anglechasing

Angle chasing is a problem-solving method in Euclidean geometry that derives unknown angle measures by systematically applying established angle relationships. Practitioners build a chain of inferences from given angles and invariant properties such as sums, equalities, parallelisms, and symmetry. The goal is to transform a geometric configuration into solvable angle equations rather than constructing new elements.

Common tools include the angle sum in a triangle (three angles total 180 degrees), the linear pair criterion (adjacent angles on a straight line sum to 180 degrees), and the fact that vertically opposite angles are equal. Parallel lines yield equal corresponding or alternate interior angles, while an exterior angle equals the sum of the two remote interior angles. In cyclic figures, the opposite angles of a cyclic quadrilateral sum to 180 degrees, and inscribed angles subtend equal arcs.

The typical workflow is to identify what is known, mark equal angles, and progressively substitute these relations to reduce the unknowns to a solvable set of equations. Diagrams are essential, but the reasoning should be logic-based and independent of measurement, relying on angle properties rather than lengths.

Angle chasing is especially common in contest and olympiad geometry, where problems often demand a synthetic, diagram-driven approach rather than coordinate or analytic methods. It is not always possible to determine all angles from the given data alone; some configurations admit multiple solutions or require additional theorems or constructions to resolve ambiguities.