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

 May 31, 2023
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Since B, A, and D are consecutive vertices of a regular n-gon, we know that angle BAD is equal to (n-2)/n × 180 degrees. Since the n-gon is regular, we know that angle ABD is equal to 180 degrees / n, since there are n total angles in the n-gon that are equal in measure.

Therefore, angle BAC is equal to angle BAD + angle ABD, which is:

(n-2)/n × 180 + 180/n

Simplifying, we get:

(2n - 2)/n × 90

Since the heptagon is regular, angle BAC is also equal to 180 degrees / 7, since there are 7 total angles in the heptagon that are equal in measure.

Setting these two expressions equal to each other, we get:

(2n - 2)/n × 90 = 180/7

Multiplying both sides by 7n and simplifying, we get:

14n - 14 = 180n/7

Multiplying both sides by 7 and simplifying, we get:

98n - 98 = 180n

Solving for n, we get:

n = 14

Therefore, the given n-gon is a regular 14-gon. Since angle BCD is an exterior angle of the heptagon, it is equal to the sum of the remote interior angles,which are angles ACB and BAC. 

Angle ACB is an interior angle of the regular heptagon, so it is equal to (7-2)/7 × 180 degrees, which simplifies to 128.57 degrees.

Angle BAC was already calculated above to be (2n - 2)/n × 90 degrees, which simplifies to 140 degrees.

Therefore, angle BCD is equal to:

angle ACB + angle BAC = 128.57 degrees + 140 degrees = 268.57 degrees

So angle BCD measures 268.57 degrees. 

 Jun 1, 2023

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