Is this statement true or false? In a regular polygon, two consecutive radii and their included side form an equilateral triangle.

To determine whether the statement is true or false, let's analyze the properties of a regular polygon. 1. **Definition of a Regular Polygon**: A regular polygon is a polygon with all sides and angles equal. 2. **Radii of a Regular Polygon**: The radii of a regular polygon are the line segments from the center of the polygon to each vertex. 3. **Consecutive Radii**: When we take two consecutive radii, they connect the center of the polygon to two adjacent vertices. 4. **Included Side**: The side of the polygon that connects these two vertices is the side of the polygon itself. 5. **Forming a Triangle**: The two consecutive radii and the included side form a triangle. Now, let's consider the angles in this triangle: - The angle at the center of the polygon (formed by the two radii) is equal to the central angle, which is given by \( \frac{360^\circ}{n} \), where \( n \) is the number of sides (or vertices) of the polygon. - The lengths of the two radii are equal (since they are both radii of the same circle). - The side of the polygon is not necessarily equal to the lengths of the radii unless the polygon is specifically an equilateral triangle. For a regular polygon with more than three sides (e.g., a square, pentagon, hexagon, etc.), the included side will not be equal to the lengths of the radii, and the angles will not be \( 60^\circ \) (which is required for an equilateral triangle). Thus, the statement is **false**. In a regular polygon, two consecutive radii and their included side do not generally form an equilateral triangle.