External Angles in Polygons

External Angles

• Definition: An external angle is formed outside of a polygon when one side is extended.

• Properties:

  • External angles are supplementary to the internal angles at that vertex.
  • The sum of the external angles of any polygon, one at each vertex, is always 360 degrees.

• Calculation:

  • For a polygon with ( n ) sides:
    • Measure of each external angle = ( \frac{360}{n} ) degrees (for regular polygons).

• Relation to Internal Angles:

  • If an internal angle is ( x ) degrees, the corresponding external angle is ( 180 - x ) degrees.

• Types of External Angles:

  • Regular Polygon: All external angles are equal.
  • Irregular Polygon: External angles vary based on the shape.

• Applications:

  • Useful in geometry for solving problems related to polygons.
  • Helps in determining properties of geometric figures and understanding their behavior in transformations.

• Illustration:

  • Visualize a triangle: Extend one side; the angle formed outside the triangle is the external angle at that vertex.

• Key Formulae:

  • Sum of external angles = 360 degrees
  • Each external angle (regular polygon) = ( \frac{360}{n} )

Keep these concepts in mind as you study external angles and their significance in geometry.

Definition of External Angles

• External angle is created when a side of a polygon is extended outward.

Properties of External Angles

• External angles are supplementary to their corresponding internal angles at the same vertex.
• The total of the external angles around a polygon, when one is taken at each vertex, equals 360 degrees.

Calculation of External Angles

• For a polygon with ( n ) sides, the measure of each external angle in a regular polygon can be found using the formula ( \frac{360}{n} ) degrees.

Relation to Internal Angles

• An internal angle measuring ( x ) degrees leads to a corresponding external angle of ( 180 - x ) degrees.

Types of External Angles

• In a regular polygon, all external angles are identical in measure.
• In irregular polygons, external angles differ depending on the specific shape.

Applications of External Angles

• Essential for solving geometric problems related to polygons.
• Aids in identifying geometric properties and understanding transformations of figures.

Illustration of External Angles

• For instance, in a triangle, extending one side will form an external angle at the extended vertex.

Key Formulae

• The overall sum of all external angles in any polygon is consistently 360 degrees.
• Each external angle in a regular polygon can be calculated with ( \frac{360}{n} ).

Make sure to grasp these concepts for a comprehensive understanding of external angles and their relevance in geometry.