Corresponding Angles
Corresponding angles are pairs of angles in matching positions when a transversal crosses two lines, and they are equal when the lines are parallel.
Formula
\text{angle } A = \text{angle } B \text{ (when lines are parallel)}
Definition
Corresponding angles are pairs of angles in matching positions at each intersection when a transversal crosses two lines, lying on the same side of the transversal and in the same position (both above-left, both above-right, etc.); when the two lines are parallel, corresponding angles are congruent (the Corresponding Angles Postulate), and the converse also holds. This postulate is, in many axiom systems, equivalent to Euclid's parallel postulate: a transversal $t$ crossing parallel lines $l_1$ and $l_2$ creates a translation along $t$ that maps $l_1$ to $l_2$ and carries each angle to its corresponding angle, proving congruence via this isometry.
Example
If a transversal crosses two parallel lines, the angle in the upper-right at the first crossing equals the angle in the upper-right at the second, forming an "F" shape (forwards or backwards). If the upper-right angle at line $l$ is $65^\circ$, the corresponding angle at line $m$ is also $65^\circ$, and the other six angles are either $65^\circ$ or $115^\circ$ (its supplement).
Key Insight
An easy way to spot corresponding angles is the "F" shape they form. Corresponding angles form the basis for proving the alternate interior and co-interior angle theorems: accepting that they are equal for parallel lines lets you derive every other transversal angle relationship from vertical angles and linear pairs. In non-Euclidean geometries, parallel lines (if defined) do not produce equal corresponding angles, and in hyperbolic geometry the angle sum of a triangle is less than $180^\circ$, showing how deeply this result depends on the parallel postulate.