Reflex Angle
A reflex angle measures greater than 180 degrees and less than 360 degrees.
Formula
180 < \text{angle} < 360^\circ
Definition
A reflex angle is greater than $180^\circ$ but less than a full $360^\circ$ circle ($\pi < \theta < 2\pi$ radians), the "big" angle on the outside when two rays are drawn from a point. Every non-straight angle actually creates two angles, one less than $180^\circ$ and its reflex counterpart equal to $360^\circ$ minus the original, always summing to $360^\circ$; reflex angles appear in concave polygons (where at least one interior angle is reflex) and in circle geometry when discussing major arcs.
Example
The remaining large portion of a pizza after cutting a small slice forms a reflex angle at the center; a $270^\circ$ angle is reflex, as is the large angle swept the long way from 12 to 7 on a clock. If an angle measures $70^\circ$, its reflex angle is $360 - 70 = 290^\circ$; a concave polygon has at least one interior reflex angle. In circle geometry, the central angle for a major arc is a reflex angle, and the inscribed angle theorem still holds, an inscribed angle is half its intercepted arc, even when the arc is major.
Key Insight
Reflex angles highlight the importance of orientation in geometry. When computing areas with the shoelace formula or defining winding numbers, the distinction between a reflex angle and its non-reflex counterpart determines the sign of a contribution, and the exterior angles of a concave polygon can be negative at reflex vertices even though the exterior angle sum of any simple polygon is still $360^\circ$, connecting angle measure to signed area.