Final answer:
The Arc Addition Postulate is most relevant for proving that arc AB is a subset of arc CD and arc AC is a subset of arc BD when given parallel chords AD and BC in a circle.
Step-by-step explanation:
To address the question regarding proving that arc AB is a subset of arc CD and that arc AC is a subset of arc BD in a circle with parallel chords AD and BC, the most relevant theorem is the Arc Addition Postulate (option C). This postulate allows us to add together the measures of adjacent arcs to get the measure of the larger arc encompassing them. If AD and BC are parallel, then arc AB and arc AC are parts of arc AD and arc BC respectively. Because AD and BC are parallel, angles formed by the chords and intersecting radii or tangents will be congruent, implying that the arcs subtended by these angles will have a relationship in their measures that can be addressed by the Arc Addition Postulate.
In this context, the Arc Addition Postulate states that if a point B lies on arc CD, then the measure of arc CD is the sum of the measures of arcs CB and BD. Hence, since chords AD and BC are parallel, arc AB is a part of arc CD, and similarly, arc AC is a part of arc BD.