The total of the degree measures of any ten-sided polygon is

\(\displaystyle \left (10-2 \right )180^{\circ} =1,440^{\circ}\).

In an arithmetic sequence, the terms are separated by a common difference, which we will call \(\displaystyle d\). Since the least of the degree measures is \(\displaystyle 100^{\circ }\), the measures of the angles are

\(\displaystyle 100, 100+d, 100+2d, ...100+9d\)

Their sum is 

\(\displaystyle 100+ \left (100+d \right )+\left (100+2d \right ) ...\left (100+9d \right )=1,440\)

\(\displaystyle 1,000+ 45d =1,440\)

\(\displaystyle 45d = 440\)

\(\displaystyle d = 9\frac{7}{9}\)

The greatest of the angle measures is 

\(\displaystyle 100+9d = 100 + 9\left ( 9\frac{7}{9}\right )= 100+88 = 188\)

However, an angle measure cannot exceed \(\displaystyle 180^{\circ}\). The correct choice is that this polygon cannot exist.

In the diagram below, some other angles have been numbered for the sake of convenience.

An interior angle of a regular heptagon has measure 

\(\displaystyle \frac{(7-2)180^{\circ }}{7}= 128\frac{4}{7}^{\circ }\).

This is the measure of \(\displaystyle \angle 2\).

As a result of the Isosceles Triangle Theorem, \(\displaystyle \angle 3 \cong \angle 4\), so

\(\displaystyle m \angle 3 = \frac{180^{\circ} - m \angle 2}{2} = \frac{180^{\circ} - 128\frac{4}{7}^{\circ }}{2}= 25\frac{5}{7}^{\circ }\).

This is also the measure of \(\displaystyle \angle 5\).

By angle addition, 

\(\displaystyle m\angle 6 = 128\frac{4}{7}^{\circ } - 2 \cdot 25\frac{5}{7}^{\circ } = 77\frac{1}{7}^{\circ }\)

Again, as a result of the Isosceles Triangle Theorem, \(\displaystyle \angle 1 \cong \angle 7\), so

\(\displaystyle m \angle 1 = \frac{180^{\circ} - m \angle 6}{2} = \frac{180^{\circ} - 77\frac{1}{7}^{\circ }^{\circ }}{2}= 51\frac{3}{7}^{\circ }\)

The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides. 

\(\displaystyle t=(n-2)(180)\)

\(\displaystyle t=(10-2)(180)\)

\(\displaystyle t=8\cdot 180\)

\(\displaystyle t=1,440\)

The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides. 

\(\displaystyle t=(5-2)(180)\)

Given that a hexagon has 6 angles, the total number of angles will be:

\(\displaystyle t=(5-2)(180)\)

\(\displaystyle t=3\cdot 180\)

\(\displaystyle t=540\)

To find the value of each angle, we divide 540 by 5. This results in 108 degrees. 

Thus, 

\(\displaystyle x=108\)

\(\displaystyle \frac{1}{2}x=54\)

The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides. 

\(\displaystyle t=(n-2)(180)\)

Given that a hexagon has 6 angles, the total number of angles will be:

\(\displaystyle t=(22-2)(180)\)

\(\displaystyle t=20\cdot 180\)

\(\displaystyle t=3,600\)

Given that there are 3,600 degrees total in a polygon with 22 sides, the number of degrees in each angle can be found by dividing 3,600 by 22. To the nearest degree, this results in 164 degrees. Therefore, 164 is the correct answer.