Linear Pair
GeometryA 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 angles that are next to each other (adjacent) and together form a straight line. Linear pairs always add up to $180^\circ$.
Example
When a ray sits on a line, it creates two angles on either side. Those two angles are a linear pair. If one is $60^\circ$, the other must be $120^\circ$ (because $60 + 120 = 180$).
Key Insight
Linear pairs are supplementary by definition - they always add to $180^\circ$ because together they form a straight (linear) angle. This is why "linear pair" has the word "linear" in it.
Definition
A linear pair consists of two adjacent angles whose non-common sides are opposite rays (forming a straight line). Linear pairs are always supplementary. The Linear Pair Postulate states: if two angles form a linear pair, then they are supplementary.
Example
Ray $OC$ intersects line $AB$ at $O$. Angles $AOC$ and $COB$ form a linear pair. If angle $AOC = 130^\circ$, then angle $COB = 50^\circ$. Every pair of adjacent angles formed when two lines cross is a linear pair.
Key Insight
The Linear Pair Postulate is one of the fundamental postulates used to derive angle theorems. From it, we can prove the Vertical Angles Theorem and the angle relationships with parallel lines, making it a gateway postulate in formal geometry.
Definition
A linear pair is two adjacent angles whose union is a half-plane together with its boundary ray - equivalently, whose non-shared rays are antipodal from the vertex. The Linear Pair Postulate (a theorem in some axiomatic systems) follows from the definition of angle measure and the straight angle axiom: $m(\text{angle } A) + m(\text{angle } B) = 180$ whenever they form a linear pair.
Example
In Hilbert's axioms, the linear pair supplementarity follows from the axioms of order and congruence without a separate postulate. In Birkhoff's system, it follows from the protractor postulate and the definition of a straight angle.
Key Insight
Different axiomatic systems handle the linear pair relationship differently: some adopt it as a postulate (as in many high school curricula), while others derive it as a theorem. This illustrates how the choice of axioms shapes the logical structure of geometry.