Consider drawing multiple chords of a circle, none of which share an endpoint.

The circle is thereby divided into multiple arcs.

To each arc assign a number as follows.

If the two endpoints of the arc are from the same chord, number the arc 180.
If the endpoints of the arc are from parallel chords, number the arc 0.
If the endpoints of the arc come from chords that intersect inside the circle, number the arc by the number of degrees of the angle formed.
If the endpoints of the arc come from chords that, if extended, intersect outside the circle but on the same side as the arc, number the arc by the number of degrees of the angle formed by the extended chords.
If the endpoints of the arc come from chords that, if extended, intersect outside the circle but on the opposite side from the arc, number the arc by the negative number of degrees of the angle formed by the extended chords.
Prove or give a counterexample to the following:

The sum of these numbers is 360.

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