An angle in geometry refers to the figure formed by two rays sharing a common endpoint, which is called the vertex of the angle. The magnitude of an angle is usually measured in degrees or radians, and it represents the rotation needed to align one ray (the initial side) with the other (the terminal side).

Here are some key aspects of angles:

Measurement:

  • Degrees: One full rotation is 360 degrees.
  • Radians: One full rotation is (2\pi) radians.

Types of Angles

based on their measurement:

  • Acute Angle: An angle less than 90°.
  • Right Angle: An angle equal to 90°.
  • Obtuse Angle: An angle greater than 90° but less than 180°.
  • Straight Angle: An angle equal to 180°.
  • Reflex Angle: An angle greater than 180° but less than 360°.
  • Full Rotation: An angle equal to 360°.

Basic Relationships:

  • Complementary Angles: Two angles whose sum is 90°.
  • Supplementary Angles: Two angles whose sum is 180°.
  • Adjacent Angles: Two angles that share a common side and vertex, but do not have any common interior points.
  • Vertical Angles: Pairs of opposite angles formed by intersecting lines; they are always equal.

Angle Bisector: A line or ray that divides an angle into two equal angles.

Central and Inscribed Angles: Central angles have their vertex at the center of a circle, while inscribed angles have their vertex on the circle itself.

Exterior and Interior Angles: In polygons, exterior angles are outside the polygon formed by extending one side, while interior angles are inside the polygon.

Naming Angles: Angles can be named using three points, with the vertex point listed in the middle (e.g., (\angle ABC)), or by a single letter corresponding to the vertex (e.g., (\angle A)).

Angle Addition Postulate: The Angle Addition Postulate states that if point D lies in the interior of (\angle ABC), then (\angle ABD + \angle DBC = \angle ABC).

Angles are fundamental in geometry, providing a way to describe and measure the space between two intersecting lines or rays, and they serve as the basis for many geometric theorems and constructions.