1. The problem is to find the remaining interior angle of a polygon when the sum of the interior angles is 540 degrees. 2. The formula for the sum of interior angles of a polygon with $n$ sides is $$\text{Sum of interior angles} = (n-2) \times 180$$ degrees. 3. Since the sum is given as 540 degrees, we set up the equation: $$(n-2) \times 180 = 540$$. 4. Divide both sides by 180 to isolate $(n-2)$: $$\cancel{180} \times (n-2) = \frac{540}{\cancel{180}}$$ $$n-2 = 3$$ 5. Add 2 to both sides to solve for $n$: $$n = 3 + 2 = 5$$ 6. This means the polygon has 5 sides, which is a pentagon. 7. To find the remaining angle, if some angles are known, subtract the sum of known angles from 540 degrees. 8. For example, if four angles sum to $x$, then the remaining angle is $$540 - x$$ degrees. Final answer: The polygon has 5 sides, and the remaining angle is $$540 - \text{sum of known angles}$$ degrees.