1. **State the problem:** We have an isosceles triangle with an exterior angle of 128° adjacent to one vertex, and we need to find the interior angle $z$ opposite this exterior angle. 2. **Recall the exterior angle theorem:** The exterior angle of a triangle is equal to the sum of the two opposite interior angles. 3. **Identify the angles:** Since the triangle is isosceles, the two base angles are equal. Let each base angle be $z$ degrees. 4. **Use the exterior angle theorem:** The exterior angle 128° equals the sum of the two opposite interior angles, which are both $z$. $$128 = z + z = 2z$$ 5. **Solve for $z$:** $$2z = 128$$ $$z = \frac{128}{2} = 64$$ 6. **Conclusion:** The value of $z$ is 64 degrees.