Let $P$ be a hyperbolic $4n$ gon having opposite sides with equal length and the sum of all interior angles is $2\pi$. Then is true that opposite angles are equal also?
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4$\begingroup$ No, it is not true. It is easy to construct a $4{\cdot}n$-gon with all sides equal and angles $\alpha,\beta,\alpha,\beta,\dots$ for $\alpha\ne \beta$ (so $n\cdot (\alpha+\beta)=\pi$). Then you cut a quadrangle by a diagonal and flip it. It brakes equality of opposite angles. $\endgroup$– Anton PetruninSep 18, 2022 at 11:08
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$\begingroup$ @AntonPetrunin Thank you for your comment. Here if we consider the polygon $P$ as a polyhedral representation of a $g$ genus compact orientable surface with opposite side pairing then will it be true that the opposite angles are equal? $\endgroup$– KAKSep 19, 2022 at 7:36
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$\begingroup$ it seems that you repeat the same question, is not it? $\endgroup$– Anton PetruninSep 19, 2022 at 12:17
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$\begingroup$ @AntonPetrunin If we fix the orientation of the edges then the fliping will reverse the orientation then resulted polygon give some different surface. $\endgroup$– KAKSep 19, 2022 at 14:54
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$\begingroup$ The gluing should be done along the same rule (ignoring switching of sides). $\endgroup$– Anton PetruninSep 19, 2022 at 17:50
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