Multiple choice

$AB$, $BC$ and $CD$ are the three consecutive sides of a regular polygon. If $\angle BAC\, =\, 15^{\circ}$, find the measure of each exterior angle of the polygon.

  1. $20^{\circ}$
  2. $30^{\circ}$
  3. $40^{\circ}$
  4. $70^{\circ}$
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B Correct answer
Explanation

In a regular polygon, the triangle formed by the center and two adjacent vertices is isosceles. The angle at the center is 360/n. The base angles are (180 - 360/n)/2 = 90 - 180/n. Given the angle BAC is 15, and using properties of regular polygons, n=12. Exterior angle = 360/12 = 30 degrees.

AI explanation

Let the regular polygon have n sides. Triangle ABC is isosceles with AB equal to BC, so the base angles are 15 degrees and the vertex angle ABC is 180 minus 15 minus 15, which is 150 degrees. The interior angle of a regular polygon equals 150 degrees. The exterior angle is 180 minus the interior angle, giving 30 degrees.

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