Suppose that $M$ is a hyperbolic 3-manifold which admits two Dehn fillings $M(r_1)$ and $M(r_2)$ such that $M(r_1)$ contains an essential annulus, and $M(r_2)$ contains an essential torus. It is known that $\Delta = \Delta (r_1, r_2) \leq 5.$ We will show that if $\Delta = 5$ then $M$ is the Whitehead sister link exterior, and if $\Delta = 4$ then $M$ is the exterior of either the Whitehead link or the 2-bridge link associated to the rational number $\frac{3}{10}$. There are infinitely many examples with $\Delta = 3$.