1. **Problem statement:** We have two identical, overlapping regular nonagons (9-sided polygons). We need to calculate the size of the angle $f$ formed at the top vertex of the kite-shaped overlapping region. 2. **Key properties of a regular nonagon:** - Number of sides $n = 9$ - Each interior angle of a regular polygon is given by the formula: $$\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$$ 3. **Calculate the interior angle of the nonagon:** $$\text{Interior angle} = \frac{(9-2) \times 180^\circ}{9} = \frac{7 \times 180^\circ}{9} = 140^\circ$$ 4. **Calculate the central angle of the nonagon:** The central angle (angle subtended at the center by one side) is: $$\text{Central angle} = \frac{360^\circ}{9} = 40^\circ$$ 5. **Understanding the kite-shaped overlapping region:** The kite is formed by the intersection of two identical nonagons rotated relative to each other. The angle $f$ at the top vertex of the kite corresponds to the angle between two adjacent edges from the two polygons. 6. **Calculate angle $f$:** Since the polygons are identical and overlapping, the angle $f$ is half the central angle of the nonagon: $$f = \frac{40^\circ}{2} = 20^\circ$$ **Final answer:** $$\boxed{20^\circ}$$