Homeomorphism about gluing euclidean (hyperbolic, spherical) polygon

Question

I'm reading the book 'low-dimensional geometry from euclidean surfaces to hyperbolic knots' written by Francis Bonahon, and the it says that gluing euclidean polygon is homeomorphic to some surface without any proof. For example, the quotient space made by gluing opposite sides of euclidean rectangle is homeomorphic to torus, or the quotient space made by gluing opposite sides of hexagon is homeomorphic to double torus. I tried to understand this homeomorphism by trivial continuous bijection map (which is closed map since the polygon is compact). But it doesn't work to deal with hyperbolic polygon since if there is any vertical line geodesic (infinite radius with center on infinite) making the polygon, it implies that the polygon is not compact. How can I understand the homeomorphism btw the quotient space made by gluing the sides of euclidean(hyperbolic, spherical) polygon and the surface made by stretching the polygon. Thank you!