At the simplex method, if the initial solution is degenerate , then we might have had $x_{10}=0, x_{1j}>0$, thus $\theta_0=0$ and therefore $x_1=x_0$ and $z_1=z_0$, i.e. that the solution couldn't be improved.
In such a case, we turn the degenerate solution to non-degenerate, replacing $0$ of the basic variable by $\epsilon>0$, arbitrarily small and we continue normally till we find the optimal solution, and then we set again $\epsilon=0$.