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Let $B,$ $A,$ and $D$ be three consecutive vertices of a regular $20$-gon. A regular heptagon is constructed on $\overline{AB},$ with a vertex $C$ next to $A.$ Find $\angle BCD,$ in degrees.

 May 23, 2023
 #1
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The sum of the interior angles of a regular 20-gon is (20-2)*180 = 3600 degrees. Each angle of a regular 20-gon is 3600/20 = 180 degrees.

The sum of the interior angles of a regular heptagon is (7-2)*180 = 900 degrees. Each angle of a regular heptagon is 900/7 degrees.

Angle BCD is the angle between two consecutive sides of a regular 20-gon, so it is 180 degrees. Angle ACB is an interior angle of a regular heptagon, so it is 900/7 degrees. Therefore, angle BCD is 180 - 900/7 = 360/7 degrees.

 May 23, 2023
 #2
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which is ~51.4 degrees

 May 23, 2023

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