Types, Measures and Properties of Angles in Geometry

Definition of an Angle

An angle is the plane region between two rays that share the same endpoint, called the vertex. In geometry, an angle is the figure formed by two lines or rays with a common origin. The angle between two curves is the angle between their tangents at the point of intersection.

Geometric definition

Geometric shape: The angle (often described as its measure) is the magnitude between two lines of any type that converge at a common point called the vertex. Colloquially, the angle is the figure formed by two lines with a common origin.

Trigonometric definition

Trigonometric form: The angle may describe a rotation of a ray or a line segment around one endpoint taken as the vertex, from a starting position to a final position. If the rotation is counterclockwise, the angle is considered positive. If the rotation is clockwise, the angle is considered negative.

Convex and Concave Angles

• Convex angle: an angle greater than 0° and less than 180° (0° < angle < 180°).
• Concave angle: an angle greater than 180° and less than 360° (180° < angle < 360°).

Related Angles

Depending on their position relative to each other, angles are called:

• Adjacent angles: angles that share a vertex and a common side but have no common interior point.
• Consecutive angles: angles that share one side and a common vertex (often used in polygons to refer to angles next to each other).
• Vertical (vertex) angles: angles whose sides are opposite rays; they are formed by intersecting lines and are congruent.

Depending on their size, angles are called:

• Congruent angles: angles that have the same measure.
• Complementary angles: two angles whose measures sum to π/2 radians or 90°.
• Supplementary angles: two angles whose measures sum to π radians or 180°.
• Conjugate angles: angles whose measures add to 2π radians or 360°.

Interior and Exterior Angles of Polygons

The interior angle of a polygon is formed by adjacent sides measured toward the interior of the polygon. The exterior angle of a polygon is formed by one side and the extension of an adjacent side, measured outside the polygon.

Angles Related to Circles

An angle with respect to a circle may be classified as follows:

• Central angle: an angle whose vertex is at the center of the circle. The measure of a central angle is equal to the measure (in degrees or radians) of the arc it intercepts.
• Inscribed angle: an angle whose vertex lies on the circumference and whose sides intercept the circle at two points. The measure of an inscribed angle is half the measure of its intercepted arc.
• Angle formed by a chord and a tangent: an angle whose vertex is on the circle, with one side a chord and the other a tangent at the vertex. The measure of this angle is half the measure of the intercepted arc.
• Interior angle (formed by two chords): an angle whose vertex is inside the circle, formed by the intersection of two chords. The measure of this interior angle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle (i.e., half the sum of the arcs covered by its sides and the arcs spanned by their extensions).
• Exterior angle (vertex outside the circle): an angle with vertex outside the circle formed by two secants, a secant and a tangent, or two tangents. The measure of this exterior angle is half the difference of the measures of the intercepted arcs.

Note on units: angles are commonly measured in degrees (°) or radians (where π radians = 180°). Common special sums include π/2 (90°), π (180°), and 2π (360°).