Find the measure of each interior and exterior angle of a regular hexagon and a regular octagon.

    The formula for each interior angle of a regular polygon is $\frac{(n-2) \times 180^\circ}{n}$.
    $$
    \text{Interior angle} = \frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ
    $$

    The formula for each exterior angle of a regular polygon is $\frac{360^\circ}{n}$.
    $$
    \text{Exterior angle} = \frac{360^\circ}{6} = 60^\circ
    $$
    Alternatively, the exterior angle is $180^\circ - \text{interior angle}$.
    $$
    \text{Exterior angle} = 180^\circ - 120^\circ = 60^\circ
    $$
    **Answer:** Each interior angle is $\boxed{120^\circ}$ and each exterior angle is $\boxed{60^\circ}$.

*   **ii) a regular octagon**
    A regular octagon has $n=8$ sides.

    $$
    \text{Interior angle} = \frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ
    $$

    $$
    \text{Exterior angle} = \frac{360^\circ}{8} = 45^\circ
    $$
    Alternatively, the exterior angle is $180^\circ - \text{interior angle}$.
    $$
    \text{Exterior angle} = 180^\circ - 135^\circ = 45^\circ
    $$
    **Answer:** Each interior angle is $\boxed{135^\circ}$ and each exterior angle is $\boxed{45^\circ}$.

The sum of an interior angle and its corresponding exterior angle is $180^\circ$.
$$
\text{Exterior angle} = 180^\circ - \text{Interior angle} = 180^\circ - 135^\circ = 45^\circ
$$

The formula for each exterior angle of a regular polygon is $\frac{360^\circ}{n}$.
$$
45^\circ = \frac{360^\circ}{n}
$$
$$
n = \frac{360^\circ}{45^\circ} = 8
$$

A polygon with 8 sides is an octagon.  
**Answer:** The name of this polygon is a regular $\boxed{\text{octagon}}$.

We are given $I = 5E$.  
We also know that $I + E = 180^\circ$.

Substitute $I = 5E$ into the second equation:
$$
5E + E = 180^\circ
$$
$$
6E = 180^\circ
$$
$$
E = \frac{180^\circ}{6} = 30^\circ
$$

The formula for each exterior angle of a regular polygon is $E = \frac{360^\circ}{n}$.
$$
30^\circ = \frac{360^\circ}{n}
$$
$$
n = \frac{360^\circ}{30^\circ} = 12
$$
**Answer:** The number of sides of this polygon is $\boxed{12}$.

From question 1 i), the interior angle of a regular hexagon is $120^\circ$.

From question 1 ii), the interior angle of a regular octagon is $135^\circ$.

The angles around a point sum to $360^\circ$. The angle $x$ along with the interior angles of the hexagon and octagon form a complete circle at their common vertex.
$$
120^\circ + 135^\circ + x = 360^\circ
$$
$$
255^\circ + x = 360^\circ
$$
$$
x = 360^\circ - 255^\circ
$$
$$
x = 105^\circ
$$
**Answer:** The size of the angle marked $x$ is $\boxed{105^\circ}$.