  Recent experimental realizations of the topological semimetal states in
several monolayer systems are very attractive because of their exotic quantum
phenomena and technological applications. Based on first-principles
density-functional theory calculations including spin-orbit coupling, we here
explore the drastically different two-dimensional (2D) topological semimetal
states in three monolayers Cu$_2$Ge, Fe$_2$Ge, and Fe$_2$Sn, which are
isostructural with a combination of the honeycomb Cu or Fe lattice and the
triangular Ge or Sn lattice. We find that (i) the nonmagnetic (NM) Cu$_{2}$Ge
monolayer having a planar geometry exhibits the massive Dirac nodal lines, (ii)
the ferromagentic (FM) Fe$_2$Ge monolayer having a buckled geometry exhibits
the massive Weyl points, and (iii) the FM Fe$_2$Sn monolayer having a planar
geometry and an out-of-plane magnetic easy axis exhibits the massless Weyl
nodal lines. It is therefore revealed that mirror symmetry cannot protect the
four-fold degenerate Dirac nodal lines in the NM Cu$_{2}$Ge monolayer, but
preserves the doubly degenerate Weyl nodal lines in the FM Fe$_{2}$Sn
monolayer. Our findings demonstrate that the interplay of crystal symmetry,
magnetic easy axis, and band topology is of importance for tailoring various 2D
topological states in Cu$_2$Ge, Fe$_2$Ge, and Fe$_2$Sn monlayers.
