The fission fragment mass angle correlations and mass ratio distributions have been investigated for the two systems $^{16}\mathrm{O}+^{197}\mathrm{Au}$ and $^{27}\mathrm{Al}+^{186}\mathrm{W}$, leading to the same compound nucleus $^{213}\mathrm{Fr}$ around the Coulomb barrier energies. Systematic analysis of the variance of the mass distributions as a function of temperature and angular momentum suggests true compound nuclear fission for both the reactions, indicating the absence of nonequilibrium fission processes.