The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by
E
1
,
1
,
g
λ
1
,
λ
2
, where
λ
1
≥
λ
2
>
0
. It provides a natural
2
-parametric deformation family of the Riemannian homogeneous manifold
Sol
3
=
E
1
,
1
,
g
1
,
1
which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean
C
2
-smooth surface in
E
1
,
1
,
g
L
λ
1
,
λ
2
away from characteristic points and signed geodesic curvature for the Euclidean
C
2
-smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.
A note on constant geodesic curvature curves on surfaces