What Is the Exterior Angle at Infinity for a Semi-Infinite Strip?

[rsq_a]

Suppose we have the following shape in the complex plane: The EXTERIOR of the semi-infinite strip bounded by 0 < y < 1 and x > 0. The two physical angles making up the rectangle have interior angles of 3*pi/2 and thus exterior angles of -pi/2.

Now, because the sum of the exterior angles of a polygon have to sum to 2pi, we can claim that the exterior angle at infinity is simply 2pi - (-pi/2 - pi/2) = 3pi.

However, I'm having trouble justifying this geometrically. I've looked at projecting the surface onto the Riemann sphere, but the exterior angle at infinity simply seems to be either pi or -pi. I see no reason why it has to do an additional orbit of 2pi.

 

[symbolipoint]

How many non-overlapping triangles can you determine in the polygon? Each triangle contains sum of the interior angles being $$\pi$$ radians.

 

[rsq_a]

How many non-overlapping triangles can you determine in the polygon? Each triangle contains sum of the interior angles being $$\pi$$ radians.

How do you imagine triangles on an infinite domain? I don't see how this helps...