Line
A 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, with no endpoints; we draw arrows on both ends to show it never stops, and name it using two points on it, like line $AB$, or with a single lowercase letter. It has only one dimension, length, with no width or thickness, and it is determined by any two distinct points on it. 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, and a general line is an affine subspace of dimension $1$.
Example
Imagine a laser beam going 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 forever. Line $AB$ passing through points $A(1, 2)$ and $B(4, 6)$ is the unique line through those two points; parallel lines never intersect, while perpendicular lines intersect at $90^\circ$. The line through points $(x_1, y_1)$ and $(x_2, y_2)$ has equation $(y - y_1)/(y_2 - y_1) = (x - x_1)/(x_2 - x_1)$, and 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
Two distinct points determine exactly one line, one of the core postulates of Euclidean geometry, and this uniqueness underlies many geometric proofs. Euclid's fifth postulate (the parallel postulate) concerns lines: through a point not on a line, exactly one parallel line exists. Changing this single axiom yields hyperbolic geometry (many parallels) or elliptic geometry (no parallels), showing how one assumption about lines expands geometry beyond Euclid.