To solve this problem, we need to work with the relationships between the sum of the interior angles, the measure of each interior angle, the measure of each exterior angle, and the number of sides of a polygon.

Step 1: Use the sum of the interior angles formula

The sum of the interior angles of a polygon is given by: $\text{Sum of interior angles} = 180^\circ \times (n - 2)$ Given that the sum of the interior angles is $1620^\circ$, we can set up the equation: $180^\circ \times (n - 2) = 1620^\circ$

Step 2: Solve for $n$

Divide both sides by $180^\circ$: $n - 2 = \frac{1620^\circ}{180^\circ} = 9$ Adding 2 to both sides: $n = 11$

So, the polygon has $11$ sides.

Step 3: Calculate the measure of each interior angle

The measure of each interior angle of a regular polygon is given by: $\text{Each interior angle} = \frac{\text{Sum of interior angles}}{n} = \frac{1620^\circ}{11} = 147.27^\circ$

Step 4: Calculate the measure of each exterior angle

The measure of each exterior angle of a regular polygon is given by: $\text{Each exterior angle} = 180^\circ - \text{Each interior angle} = 180^\circ - 147.27^\circ = 32.73^\circ$

However, since it is stated that the measure of each interior angle, each exterior angle, and the number of sides are all equal to $n$, it must be interpreted as:

There is no single value of $n$ that can satisfy all these conditions exactly. The question likely contains a contradiction or misinterpretation in how the values relate.

Would you like to explore more detailed aspects of this problem or clarify any of these concepts?

Related Questions:

1. What is the relationship between the number of sides of a polygon and the measure of its exterior angles?
2. How can the measure of interior angles help determine the type of polygon?
3. What is the formula to calculate the sum of the exterior angles of any polygon?
4. How does the number of sides affect the regularity of a polygon's angles?
5. Can you find a real-world example where understanding the angles of a polygon is essential?

Tip: The exterior angles of any polygon always sum up to $360^\circ$, regardless of the number of sides.