I am studying the classification of Surfaces, and run into the notion of connected sum. We define it in terms of triangulations. I want to show the following. Let $S$ be a triangulated surface. I want to show that $S\#T$ with $T$ the torus is combinatorially homeomorphic to attaching a handle to $S$. I am not allowed to use the classification of Surfaces, since this is what I want to prove.
So, Let $S'$ denote the complement of a combinatorial disk in $S$, Then I want to glue the boundary of $S'$ to the complement of a combinatorial disk in $T$. How is this combinatorially homeomorphic to $S$ with a handle glued in?
EDIT: Possible solution: Since $S\#S^2\cong S$, and $T$ is $S^2$ with a handle glued in, $S\#T\cong S\#(S^2\text{with handle})\cong (S\# S^2)\text{with a handle glued in}$. Is this last line correct?