A regular polygon is a polygon that has the following properties:

• All sides have the same length (i.e., congruent).

• All angles have the same measure (i.e., congruent).

With that in mind, let's look at each polygon in turn:

• Polygon I is not regular. Its sides and angles are not all congruent.

• Polygon II is regular. Its sides and angles are all congruent.

• Polygon III is not regular. Its sides and angles are not all congruent.

Therefore the answer is "II only."

A regular polygon is a polygon that has the following properties:

• All sides have the same length (i.e., congruent).

• All angles have the same measure (i.e., congruent).

With that in mind, let's look at each polygon in turn:

• Polygon I is not regular. Its sides are not all congruent.

• Polygon II is regular. Its sides and angles are all congruent.

• Polygon III is regular. Its sides and angles are all congruent.

Therefore the answer is "I only."

A regular hexagon is a $6$-sided polygon where all the interior angles have the same measure.

The sum of the measures of its interior angles is

\[ 180^\circ \times (6 - 2) = 720^\circ. \]

Therefore, the measure of each interior angle is

\[ \dfrac{720^\circ}{6} = 120^\circ. \]

A regular octagon is an $8$-sided polygon where all the interior angles have the same measure.

The sum of the measures of its interior angles is

\[ 180^\circ \times (8 - 2) = 1\,080^\circ. \]

Therefore, the measure of $\angle HAB$ is

\[ m\angle HAB = \dfrac{1\,080^\circ}{8} = 135^\circ. \]

The angles $\angle BAP, $ $\angle HAB,$ and $\angle QAH$ form a straight angle. Therefore, \begin{align*} m\angle BAP + m\angle HAB + m\angle QAH & = 180^\circ\\[3pt] y + m \angle HAB + y & = 180^\circ\\[3pt] 2y + 135^\circ & = 180^\circ\\[3pt] 2y & = 45^\circ\\[3pt] y & = 22.5^\circ. \end{align*}

For any polygon, the exterior angles sum to $360^\circ.$ In a regular hexagon, there are $6$ equivalent exterior angles, so each has a measure of \[ \dfrac{360^\circ}{6} = 60^\circ . \]

The sum of the exterior angles of any polygon is $360^\circ.$

For a regular polygon, all exterior angles have the same measure.

Therefore, the number of exterior angles is \[ \dfrac{360^\circ}{45^\circ} = 8\,. \]

The number of sides, then, is also $8.$

The shaded angle is the sum of the measures of exterior angles of a regular octagon and a square.

For any polygon, the exterior angles sum to $360^\circ.$ In a regular octagon, there are $8$ equivalent exterior angles, so each has a measure of \[ \dfrac{360^\circ}{8} = 45^\circ . \]

Likewise, in a square, there are $4$ equivalent exterior angles, so each has a measure of \[ \dfrac{360^\circ}{4} = 90^\circ . \]

Therefore, the given angle has the following measure:

\[ 45^\circ + 90^\circ = 135^\circ \]

Note, that the full angle is the sum of the measure of interior angles of an equilateral triangle, a regular hexagon, and a regular octagon and the measure of the shaded angle.

For a regular polygon with $n$ sides, the measure of each interior angle must be \[ \dfrac{180^\circ \cdot (n-2)}{n}. \]

In an equilateral triangle, there are $3$ equivalent interior angles, so each has a measure of \[ \dfrac{180^\circ (3-2)}{3} = 60^\circ . \]

Likewise, in a regular hexagon, there are $6$ equivalent interior angles, so each has a measure of \[ \dfrac{180^\circ (6-2)}{6} = 120^\circ . \]

A regular octagon has $8$ equivalent interior angles, so each has a measure of \[ \dfrac{180^\circ (8-2)}{8} = 135^\circ . \]

Let us call $x$ the measure of the shaded angle. So, we have

\begin{align*} 60^\circ + 120^\circ+ 135^\circ + x &= 360^\circ\\ 315^\circ + x &= 360^\circ\\ x &= 45^\circ. \end{align*}