  Through the measurements of magnetic field dependence of specific heat in
$La_{2-x}Sr_xCuO_4$ in zero temperature limit, we determined the nodal slope
$v_\Delta$ of the quasiparticle gap. It is found that $v_\Delta$ has a very
similar doping dependence of the pseudogap temperature $T^*$ or value
$\Delta_p$. Meanwhile the virtual maximum gap at ($\pi,0$) derived from
$v_\Delta$ is found to follow the simple relation $\Delta_q=0.46k_BT^*$ upon
changing the doping concentration. This strongly suggests a close relationship
between the pseudogap and superconductivity. It is further found that the
superconducting transition temperature is determined by both the residual
density of states of the pseudogap phase and the nodal gap slope in the zero
temperature limit, namely, $T_c \approx \beta v_\Delta \gamma_n(0)$, where
$\gamma_n(0)$ is the extracted zero temperature value of the normal state
specific heat coefficient which is proportional to the size of the residual
Fermi arc $k_{arc}$. This manifests that the superconductivity may be formed by
forming a new gap on the Fermi arcs near nodes below $T_c$. These observations
mimic the key predictions of the SU(2) slave boson theory based on the general
resonating-valence-bond (RVB) picture.
