Outside the Nonagon
Extend two of the sides of a nonagon to form an angle. How large is this acute angle?
The diagram shows a regular 9-sided polygon (a nonagon or an enneagon) with two of the sides extended to meet at the point X.
What is the size of the acute angle at X?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
The exterior angles of a regular nonagon are $360^{\circ}\div 9 = 40^{\circ}$, whence the interior angles are $180^{\circ} - 40^{\circ}= 140 ^{\circ}$.
In the arrowhead quadrilateral whose rightmost vertex is X, three of the angles are $40^{\circ}$, $40^{\circ}$ and $360^{\circ} - 140^{\circ}=220^{\circ}$ and these add up to $300^{\circ}$.
So the angle at X is $60^{\circ}$.
[It is now posible to see that the entire nonagon can fit neatly inside an equilateral triangle and so the angle X is $60^{\circ}$ ]
