 Solution

Take a collar ${\partial }_{i}X\times [0,\epsilon ]$. In the quotient, the two annuli $A_1,A_2$ are folded together by $f$, while their common boundary curves ${\beta }_1,{\beta }_2$ become the boundary of a meridional disk. Because $f({\alpha }_1)={\alpha }_2$, the two halves of $\alpha$ join across this disk and bound it. The quotient of the collar is therefore a solid torus whose meridian is attached along $\alpha$; outside the collar nothing changes. Hence the annulus-quotient model of Dehn filling gives