Piecewise smooth? triangles on a Compact Riemann Surface

Question

For integrate on a Riemann Surface $X$ we need a holomorphic $1$-form $\omega$ and a piecewise smooth path $\gamma : [a,b] \to X$ (This means $\phi \circ \gamma(t) : [a,b] \to \mathbb{C}$ is smooth for $\phi$ a chart on $X$). Then we have this tiny version of Stoke's theorem: If $T\subset X$ is a triangle on the domain of a chart then $$\int_{\partial T} \omega = 0 $$ if we consider the triangle $T$ as a homeomorphic image of a triangle in $\mathbb{R}^2$. But, Why can I say that $\partial T$ is a piecewise smooth path?