The figure is made up of a regular pentagon ABCDE and a regular hexagon CDFGHI.

1. Identify the two known interior angles at vertex C

At $C$, the interior angle of the regular pentagon is
$\angle BCD=108^\circ.$
The interior angle of the regular hexagon is
$\angle DCI=120^\circ.$

2. Use angles at a point to find angle BCI

Angles around point $C$ add up to $360^\circ$.

Thus,
$\angle BCI=360^\circ-108^\circ-120^\circ=132^\circ.$

1. Relate angle BCI to the interior angle of the possible regular polygon

From part (a),
$\angle BCI=132^\circ.$
If $BC$ and $CI$ are two adjacent sides of another regular polygon, then $132^\circ$ would be its interior angle.

2. Check whether the angle gives a whole number of sides

Exterior angle of the possible regular polygon:
$180^\circ-132^\circ=48^\circ.$
For a regular polygon,
$\text{number of sides}=\frac{360^\circ}{\text{exterior angle}}=\frac{360}{48}=7.5.$
Since $7.5$ is not a whole number, such a regular polygon cannot exist.