1. **Problem Statement:** Given a star with 10 points, the sum of the outer angles labeled $\angle 1 + \angle 2 + \cdots + \angle 9 = 266^\circ$. Find the sum of the inner angles $\angle a + \angle b + \cdots + \angle i$. 2. **Key Concept:** For any polygon, the sum of the exterior angles is always $360^\circ$. Here, the outer angles $\angle 1$ through $\angle 9$ are part of the exterior angles of the star polygon. 3. **Calculation:** Since the star has 10 points, there are 10 exterior angles in total. The sum of all exterior angles is $360^\circ$. 4. Given $\angle 1 + \angle 2 + \cdots + \angle 9 = 266^\circ$, the tenth exterior angle is: $$\angle 10 = 360^\circ - 266^\circ = 94^\circ$$ 5. The inner angles $\angle a$ through $\angle i$ correspond to the angles inside the star between points, which are related to the exterior angles by: $$\text{Inner angle} = 180^\circ - \text{Exterior angle}$$ 6. Since there are 9 inner angles $a$ through $i$, and 9 corresponding exterior angles $1$ through $9$, their sums relate as: $$\sum_{k=a}^i \angle k = 9 \times 180^\circ - \sum_{k=1}^9 \angle k = 1620^\circ - 266^\circ = 1354^\circ$$ **Final answer:** $\angle a + \angle b + \angle c + \angle d + \angle e + \angle f + \angle g + \angle h + \angle i = 1354^\circ$.