Question 3234
Begin by constructing the nonagon. To do this we pick a center point from which we draw nine radiating lines. Each of the radiating lines must be straight and of equal length. Also, the angle between any pair of adjacent radiating lines must be equal to the angle between every other pair of adjacent radiating lines. This will form a star-like pattern. The sum of the angles formed by adjacent lines is 360 degrees, so each of these angles must be {{{360/9 = 40}}} degrees<br>
Pick two adjacent radiating lines and call them l<sub>1</sub> and l<sub>2</sub>. The endpoints of l<sub>1</sub> are C (the center) and E<sub>1</sub> and the endpoints of l<sub>2</sub> are C (the center) and E<sub>2</sub>. If we draw a line from E<sub>1</sub> to E<sub>2</sub>, we form a triangle CE<sub>1</sub>E<sub>2</sub>. If we repeat this process for every pair of adjacent lines, the lines E<sub>1</sub>E<sub>2</sub>, E<sub>2</sub>E<sub>3</sub>, E<sub>3</sub>E<sub>4</sub>, E<sub>4</sub>E<sub>5</sub>, E<sub>5</sub>E<sub>6</sub>, E<sub>6</sub>E<sub>7</sub>, E<sub>7</sub>E<sub>8</sub>, E<sub>8</sub>E<sub>9</sub> and E<sub>9</sub>E<sub>1</sub>form a regular nonagon. Moreover, the nonagon is formed from similar isoceles triangles.<br>
We know that the angle E<sub>i</sub>CE<sub>j</sub> = 40 degrees, where i and j are the end points of two adjacent radiating lines. Given that the sum of the interior angles of a triangle = 180 and the angle CE<sub>i</sub>E<sub>j</sub> equals the angle CE<sub>j</sub>E<sub>i</sub> we conclude that  CE<sub>i</sub>E<sub>j</sub> = CE<sub>j</sub>E<sub>i</sub> = {{{(1/2)(180-40) = 70}}}.<br>
Now, consider the point E<sub>1</sub>. This is a vertex on the nonagon, the triangle CE<sub>1</sub>E<sub>2</sub> and the triangle CE<sub>9</sub>E<sub>1</sub>. The interior angle of the nonagon at this point is E<sub>9</sub>E<sub>1</sub>E<sub>2</sub>. This angle is formed by adding the angles CE<sub>1</sub>E<sub>2</sub> and CE<sub>1</sub>E<sub>9</sub>. We just deteremined that each of these angles is 70 degrees, so the interior angle formed by E<sub>9</sub>E<sub>1</sub>E<sub>2</sub> is 140 degrees.<br>
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A exterior angle is the supplementary angle to the interior angle. That is, the sum of the exterior and interior angle is 180 degrees. To find the exterior angle, we compute 180 less the interior angle or {{{180 - 140 = 40}}} degrees.