To solve the problem of finding the number of sides of a regular polygon where each interior angle doubles its corresponding exterior angle, we need to understand the relationship between interior and exterior angles in polygons. ### Step-by-Step Explanation 1. **Understanding Interior and Exterior Angles**: - In any polygon, the sum of the exterior angles is always \(360^\circ\). - The measure of each exterior angle of a regular polygon (where all sides and angles are equal) can be calculated using the formula: \[ \text{Exterior Angle} = \frac{360^\circ}{n} \] where \(n\) is the number of sides of the polygon. 2. **Relationship Between Interior and Exterior Angles**: - The interior angle of a polygon is related to its exterior angle by the formula: \[ \text{Interior Angle} = 180^\circ - \text{Exterior Angle} \] - Given the problem states that each interior angle doubles its corresponding exterior angle, we can express this relationship mathematically: \[ \text{Interior Angle} = 2 \times \text{Exterior Angle} \] 3. **Setting Up the Equation**: - Substituting the expression for the interior angle into the equation gives: \[ 180^\circ - \text{Exterior Angle} = 2 \times \text{Exterior Angle} \] - Let’s denote the exterior angle as \(E\): \[ 180^\circ - E = 2E \] - Rearranging this equation: \[ 180^\circ = 3E \] - Solving for \(E\): \[ E = \frac{180^\circ}{3} = 60^\circ \] 4. **Finding the Number of Sides**: - Now that we have the exterior angle, we can find the number of sides \(n\) using the exterior angle formula: \[ E = \frac{360^\circ}{n} \] - Substituting \(E = 60^\circ\) into the equation: \[ 60^\circ = \frac{360^\circ}{n} \] - Rearranging to solve for \(n\): \[ n = \frac{360^\circ}{60^\circ} = 6 \] ### Conclusion The number of sides of the polygon is **6**. ### Explanation of Other Options - **Option A (8)**: If the polygon had 8 sides, the exterior angle would be \( \frac{360^\circ}{8} = 45^\circ\). The corresponding interior angle would be \(180^\circ - 45^\circ = 135^\circ\), which does not satisfy the condition of the interior angle being double the exterior angle. - **Option C (4)**: For a polygon with 4 sides (a square), the exterior angle is \( \frac{360^\circ}{4} = 90^\circ\). The interior angle would be \(180^\circ - 90^\circ = 90^\circ\), which again does not satisfy the condition. - **Option D (3)**: For a triangle, the exterior angle is \( \frac{360^\circ}{3} = 120^\circ\). The interior angle would be \(180^\circ - 120^\circ = 60^\circ\), which does not satisfy the condition either. ### Revision Summary - The relationship between interior and exterior angles is crucial in polygon problems. - The exterior angle can be calculated as \( \frac{360^\circ}{n} \). - The interior angle is \( 180^\circ - \text{Exterior Angle} \). - For this problem, the correct number of sides of the polygon is **6**.