Angles · Superchallenge problem

Five rays $OA$, $OB$, $OC$, $OD$, $OE$ start at point $O$, in that order. The angles between neighbouring rays are: angle $AOB = 10^\circ$, angle $BOC = 20^\circ$, angle $COD = 30^\circ$, angle $DOE = 40^\circ$. Every pair of rays makes an angle (for example angle $AOC$ or angle $BOE$). What is the sum of the angles made by all 10 pairs of rays?