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By Lawrence Hurley WASHINGTON (Reuters) - The U.S. Supreme Court appeared sharply divided on Tuesday on whether the Constitution guarantees a right to same-sex marriage, but pivotal Justice Anthony Kennedy seemed to inch cautiously toward legalizing gay nuptials nationwide. Kennedy, a conservative who often casts the deciding vote in close cases and has a history of backing gay rights, posed tough questions to both sides but seemed to give hope to gay marriage advocates by stressing the nobility and dignity of same-sex couples. At one point during the arguments in the decorous white-marble, crimson-draped courtroom, a protester shouted that the justices would "burn in hell" if they backed gay marriage. The court's four liberal justices seemed willing to vote in favor of gay marriage, while the court's conservatives, including Chief Justice John Roberts, appeared inclined to back the right of states to restrict the definition of marriage.