If the difference between an interior angle of a regular polygon of (n + 1) sides and an interior angle of a regular polygon of n sides is` 4^(@) `, find the value of n. Also, state the difference between their exterior angles.

Answer

To solve the problem step by step, we will find the value of \( n \) such that the difference between the interior angles of a regular polygon with \( n+1 \) sides and a polygon with \( n \) sides is \( 4^\circ \). We will also find the difference between their exterior angles.

### Step 1: Write the formula for the interior angle of a regular polygon
The interior angle \( A \) of a regular polygon with \( n \) sides is given by the formula:
\[ A = \frac{180(n - 2)}{n} \]

### Step 2: Write the interior angles for \( n \) and \( n + 1 \) sides
- For a polygon with \( n \) sides:
\[ A_n = \frac{180(n - 2)}{n} \]
- For a polygon with \( n + 1 \) sides:
\[ A_{n+1} = \frac{180((n + 1) - 2)}{n + 1} = \frac{180(n - 1)}{n + 1} \]

### Step 3: Set up the equation for the difference of the interior angles
According to the problem, the difference between the interior angles is \( 4^\circ \):
\[ A_{n+1} - A_n = 4 \]
Substituting the expressions for \( A_{n+1} \) and \( A_n \):
\[ \frac{180(n - 1)}{n + 1} - \frac{180(n - 2)}{n} = 4 \]

### Step 4: Simplify the equation
Taking \( 180 \) common:
\[ 180 \left( \frac{n - 1}{n + 1} - \frac{n - 2}{n} \right) = 4 \]
Dividing both sides by \( 180 \):
\[ \frac{n - 1}{n + 1} - \frac{n - 2}{n} = \frac{4}{180} = \frac{1}{45} \]

### Step 5: Find a common denominator and simplify
The common denominator for the fractions is \( n(n + 1) \):
\[ \frac{(n - 1)n - (n - 2)(n + 1)}{n(n + 1)} = \frac{1}{45} \]
Expanding the numerator:
\[ (n^2 - n) - (n^2 + n - 2) = n^2 - n - n^2 - n + 2 = -2n + 2 \]
So we have:
\[ \frac{-2n + 2}{n(n + 1)} = \frac{1}{45} \]

### Step 6: Cross-multiply to solve for \( n \)
Cross-multiplying gives:
\[ -2n + 2 = \frac{n(n + 1)}{45} \]
Multiplying both sides by \( 45 \):
\[ -90n + 90 = n(n + 1) \]
Rearranging the equation:
\[ n^2 + 91n - 90 = 0 \]

### Step 7: Factor the quadratic equation
Factoring:
\[ (n - 9)(n + 10) = 0 \]
Thus, \( n = 9 \) or \( n = -10 \). Since \( n \) must be positive, we have:
\[ n = 9 \]

### Step 8: Find the difference in exterior angles
The exterior angle \( E \) of a regular polygon with \( n \) sides is given by:
\[ E = \frac{360}{n} \]
For \( n = 9 \):
\[ E_n = \frac{360}{9} = 40^\circ \]
For \( n + 1 = 10 \):
\[ E_{n+1} = \frac{360}{10} = 36^\circ \]
The difference in exterior angles is:
\[ E_n - E_{n+1} = 40^\circ - 36^\circ = 4^\circ \]

### Final Answer
Thus, the value of \( n \) is \( 9 \), and the difference between their exterior angles is \( 4^\circ \).

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