  The classical Chung-Feller theorem [2] tells us that the number of Dyck paths
of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on
$m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In
this paper, we find the connections between these two Chung-Feller theorems. We
focus on the weighted versions of three classes of lattice paths and give the
generalizations of the above two theorems. We prove the Chung-Feller theorems
of Dyck type for these three classes of lattice paths and the Chung-Feller
theorems of Motzkin type for two of these three classes. From the obtained
results, we find an interesting fact that many lattice paths have the
Chung-Feller properties of both Dyck type and Motzkin type.
