1. The problem involves determining the values of angles \(\angle 1\) through \(\angle 27\) in a circle with many chords and points. 2. Given the complexity and the provided formula on the chalkboard: $$m\angle j = \frac{\angle O + VE}{2}$$ This suggests angles formed by chords intersecting inside the circle can be calculated as half the sum of intercepted arcs. 3. Without explicit arc measures or additional angle relationships, individual angle values cannot be uniquely determined. 4. Key geometry principles: - Angle formed by two chords intersecting inside a circle equals half the sum of the measures of the arcs intercepted by the angle and its vertical angle. - Angles subtended by the same chord are equal. - Opposite angles in cyclic quadrilaterals sum to 180°. 5. Without numerical arc lengths or segment measures, and no further data, none of the angles \(\angle 1\) through \(\angle 27\) can be assigned exact numerical values. Final answer: All angle values \(\angle 1\) to \(\angle 27\) cannot be determined with the given information.