Step 1: Find the interior angle of the triangle adjacent to the $132^\circ$ exterior angle. Angles on a straight line sum to $180^\circ$. Let the interior angle be $A$. $$A = 180^\circ - 132^\circ = 48^\circ$$ Step 2: Use the exterior angle theorem. The exterior angle of a triangle is equal to the sum of the two opposite interior angles. In this case, angle $x$ is an exterior angle, and the two opposite interior angles are $A$ and $70^\circ$. $$x = A + 70^\circ$$ Substitute the value of $A$: $$x = 48^\circ + 70^\circ$$ $$x = 118^\circ$$ The size of angle $x$ is $\boxed{\text{118}^\circ}$