Final answer:
The sum of the measures of the exterior angles of a convex 39-gon is always 360 degrees, as this is true for any convex polygon.
Step-by-step explanation:
To find the sum of the measures of the exterior angles of a convex 39-gon (a polygon with 39 sides), we need to use the fact that the sum of the exterior angles of any convex polygon is always 360 degrees, regardless of the number of sides the polygon has. This property holds true because an exterior angle is formed by extending one side of the polygon and measuring the angle between this extended line and the next side. Since one complete rotation around a point is 360 degrees, the sum of all exterior angles must be 360 degrees.