Line
GeometryA line is a straight, one-dimensional figure that extends infinitely in both directions, with no endpoints.
Definition
A line is perfectly straight and goes on forever in both directions. It has no endpoints. We draw arrows on both ends to show it never stops, and we name it using two points on it, like line $AB$.
Example
Imagine a laser beam that goes on forever in both directions - that is like a line. The edge of a ruler shows a straight path, but a true line would extend past both ends of the ruler forever.
Key Insight
A line has only one dimension - length. It has no width or thickness at all. That is what makes it different from a flat surface or a solid object.
Definition
A line is a straight, one-dimensional figure extending infinitely in both directions. It is determined by any two distinct points on it. A line has no endpoints, no width, and no thickness. Lines are named by two points (line $AB$, written with a double arrow above $AB$) or by a single lowercase letter.
Example
Line $AB$ passes through points $A(1, 2)$ and $B(4, 6)$. Any two distinct points define exactly one line. Parallel lines never intersect; perpendicular lines intersect at $90^\circ$.
Key Insight
Two distinct points determine exactly one line - this is one of the core postulates of Euclidean geometry. This uniqueness property underlies many geometric proofs.
Definition
In Euclidean geometry, a line is a primitive term satisfying axioms of incidence and order. In analytic geometry, a line in $\mathbb{R}^2$ is the solution set of $ax + by = c$ (with $a, b$ not both zero). In linear algebra, a line through the origin is a one-dimensional subspace; a general line is an affine subspace of dimension $1$.
Example
The line through points $(x_1, y_1)$ and $(x_2, y_2)$ has equation: $(y - y_1)/(y_2 - y_1) = (x - x_1)/(x_2 - x_1)$. In projective geometry, any two distinct lines in the projective plane meet in exactly one point (including a "point at infinity" for parallel lines).
Key Insight
Euclid's fifth postulate (parallel postulate) concerns lines: through a point not on a line, exactly one parallel line exists. Changing this axiom yields hyperbolic geometry (many parallels) or elliptic geometry (no parallels), expanding geometry beyond Euclid.