Plane Figures

Polygons

Interior and Exterior Angles of Polygons

A polygon is a plane figure enclosed by multiple line segments. Polygons with 3, 4, …, sides are called triangles, quadrilaterals, …, -gons, respectively. Each line segment is referred to as a side of the polygon, and the points where two sides meet are called vertices of the polygon.

  • An interior angle of a polygon is an angle formed inside the polygon by two adjacent sides.
  • An exterior angle of a polygon, corresponding to an interior angle, is formed by extending one side of the polygon beyond its vertex and measuring the angle between the extended side and the adjacent side.

Number of Diagonals in a Polygon

For an -sided polygon, the number of diagonals that can be drawn from a single vertex is , excluding the vertex itself and its two adjacent vertices. Therefore, the total number of diagonals in the polygon is . Since each diagonal is counted twice, the actual number of diagonals is given by:

Relationship Between Interior and Exterior Angles of a Triangle

Consider , where an external point lies on the extension of side . Draw a ray parallel to line , passing through vertex .
From the diagram:
Thus:
This shows that the exterior angle is equal to the sum of the two non-adjacent interior angles and .

Properties of Interior and Exterior Angles in a Triangle

  1. The sum of the measures of the three interior angles of a triangle is .
  2. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.