Transversal
A transversal is a line that intersects two or more other lines at distinct points, creating pairs of angles with special relationships.
Definition
A transversal is a line that crosses two or more other lines; when it crosses two parallel lines, it creates several pairs of angles with special equal or supplementary relationships. Formally, it is a line in the same plane as two or more given lines that intersects each at a distinct point, and crossing two parallel lines it creates $4$ pairs of corresponding angles (equal), $2$ pairs of alternate interior angles (equal), $2$ pairs of alternate exterior angles (equal), and $2$ pairs of co-interior angles (supplementary).
Example
A street crossing two parallel train tracks is like a transversal, forming eight angles with special relationships to each other. Transversal $t$ crosses parallel lines $m$ and $n$, forming angles $1$-$4$ at $m$ and $5$-$8$ at $n$: angles $1$ and $5$ are corresponding (equal), angles $3$ and $6$ are alternate interior (equal), and angles $3$ and $5$ are co-interior (supplementary). Given a transversal at angle $\theta$ crossing parallels, all eight angles are determined: four equal $\theta$ and four equal $180-\theta$.
Key Insight
Transversals are the key tool for studying parallel lines, without one there would be no angles to compare, and the relationships can be reversed: equal corresponding angles prove two lines are parallel. In fact, the transversal angle theorems are equivalent to the parallel postulate itself, assuming any one of them forces all the others and forces the lines to be parallel, a logical equivalence that fails in non-Euclidean geometry, reflecting the deep connection between angle sums and the curvature of space.