Diagonals and Angles

Since every two vertices of $n$ -sided polygon form a line. The total number of lines formed by $n$ vertices is $\dbinom n2$ , subtract away the $n$ sides, we get the number of diagonals. Therefore we have:

$\begin{aligned} \binom n2 - n & = 2774 \\ \dfrac {n(n-1)}2-n & = 2774 \\ n^2 - 3n - 5548 & = 0 \\ (n-76)(n+73) & = 0 & \small \blue{\text{Since }n > 0} \\ \implies n & = 76 \end{aligned}$

An $n$ -sided regular polygon is formed by $n$ congruent isosceles triangles with a common vertex which is the center of the regular polygon. Each the sum of three interior angles of each triangle is $180^\circ$ . Therefore the sum for $n$ triangles is $180^\circ n$ , subtract away $350^\circ$ at the center, we get the measure of the $n$ interior angles of the regular polygon. Let the measure of interior angle be $\theta^\circ$ . Then we have:

$\theta = \frac {180n - 360}n = \frac {74 \cdot 180}{76} = \frac {3330}{19} \approx \boxed{175^\circ}$

√(2 × 2774) ≈ 74.485
2774 = n(n – 3) / 2
n = 76

Answer
= 180° – (360° ÷ n)
= 180° – (360° / 76)
= 175.263158°