Below is the official solution of above problem. I have a doubt in the solution: My doubt is: After we sum up all such equalities, we obtain \begin{align*} 2 \left( \widehat{M_1H_1} + \cdots + \widehat{M_nH_n} \right) \equiv 0^\circ \pmod{360^\circ} \\ \implies \widehat{M_1H_1} + \cdots + \widehat{M_nH_n} \equiv 0^\circ \pmod{180^\circ} \end{align*}So we only get that sum of all directed arcs is $0$ $\mod 180^\circ$, and not $\mod 360^\circ$. I can't see why $\mod 180^\circ$ is enough to imply our desired result. Like, it might also happen that (in undirected sense) $$ \epsilon_1 \cdot \widehat{M_1H_1} + \cdots + \epsilon_n \cdot \widehat{M_nH_n} = \pm 180^\circ $$where each $\epsilon_i \in \{+1,-1\}$ is a sign. But we instead require $$ \epsilon_1 \cdot \widehat{M_1H_1} + \cdots + \epsilon_n \cdot \widehat{M_nH_n} = 0^\circ $$I can't get what I am missing.