Understanding Regular Polygon Properties

The problem asks us to find the difference between the number of sides of two regular polygons, $P_1$ and $P_2$, based on information about their interior and exterior angles. We need to calculate the number of sides for each polygon first.

Calculating Sides of Polygon $P_1$

We are given that the sum of all the interior angles of the regular polygon $P_1$ is $1800^\circ$. The formula for the sum of the interior angles of a polygon with $n$ sides is:

Sum of Interior Angles = $ (n - 2) \times 180^\circ $

Let $n_1$ be the number of sides of polygon $P_1$. We can set up the equation:

$ (n_1 - 2) \times 180^\circ = 1800^\circ $

To find $n_1$, we can solve this equation:

1. Divide both sides by $180^\circ$: $ n_1 - 2 = \frac{1800^\circ}{180^\circ} $ $ n_1 - 2 = 10 $
2. Add 2 to both sides: $ n_1 = 10 + 2 $ $ n_1 = 12 $

So, the regular polygon $P_1$ has 12 sides.

Calculating Sides of Polygon $P_2$

For the regular polygon $P_2$, we are told that each interior angle exceeds its exterior angle by $120^\circ$. Let $i_2$ represent the measure of each interior angle and $e_2$ represent the measure of each exterior angle of $P_2$.

We have two key relationships for any regular polygon:

• The interior angle exceeds the exterior angle by $120^\circ$: $ i_2 = e_2 + 120^\circ $
• The sum of an interior angle and its corresponding exterior angle is $180^\circ$: $ i_2 + e_2 = 180^\circ $

Now, we can substitute the first equation into the second equation:

$ (e_2 + 120^\circ) + e_2 = 180^\circ $

Let's solve for $e_2$:

1. Combine like terms: $ 2e_2 + 120^\circ = 180^\circ $
2. Subtract $120^\circ$ from both sides: $ 2e_2 = 180^\circ - 120^\circ $ $ 2e_2 = 60^\circ $
3. Divide by 2: $ e_2 = \frac{60^\circ}{2} $ $ e_2 = 30^\circ $

The measure of each exterior angle of $P_2$ is $30^\circ$. The number of sides of a regular polygon can be found using the formula:

Number of sides ($n_2$) = $ \frac{360^\circ}{\text{Exterior Angle}} $

So, for polygon $P_2$:

$ n_2 = \frac{360^\circ}{30^\circ} $ $ n_2 = 12 $

Therefore, the regular polygon $P_2$ also has 12 sides.

Finding the Difference Between the Number of Sides

We found that $P_1$ has $n_1 = 12$ sides and $P_2$ has $n_2 = 12$ sides. The problem asks for the difference between the number of sides of $P_1$ and $P_2$.

Difference = $ n_1 - n_2 $

Difference = $ 12 - 12 $

Difference = $ 0 $

The difference between the number of sides of $P_1$ and $P_2$ is 0.