page_title: Angle
section_title: Polygon-related angles
hierarchical_section_title: Angle / Types of angles / Polygon-related angles
context_page_description: In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
  Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements also hold in space. For example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.
  Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.
context_section_description: An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle that is a reflex angle.
  In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, 180°, or  turn; the measures of the interior angles of a simple convex quadrilateral add up to 2π radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with n sides add up to (n − 2)π radians, or 180(n − 2) degrees, (2n − 4) right angles, or ( − 1) turn.
  The supplement of an interior angle is called an exterior angle, that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical angles and hence are equal. An exterior angle measures the amount of rotation one has to make at a vertex to trace out the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure.
  In Euclidean geometry, the sum of the exterior angles of a simple convex polygon will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons.
  In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).
  In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.
  In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.
  Some authors use the name exterior angle of a simple polygon to simply mean the explement exterior angle (not supplement!) of the interior angle. This conflicts with the above usage.
caption: Internal and external angles.