Question

What is the measure of one interior angle of a regular nonagon?

Answer 1

Each interior angles of a regular nonagon measures `140^o`

Explanation:

Sum of all interior angles of any convex `N`-sided polygon equals to `(N-2)180^o`

The proof of this is simple. Pick an initial point `O` inside a polygon, connect it with all vertices, forming `N` triangles.
Sum of all angles of these triangles is `N*180^o`.
To get a sum of all interior angles we should subtract a sum of all angles lying around that initial point `O`, that is we have to subtract `360^o`.
The result for a sum of all interior angles for `N`-sided convex polygon is
`N*180^o - 360^o = (N-2)*180^o`

In `N`-sided regular polygon all `N` angles are equal, so each is equal to
`((N-2)/N)*180^o`

For `N=9` this gives a measure of one interior angle of a regular 9-sided polygon:
`((9-2)/9)*180^o = 140^o`