1. **State the problem:** We need to find the value of angle $z$ in a geometric sketch where three angles are given: $72^\circ$, $39^\circ$, and $105^\circ$. These angles are part of triangles formed by the man's arms and legs. 2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always $180^\circ$. 3. **Identify the triangle containing $z$:** Since $z$ is an unknown angle in the sketch, it is likely part of a triangle with the given angles. We can use the sum of angles to find $z$. 4. **Calculate $z$:** Assuming $z$ is in the triangle with angles $72^\circ$ and $39^\circ$, we have: $$z + 72 + 39 = 180$$ 5. **Simplify the equation:** $$z + 111 = 180$$ 6. **Solve for $z$:** $$z = 180 - 111$$ $$z = 69$$ 7. **Conclusion:** The value of $z$ is $69$ degrees.