Polygons

In the figure below, a number of congruent (identical) regular \(k\)-gons are placed in a ring to enclose a regular decagon, of which \(ABCDE\) is a part.

1. Find the size of angle \(ABC\) and the value of \(k\).
2. If a number of congruent regular \(n\)-gons are placed in a ring to enclose a regular \(m\)-gon, find the relationship between \(m\) and \(n\) by expressing \(m\) in terms of \(n\).
3. Hence, or otherwise, explain whether a nonagon can be enclosed by placing a number of congruent regular polygons in a ring.

In (b) and (c), use the same contact pattern as the diagram: exactly two outer regular polygons and the enclosed polygon meet at each vertex, without overlap or gaps. The side counts \(m,n\) are integers at least 3.