4. (20 points) Before the geometry lesson, the teacher wrote on the board the values of all the angles (in degrees) of a certain convex polygon. However, the duty students erased one of the written numbers. When the lesson began, it turned out that the sum of the remaining numbers was 1703. What number did the duty students erase?
Answer: 97.

Solution: Let the polygon have $n$ vertices. Since the $n$-gon is convex, each of its angles is less than $180^{\circ}$, and the sum of all angles is $(n-2) \cdot 180^{\circ}$. Therefore, the sum of all angles of the polygon minus one lies in the interval from $180(n-3)$ to $180(n-2)$. Then

$$
180(n-3)<1703<180(n-2), \quad \text { hence } \quad n=\left[\frac{1703}{180}\right]+3=12 \text {. }
$$

Therefore, the missing angle is $180 \cdot 10-1703=97$ degrees.