 Solution

Form the dual graph of the finite triangulation: its vertices are the triangles, and two vertices are joined when their triangles share an edge. Connectedness of the surface makes this graph connected. Choose a spanning tree and begin with one disjoint triangle. Attach the remaining triangles along the tree edges, one at a time, always joining a new triangle along one boundary edge. At every step the result is a topological disk, so the final union is a polygonal disk.

The edges not used for these attachments still occur twice on the boundary, once for each incident triangle. Identify each pair as in the original triangulation. The result recovers the original surface. This proves a compact connected closed surface is a polygon with its boundary edges identified in pairs. Closedness is the convention here; a boundary would leave some edges unpaired.