1. **State the problem:** We need to find the size of angle $q$ at the top apex of a concave polygon. 2. **Recall the polygon angle sum rule:** The sum of the interior angles of an $n$-sided polygon is given by $$\text{Sum of interior angles} = (n-2) \times 180^\circ$$ 3. **Identify the number of sides:** The polygon has 4 angles: $q$, $47^\circ$, $235^\circ$, and $24^\circ$, so it is a quadrilateral ($n=4$). 4. **Calculate the sum of interior angles:** $$\text{Sum} = (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ$$ 5. **Set up the equation for $q$:** $$q + 47^\circ + 235^\circ + 24^\circ = 360^\circ$$ 6. **Sum the known angles:** $$47^\circ + 235^\circ + 24^\circ = 306^\circ$$ 7. **Solve for $q$:** $$q = 360^\circ - 306^\circ = 54^\circ$$ **Final answer:** $$\boxed{54^\circ}$$