1. **Stating the problem:** We have a cyclic quadrilateral WXYZ inscribed in a circle with given angles: at X, angles 70° and 28°; at Z, angles 31° and 51°; and unknown angles a°, b° at W and c°, d° at Y. We need to find the values of a°, b°, c°, and d°. 2. **Key property:** Opposite angles of a cyclic quadrilateral sum to 180°. 3. **At vertex X:** angles 70° and 28° are adjacent, so the total angle at X is $$70° + 28° = 98°$$. 4. **At vertex Z:** angles 31° and 51° are adjacent, so the total angle at Z is $$31° + 51° = 82°$$. 5. **Using the cyclic quadrilateral property:** - Angles at W and Y are opposite to angles at Y and W respectively. - So, $$a° + b° = \text{angle at W}$$ and $$c° + d° = \text{angle at Y}$$. 6. Since W and Y are opposite vertices, their angles sum to 180°: $$ (a° + b°) + (c° + d°) = 180° $$ 7. Also, angles at X and Z are opposite and sum to 180°: $$ 98° + 82° = 180° $$ (which confirms the quadrilateral is cyclic). 8. To find a°, b°, c°, d°, we use the fact that angles around a point sum to 360°. 9. At vertex W, angles a° and b° are adjacent and together with the arc they subtend, but since no other angles at W are given, we consider the sum of angles at W and Y is 180°. 10. At vertex Y, angles c° and d° are adjacent. Since no other angles are given, we use the fact that the sum of all internal angles in quadrilateral WXYZ is 360°: $$ (a° + b°) + 98° + (c° + d°) + 82° = 360° $$ 11. Substitute known sums: $$ (a° + b°) + 98° + (c° + d°) + 82° = 360° $$ $$ (a° + b°) + (c° + d°) + 180° = 360° $$ $$ (a° + b°) + (c° + d°) = 180° $$ 12. This matches the cyclic quadrilateral property, so we need more information to find individual angles a°, b°, c°, d°. 13. However, since angles at Z are 31° and 51°, and at X are 70° and 28°, and the sum of angles at W and Y is 180°, we can assign: - $$a° + b° = 98°$$ (opposite to angle at Y) - $$c° + d° = 82°$$ (opposite to angle at W) 14. Without additional information, we cannot separate a° and b°, or c° and d° individually. **Final answers:** $$a° + b° = 98°$$ $$c° + d° = 82°$$