Cyclic Quadrilaterals with Sides 1 and 2

If all the side lengths are the same, then opposite angles must be congruent (as a rhombus) and opposite angles must be supplementary (as a cyclic quadrilateral), and the quadrilateral must then be a square with angles of $90°$ . There are two squares possible, one square with side lengths of $1$ and one square with side lengths of $2$ .

If $3$ of the side lengths are the same and $1$ is different, then the angles formed by $2$ congruent side lengths are congruent (since an angle formed by $2$ chords is congruent to the angle formed by $2$ chords congruent to the first $2$ chords), and the angles formed by the $2$ different side lengths are congruent (since an angle formed by $2$ chords is congruent to the angle formed by $2$ chords congruent to the first $2$ chords) but different from the other congruent angles, and opposite angles must be supplementary (as a cyclic quadrilateral), and the quadrilateral must then be an isosceles trapezoid. Each acute base angle is then $\cos^{-1}\frac{b - a}{2c}$ and each obtuse base angle is $90° + \sin^{-1}\frac{b - a}{2c}$ , where $a$ is the length of the shorter base, $b$ is the length of the longer base, and $c$ is the length of the leg of the isosceles triangle. There are two isosceles trapezoids possible, one trapezoid with $3$ side lengths of $2$ and $1$ side length of $1$ , which would give angles of $\cos^{-1}\frac{2 - 1}{2\cdot2} \approx 75.5°$ and $90° + \sin^{-1}\frac{2 - 1}{2\cdot2} \approx 104.5°$ , and one trapezoid with $3$ side lengths of $1$ and $1$ side length of $2$ , which would give angles of $\cos^{-1}\frac{2 - 1}{2\cdot1} = 60°$ and $90° + \sin^{-1}\frac{2 - 1}{2\cdot1} = 120°$ .

If there are $2$ pairs of congruent side lengths and they are alternating, then all the angles formed must be congruent (since an angle formed by $2$ chords is congruent to the angle formed by $2$ chords congruent to the first $2$ chords), and the angles must be supplementary (as a cyclic quadrilateral), and the quadrilateral must then be a rectangle with angles of $90°$ .

Finally, if there are $2$ pairs of congruent side lengths that are not alternating, then the quadrilateral must be a kite. Also, the $2$ opposite angles formed by the $2$ different side lengths must be congruent (since an angle formed by $2$ chords is congruent to the angle formed by $2$ chords congruent to the first $2$ chords), and these two angles must be supplementary (as a cyclic quadrilateral), so these $2$ angles are $90°$ . The other two angles must then be $2\tan^{-1}\frac{1}{2} \approx 53.1°$ and $2\tan^{-1}\frac{2}{1} \approx 126.9°$ .

Therefore, there are 7 unique interior angles: $90°$ , $75.5°$ , $104.5°$ , $60°$ , $120°$ , $53.1°$ , and $126.9°$ , and their sum is $\boxed{630°}$ .