# Outside the Nonagon

Weekly Problem 44 - 2010

Extend two of the sides of a nonagon to form an angle. How large is this acute angle?

Extend two of the sides of a nonagon to form an angle. How large is this acute angle?

## Problem

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?

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If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

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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 $60^{\circ}$ ]

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