1. **Stating the problem:** We have a cyclic quadrilateral WXYZ inscribed in a circle with angles at each vertex split into two parts: at W angles $a^\circ$ and $b^\circ$, at X angles 70° and 28°, at Z angles 31° and 51°, and at Y angles $c^\circ$ and $d^\circ$. We need to find the unknown angles $a$, $b$, $c$, and $d$. 2. **Key properties:** In a cyclic quadrilateral, opposite angles sum to 180°. 3. **Calculate total angles at X and Z:** - At X: $70^\circ + 28^\circ = 98^\circ$ - At Z: $31^\circ + 51^\circ = 82^\circ$ 4. **Use opposite angles sum rule:** - $\angle W + \angle Y = 180^\circ$ 5. **Express $\angle W$ and $\angle Y$ as sums:** - $\angle W = a + b$ - $\angle Y = c + d$ 6. **Use opposite angles:** - $a + b + c + d = 180^\circ$ 7. **Use the fact that the sum of all interior angles in a quadrilateral is 360°:** - $(a + b) + 98^\circ + (c + d) + 82^\circ = 360^\circ$ - Substitute $a + b + c + d = 180^\circ$ from step 6: - $180^\circ + 98^\circ + 82^\circ = 360^\circ$ - $360^\circ = 360^\circ$ (confirms consistency) 8. **Find individual angles at W and Y using the fact that angles around a point sum to 360°:** - At W, angles $a$ and $b$ are adjacent to 70° and 28° at X, so $a + b = 82^\circ$ (since $180^\circ - 98^\circ = 82^\circ$) - At Y, angles $c$ and $d$ are adjacent to 31° and 51° at Z, so $c + d = 98^\circ$ (since $180^\circ - 82^\circ = 98^\circ$) 9. **Final answers:** - $a + b = 82^\circ$ - $c + d = 98^\circ$ Without additional information, individual values of $a$, $b$, $c$, and $d$ cannot be determined. **Summary:** - $a + b = 82^\circ$ - $c + d = 98^\circ$