Understanding Internal and External Angles
The internal and external angles of a polygon are fundamental concepts in geometry. To clarify their relationship, let’s delve into their definitions and properties.
Internal Angles
- The internal angle is formed between two adjacent sides of a polygon.
- For a polygon with n sides, the sum of internal angles can be calculated using the formula: (n-2) × 180 degrees.
- For example, in a triangle (3 sides), the internal angles sum up to 180 degrees.
External Angles
- An external angle is formed by one side of the polygon and the extension of an adjacent side.
- The sum of all external angles of any polygon, regardless of the number of sides, is always 360 degrees.
- Each external angle is supplementary to its corresponding internal angle (they add up to 180 degrees).
Relationship Between Internal and External Angles
- While the sum of internal angles increases with the number of sides, the sum of external angles remains constant at 360 degrees.
- For each vertex, the internal angle plus the external angle equals 180 degrees, indicating they are related but distinct.
Conclusion
- The sum of internal angles varies based on the polygon's number of sides, while the sum of external angles is always 360 degrees.
- Therefore, the sum of external angles and the sum of internal angles are not the same; they serve different purposes in polygon geometry.
Understanding these concepts is crucial for solving problems in geometry, especially in competitive exams like UPSC.