1. **State the problem:** We need to find the size of angle $n$ in a pentagon where the other four interior angles are given as $58^\circ$, $29^\circ$, $73^\circ$, $71^\circ$, and $62^\circ$. 2. **Formula used:** The sum of interior angles of a polygon with $n$ sides is given by: $$\text{Sum of interior angles} = (n-2) \times 180^\circ$$ For a pentagon, $n=5$, so: $$\text{Sum} = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$$ 3. **Apply the formula:** The sum of all interior angles in the pentagon is $540^\circ$. We know five angles including $n$: $$58^\circ + 29^\circ + 73^\circ + 71^\circ + 62^\circ + n = 540^\circ$$ 4. **Calculate the sum of known angles:** $$58 + 29 + 73 + 71 + 62 = 293^\circ$$ 5. **Find angle $n$:** $$n = 540^\circ - 293^\circ = 247^\circ$$ 6. **Interpretation:** Angle $n$ measures $247^\circ$, which is possible if the pentagon is concave (since one angle is greater than $180^\circ$). **Final answer:** $$\boxed{247^\circ}$$