigned on her experience writing budgets in Augusta and touted her ability to stay at the negotiating table to deliver results.
She said Tuesday that her victory reflects the frustration that residents across the state have about the gridlock and dysfunction in Washington.
‘‘It just reinforces the importance of sending someone to Washington who knows how to bring people together,’’ she said in an interview.
In Bangor on Tuesday, Poliquin told supporters he was running to help his son Sam and other twentysomethings, describing ‘‘a horrible business climate in the Second District and throughout Maine.’’
He said voters want someone to help.
‘‘They want someone who'll go down to Washington and fight for their interests and not become part of the mess. And I will do that,’’ he said.
The seat is now held by Democrat Mike Michaud, who seeks to oust Republican Governor Paul LePage.
Elsewhere around the country, uncontested congressional primaries were held in North Dakota and congressional face-offs were held in Virginia and Nevada, which also held a gubernatorial primary. A runoff was held in Arkansas.
In South Carolina, Senator Lindsey Graham won the GOP primary outright, defeating six Tea Party challengers who cast him as not conservative enough for this heavily Republican state to avoid a runoff.
Graham, 58, had about 56 percent of the vote in early returns, far more than what was needed to avoid the runoff. State Senator Lee Bright came in second, with about 15 percent.
Graham, who has been in office since 2002, had a hefty fund-raising advantage: He has raised more than $12 million since his last reelection bid in 2008, while none of his opponents passed the $1 million mark.
Although Graham had no trouble dispatching his challengers and their attacks that he wasn’t conservative enough, other longtime Republican lawmakers have not had such good fortune.
In Mississippi, Senator Thad Cochran is facing a runoff with Tea Party-backed Chris McDaniel later this month after both finished just shy of the 50 percent of votes needed to win outright. And in Virginia, House majority leader Eric Cantor — the GOP’s number two in the chamber — was dealt a crushing defeat by little known rival Dave Brat, a college professor with no political experience.
The conservatism critique didn’t matter to Ben Lister, a 48-year-old financial planner from Greenville who voted for Graham. The incumbent’s opponents took several swipes at his willingness to work with Democrats, including his votes to confirm President Obama’s choices for the Supreme Court, Elena Kagan and Sonia Sotomayor, both considered liberal justices.
‘‘I know that some people are saying he should be more conservative, but what does that mean?’’ Lister asked. ‘‘I want a politician who actually thinks about the issues instead of going along with the crowd.’’
However, Graham also has been among the president’s harshest critics on foreign policy. He and fellow Senators John McCain of Arizona and Kelly Ayotte of New Hampshire have called the president’s plan to withdraw virtually all US troops from Afghanistan by 2016 a short-sighted mistake that would embolden enemies.
During an emotional victory speech in downtown Columbia, Graham thanked his supporters and pledged to move forward on several key issues and to work toward winning a Republican majority in the Senate.
‘‘To the people of South Carolina: Thank you. I will not let you down,’’ Graham said.
Meanwhile, Graham’s fellow Republican Senator Tim Scott won his primary by a wide margin, setting the stage for South Carolina to elect a black person to the Senate for the first time.
Scott was appointed to the seat in 2012 after Jim DeMint stepped down, and the general election winner will serve the remainder of DeMint’s term.
At his election night gathering in North Charleston, Scott said he hoped the election would focus more on substance rather than race.
The Democrats had two primaries, though it’s widely expected that the Senate seats will remain in the GOP’s hands.
State Senator Brad Hutto won the nomination for Graham’s seat, while Richland County Councilwoman Joyce Dickerson was nominated to face Scott.