Vertical Angles
Vertical angles are the pairs of opposite angles formed when two lines intersect, and they are always equal in measure.
Formula
\text{vertical angle } A = \text{vertical angle } B
Definition
When two lines cross, they form four angles; vertical angles are the two pairs directly across from each other at the intersection, and they are always equal, the Vertical Angles Theorem. They arise from the symmetry of two intersecting lines about their point of intersection: a $180^\circ$ rotation about that point maps each line to itself and each angle to its vertical angle, establishing their congruence via this isometry.
Example
When two roads cross in an X shape, the angles diagonally across from each other are vertical angles: if one is $50^\circ$, the angle directly across is also $50^\circ$. Lines $m$ and $n$ intersecting at $P$ form angles $1, 2, 3, 4$ in order; angles $1$ and $3$ are vertical (congruent), angles $2$ and $4$ are vertical (congruent), and angles $1$ and $2$ form a linear pair (supplementary). The point reflection $(x,y) \to (2h-x, 2k-y)$ about intersection $(h,k)$ swaps each angle with its vertical angle.
Key Insight
Vertical angles are equal because each is supplementary to the same angle: angle $1$ + angle $2$ = $180$ and angle $3$ + angle $2$ = $180$, so angle $1$ = angle $3$. Despite the name, "vertical" angles do not have to be up-down, the name comes from the shared vertex. This proof is one of the first in Euclid's Elements (Book I, Proposition 15), showing how a single postulate, that straight angles sum to $180^\circ$, can establish a non-obvious geometric fact.