Alternate Exterior Angles
Alternate exterior angles are pairs of angles outside two lines on opposite sides of a transversal, equal in measure when the lines are parallel.
Formula
\text{angle } A = \text{angle } B \text{ (when lines are parallel)}
Definition
Alternate exterior angles lie outside the two lines and on opposite sides of the transversal; when the lines are parallel, alternate exterior angles are congruent (the Alternate Exterior Angles Theorem), following from the same half-turn symmetry that proves alternate interior angles equal, since each alternate exterior angle is the vertical angle of its corresponding alternate interior angle.
Example
These are like alternate interior angles but on the outside: if the inside "Z" angles are equal, the outside Z angles are equal too when the lines are parallel. If angle $1$ (above line $l$, left of $t$) $= 110^\circ$, its alternate exterior angle (below line $m$, right of $t$) is also $110^\circ$. All eight angles formed by a transversal crossing two parallel lines take only two values, $\theta$ and $180-\theta$, and the alternate exterior pairs are among the angles equal to $\theta$.
Key Insight
Alternate exterior angles are vertical angle partners of alternate interior angles, so because vertical angles are equal and alternate interior angles are equal, alternate exterior angles must be too. Recognizing all four angle-relationship types (corresponding, alternate interior, alternate exterior, co-interior) lets you find any unknown angle in a parallel-line diagram, and the complete symmetry of the eight angles (only two distinct values) reflects the dihedral symmetry group of the configuration, generated by translation along the parallels and the half-turn about the transversal's midpoint.