WASHINGTON — In trying to decide whether the Environmental Protection Agency has the authority under two programs to regulate greenhouse gas emissions from stationary sources like power plants, the Supreme Court on Monday faced what Justice Elena Kagan called “the conundrum here.”

One part of the Clean Air Act, she said, seemed to require that such emissions be regulated. But another part set the emission thresholds so low that even schools and small businesses would be covered.

The agency’s solution was to raise those thresholds, and the resulting standards covered far fewer sources. That move was at the center of Monday’s arguments, and the justices seemed divided along ideological lines over whether it was a sensible accommodation or an impermissible exercise of executive authority.

Justice Anthony M. Kennedy, who may hold the decisive vote, made a point that did not bode well for the agency.