Alternate Interior Angles
Alternate interior angles are pairs of angles on opposite sides of a transversal between two lines, equal in measure when the lines are parallel.
Formula
\text{angle } A = \text{angle } B \text{ (when lines are parallel)}
Definition
Alternate interior angles lie between two lines and on opposite sides of a transversal, forming a "Z" shape; when the lines are parallel, alternate interior angles are congruent (the Alternate Interior Angles Theorem), and this too is equivalent to the parallel postulate. A $180^\circ$ rotation (half-turn) about the midpoint of the transversal segment between the two parallels maps each line to the other and maps each alternate interior angle to its pair, proving congruence via this point symmetry.
Example
If a transversal crosses two parallel lines, the angle inside and to the left at the top intersection equals the angle inside and to the right at the bottom, making a Z (or backward Z) shape. For parallel lines $p$ and $q$ cut by transversal $t$: angle $3$ (below $p$, left of $t$) and angle $6$ (above $q$, right of $t$) are alternate interior angles, so if angle $3 = 75^\circ$, then angle $6 = 75^\circ$ too.
Key Insight
Alternate interior angles are often called "Z angles" because of the shape the transversal and parallel segments make. They are equal because each equals a corresponding angle, and corresponding angles are equal for parallel lines, a two-step chain that models deductive geometry. The half-turn symmetry proof is more illuminating than the corresponding-angles proof: it reveals that alternate interior angle equality is a direct consequence of the $180^\circ$ rotational symmetry of the parallel line configuration, the kind of symmetry argument that is a hallmark of modern axiomatic geometry.