Linear Pair
A linear pair is a pair of adjacent angles formed when two lines intersect, with their outer sides forming a straight line and their measures summing to 180 degrees.
Formula
\text{angle } A + \text{angle } B = 180^\circ
Definition
A linear pair is two adjacent angles that together form a straight line, with non-common sides that are opposite rays; linear pairs are always supplementary, adding to $180^\circ$ (the Linear Pair Postulate). Equivalently, it is two adjacent angles whose union is a half-plane together with its boundary ray, and in some axiomatic systems this supplementarity is a postulate, while in others (Hilbert's, Birkhoff's) it follows as a theorem from the axioms of order, congruence, or the protractor postulate.
Example
When a ray sits on a line, it creates two angles on either side that form a linear pair; if one is $60^\circ$, the other must be $120^\circ$. Ray $OC$ intersecting line $AB$ at $O$ creates linear pair angles $AOC$ and $COB$: if angle $AOC = 130^\circ$, then angle $COB = 50^\circ$.
Key Insight
Linear pairs are supplementary by definition, they always add to $180^\circ$ because together they form a straight (linear) angle, which is why "linear pair" has the word "linear" in it. The Linear Pair Postulate is used to derive the Vertical Angles Theorem and the parallel-line angle relationships, making it a gateway postulate, though different axiomatic systems handle it differently, illustrating how the choice of axioms shapes the logical structure of geometry.