Angle

An angle measures separation of two directions meeting at a vertex. For nonzero vectors $u,v$, the unoriented angle $\theta \in [0,\pi ]$ satisfies $\cos \theta =(u\cdot v)/(\Vert u\Vert \Vert v\Vert )$. A zero dot product gives a right angle. An oriented planar angle also records the sense of turning, usually modulo $2\pi$. In radians, an angle is subtended arc length divided by radius, so a full turn is $2\pi$. Ray lengths do not determine the angle.^[ref]

An angle is a geometric figure formed by two rays (or line segments) that have a common endpoint, known as the vertex. The measure of an angle is typically expressed in degrees or radians and represents the amount of rotation required to align one ray with the other. Angles can be classified into several types based on their measure: 1. **Acute Angle**: Less than 90 degrees. 2. **Right Angle**: Exactly 90 degrees.