The sum of the exterior angles of a polygon is:

Question

Accepted Answer

The correct option is 4 (360∘):

The fundamental property of the exterior angles of any convex polygon, regardless of the number of sides (n), is that their sum is always a constant value: 360 degrees. This can be intuitively understood by imagining walking along the perimeter of the polygon and turning at each vertex. By the time you return to your starting point, you will have completed one full revolution, which is 360 degrees. This universal geometric principle holds true for triangles, squares, pentagons, and any other convex polygon.

The incorrect option is 1 (90∘):
The value 90 degrees is the measure of a single angle in a specific geometric shape, such as an angle in a right triangle or one of the interior angles of a square or a rectangle. It is not the sum of all the exterior angles of any polygon, as 90 degrees represents only a quarter of a full rotation. For a polygon to exist, the exterior angles must sum up to a complete circle, which is 360 degrees. Therefore, 90 degrees is far too small to be the sum of all exterior angles for any polygon.

The incorrect option is 2 (180∘):
The sum of the interior angles of a triangle is 180 degrees, which is a key concept in geometry. However, 180 degrees is precisely half of the total required sum for the exterior angles of any convex polygon (360 degrees). An exterior angle is supplementary to its corresponding interior angle, meaning they add up to 180 degrees, but the sum of all exterior angles is double the total angular change represented by a straight line (180 degrees). Hence, 180 degrees is incorrect for the sum of exterior angles.

The incorrect option is 3 (270∘):
The value 270 degrees represents three-quarters of a full rotation. While some individual angles or a combination of angles within various complex figures might sum up to 270 degrees, this value does not represent a standard, universal sum for the exterior angles of a convex polygon. The sum of exterior angles must equate to the complete rotation required to traverse the entire perimeter and return to the original orientation, which is strictly 360 degrees. Therefore, 270 degrees is an arbitrary value that does not align with this fundamental geometric rule.