Q: How do you find the measure of an exterior angle of a polygon off of one vertex?

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No. The interior angle and exterior angle at the same vertex are supplementary. Each of them is (180 degrees minus the other). In rectangles (including squares), the interior and exterior angles at each vertex are both right angles.

If an interior angle is 55 deg, the corresponding exterior angle is 125 deg. This has nothing to do with the number of sides of the polygon (unless it is a regular polygon).

The sum of an adjacent interior and its exterior angle will total to 360°. If the angles were to be equal, they would both have to be 180°. An angle of 180° is a straight line. A polygon may be composed of straight lines that intersect at vertices but a straight line has no vertex. That being the case, the answer to your question is "No".

If it's a regular polygon then each exterior angle measures 36 degrees

It is: 180-exterior angle = interior angle because there are 180 degrees on a straight line

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Very rarely.

No. The interior angle and exterior angle at the same vertex are supplementary. Each of them is (180 degrees minus the other). In rectangles (including squares), the interior and exterior angles at each vertex are both right angles.

It is: 180-vertex angle = exterior angle

If an interior angle is 55 deg, the corresponding exterior angle is 125 deg. This has nothing to do with the number of sides of the polygon (unless it is a regular polygon).

The exterior angles of a polygon always total 360 degrees. That doesn't even depend on how many sides the polygon has.

The sum of an adjacent interior and its exterior angle will total to 360°. If the angles were to be equal, they would both have to be 180°. An angle of 180° is a straight line. A polygon may be composed of straight lines that intersect at vertices but a straight line has no vertex. That being the case, the answer to your question is "No".

Theorem 6-1-2; Polygon Exterior Angle Sum Theorem:The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360 degrees.

The largest exterior angle measure is 120o. It is the exterior measure of an equilateral triangle (which is a regular polygon).

The sum of the exterior angles of a convex polygon which has sides and one angle at each vertex is 360 degrees.

Providing that it's a regular polygon then each exterior angle will measure 12 degrees.

If it's a regular polygon then each exterior angle measures 36 degrees

360/number of sides = exterior angle