  In this paper, we consider a Lie group $G$ equipped with two left-invariant
Riemannian metrics $g^1$ and $g^2$. Using these two left-invariant Riemannian
metrics we define a left-invariant Riemannian metric $\tilde{g}$ on the tangent
Lie group $TG$. The Levi-Civita connection, tensor curvature, and sectional
curvature of $(TG,\tilde{g})$ in terms of $g^1$ and $g^2$ are given. Also, we
give a sufficient condition for $\tilde{g}$ to be bi-invariant. Finally,
motivated by the recent work of D. N. Pham, using symplectic forms $\omega _1$
and $\omega _2$ on $G$ we define a symplectic form $\tilde{\omega}$ on $TG$.
