Let $\Sigma$ be a connected orientable compact surface that is not a disk.
Let $\alpha,\beta:[0,1]\rightarrow \Sigma$ be loops in $\Sigma$. We say that $\alpha$ is conjugative to $\beta$ if there is a path $\gamma:[0,1]\rightarrow \Sigma$ with $\gamma(0)=\alpha(0)$ and $\gamma(1)=\beta(0)$ such that $\alpha=\gamma^{-1}\beta\gamma$.
The following affirmations are true. How can I prove them?
(1) If $\gamma_0$ and $\gamma_1$ are two distinct boundary components of $\Sigma$ such that $\gamma_0$ commutes in $\pi_1(\Sigma)$ with a non-trivial loop conjugate of $\gamma_1$ then $\Sigma$ is an annulus.
(2) If $\gamma$ is a boundary component of $\Sigma$ then $\gamma$ can not commute in $\pi_1(\Sigma)$ with a non-trivial conjugate of itself.