An endomorphism ${alpha}$ of a ring R is called right(left) symmetric if whenever abc=0 for a, b, c ${in}$ R, $ac{alpha}(b)=0({alpha}(b)ac=0)$. A ring R is called right(left) ${alpha}-symmetric$ if there exists a right(left) symmetric endomorphism ${alpha}$ of R. The notion of an ${alpha}-symmetric$ ring is a generalization of ${alpha}-rigid$ rings as well as an extension of symmetric rings. We study characterizations of ${alpha}-symmetric$ rings and their related properties including extensions. The relationship between ${alpha}-symmetric$ rings and(extended) Armendariz rings is also investigated, consequently several known results relating to ${alpha}-rigid$ and symmetric rings can be obtained as corollaries of our results.