Suppose Hexagon is regular. Each angle of a regular polygon of sides has measure

A hexagon has 6 sides, so set ; each angle of the regular hexagon has measure

Since one angle is given to be of measure , the hexagon cannot be regular.

A regular polygon with sides has congruent angles, each of which measures

Setting , the common angle measure can be calculated to be

The statement is therefore true.

The area has no relevance to find the angle of a regular hexagon.

There are 6 sides in a regular hexagon. Use the following formula to determine the interior angle.

Substitute sides to determine the sum of all interior angles of the hexagon in degrees.

Since there are 6 sides, divide this number by 6 to determine the value of each interior angle.

Given that the hexagon is a regular hexagon, this means that all the side length are congruent and all internal angles are congruent. The question requires us to solve for the measure of an internal angle. Given the aforementioned definition of a regular polygon, this means that there must only be one correct answer.

In order to solve for the answer, the question provides additional information that isn't necessarily required. The measure of an internal angle can be solved for using the equation:

where is the number of sides of the polygon.

In this case, .

For this problem, the information about the side length may be negated.

Below is regular Hexagon with center , a segment drawn from to each vertex - that is, each of its radii drawn.

Each angle of a regular hexagon measures ; by symmetry, each radius bisects an angle of the hexagon, so

.

The angles of a rectangle must measure , so it has been disproved that Quadrilateral is a rectangle.

If one exterior angle is taken at each vertex of any polygon, and their measures are added, the sum is . Each exterior angle of a regular hexagon has the same measure, so if we let be that common measure, then

Solve for :

The statement is false.

The sum of the exterior angles of any polygon, one at each vertex, is . In a regular polygon, the exterior angles all have the same measure, so divide 360 by the number of angles, which, here, is 20, the same as the number of sides.

The sum of the interior angle measures of a hexagon is

Add the angle measures in each group.

In each case, the angle measures add up to 720, so the answer is that all of these can be the six interior angle measures of a hexagon.