The measure of each interior angle of a regular polygon is five times the measure of its exterior angle. Find :
(i) measure of each interior angle,
(ii) measure of each exterior angle and
(iii) number of sides in the polygon

Answer

To solve the problem step by step, we will follow the given information and use the formulas for the interior and exterior angles of a regular polygon.

### Step 1: Understand the relationship between interior and exterior angles
We know from the problem that the measure of each interior angle (I) is five times the measure of its exterior angle (E). This can be expressed as:
\[ I = 5E \]

### Step 2: Write the formulas for interior and exterior angles
For a regular polygon with \( n \) sides:
- The formula for each interior angle is:
\[ I = \frac{(n-2) \times 180}{n} \]
- The formula for each exterior angle is:
\[ E = \frac{360}{n} \]

### Step 3: Substitute the exterior angle formula into the interior angle equation
From the relationship established in Step 1, we can substitute \( E \) into the equation:
\[ I = 5E \]
Substituting the formula for \( E \):
\[ I = 5 \left( \frac{360}{n} \right) \]

### Step 4: Set the two expressions for I equal to each other
Now we can set the two expressions for \( I \) equal to each other:
\[ \frac{(n-2) \times 180}{n} = 5 \left( \frac{360}{n} \right) \]

### Step 5: Simplify the equation
Multiply both sides by \( n \) to eliminate the denominator:
\[ (n-2) \times 180 = 5 \times 360 \]
\[ (n-2) \times 180 = 1800 \]

### Step 6: Solve for n
Now, divide both sides by 180:
\[ n-2 = 10 \]
Adding 2 to both sides gives:
\[ n = 12 \]

### Step 7: Find the measure of each interior angle
Now that we know \( n \), we can find the measure of each interior angle using the formula:
\[ I = \frac{(n-2) \times 180}{n} \]
Substituting \( n = 12 \):
\[ I = \frac{(12-2) \times 180}{12} \]
\[ I = \frac{10 \times 180}{12} \]
\[ I = \frac{1800}{12} = 150 \text{ degrees} \]

### Step 8: Find the measure of each exterior angle
Now, we can find the measure of each exterior angle using the formula:
\[ E = \frac{360}{n} \]
Substituting \( n = 12 \):
\[ E = \frac{360}{12} = 30 \text{ degrees} \]

### Summary of Results
(i) Measure of each interior angle: **150 degrees**
(ii) Measure of each exterior angle: **30 degrees**
(iii) Number of sides in the polygon: **12**