transversal

A transversal is a line that intersects two or more lines, curves, or other segments at distinct points. In the plane, a single line that cuts across two lines is a transversal to those lines. The concept also generalizes to curves or segments that a line or curve meets.

When a transversal intersects two lines, several angle relationships arise. If the two lines are parallel, corresponding angles formed by the transversal are congruent, and alternate interior angles are also congruent. The interior angles on the same side of the transversal sum to 180 degrees. The converse holds as well: if a transversal creates equal corresponding angles or equal alternate interior angles with two lines, then the lines are parallel. These angle relationships are fundamental tools in Euclidean geometry for proving parallelism and solving angle problems.

Beyond simple planar geometry, the idea of a transversal extends to higher dimensions and different contexts. A transversal to a family of curves or submanifolds is a line or map that intersects them in a well-behaved way, and transversality is a central concept in differential topology and intersection theory. In computational geometry and related fields, transversals describe lines or curves that intersect a collection of objects, and they are used in optimization and visibility problems.

Etymology: the term comes from the Latin transversus, meaning turned across or crossing. The concept remains a fundamental tool for understanding how a line interacts with a set of geometric figures.