The measure of angles is a continuous variable, so the answer would be 180 plus the tiniest amount possible (or even a smaller amount than that!).
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The proof is pretty simple, but hard to see without the pictures. SO here is a link to the proof with some pics. http://www.apronus.com/geometry/triangle.htm The answer depends on your geometry: In Euclidean geometry, the angle sum is 180 degrees, in Hyperbolic geometry it is less than 180 degrees, and in Elliptical geometry it is greater than 180 degrees.
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No, it is not possible to have a triangle with an angle of 181 degrees. The sum of the interior angles of a triangle is always 180 degrees according to Euclidean geometry. Any triangle with an angle greater than 180 degrees would not be a valid geometric shape.