1. The problem involves analyzing angles and arcs in circles with inscribed polygons and chords. 2. Key circle theorems to use: - The measure of an inscribed angle is half the measure of its intercepted arc. - Opposite angles of a cyclic quadrilateral sum to 180°. - Angles subtended by the same arc are equal. 3. For example, in the center circle with quadrilateral D, C, E and angle 125°, if angle x is opposite to 125°, then by cyclic quadrilateral property: $$x + 125^\circ = 180^\circ$$ 4. Solving for x: $$x = 180^\circ - 125^\circ = 55^\circ$$ 5. This method applies similarly to other labeled angles and arcs using the appropriate circle theorems. Final answer for the first distinct problem (center circle quadrilateral): $$x = 55^\circ$$