Polygon Sides E1B0De

1. **State the problem:** We have an n-sided polygon with three known interior angles: 125\degree, 156\degree, and 175\degree. The remaining \(n-3\) interior angles are each 111\degree. We need to find \(n\).\n\n2. **Recall the formula for the sum of interior angles of an n-sided polygon:**\n$$\text{Sum of interior angles} = 180(n-2)$$\n\n3. **Set up the equation:** The sum of all interior angles is the sum of the three given angles plus the sum of the remaining \(n-3\) angles each measuring 111\degree.\n$$125 + 156 + 175 + 111(n-3) = 180(n-2)$$\n\n4. **Simplify the left side:**\n$$125 + 156 + 175 = 456$$\nSo the equation becomes:\n$$456 + 111(n-3) = 180(n-2)$$\n\n5. **Expand the terms:**\n$$456 + 111n - 333 = 180n - 360$$\n\n6. **Combine like terms on the left:**\n$$123 + 111n = 180n - 360$$\n\n7. **Bring all terms involving \(n\) to one side and constants to the other:**\n$$123 + 111n - 180n = -360$$\n$$123 - 69n = -360$$\n\n8. **Isolate \(n\):**\n$$-69n = -360 - 123$$\n$$-69n = -483$$\n\n9. **Divide both sides by \(-69\):**\n$$n = \frac{-483}{-69}$$\n$$n = \frac{\cancel{483}}{\cancel{69}} = 7$$\n\n10. **Final answer:**\n$$\boxed{7}$$\n\nTherefore, the polygon has \(n=7\) sides.