A line is a perfectly straight path of points that goes on forever in both directions. It has no thickness and no endpoints, so it has no length you can measure. Through any two points there is exactly one line, which is named by those two points, as in line AB, or by one lowercase letter.
The arrowheads show that a line never ends. A line segment stops at two endpoints, and a ray stops at one.
Formal definition and notation
In Euclidean geometry, a line is an undefined term: a one-dimensional set of points, straight and endless in both directions, such that exactly one line passes through any two distinct points. In the coordinate plane, every line is the set of points (x, y) satisfying a linear equationax + by = c with a and b not both 0.
Symbol
Read it as
Meaning
line AB
line A B
the line through points A and B; in print, AB with a two-headed arrow over it
ℓ or line m
line l, line m
a line named by one lowercase (often script) letter
A ray starts at one endpoint and goes forever in one direction; a line goes forever in both directions. Ray AB and ray BA differ, but line AB and line BA are the same.
A line is the geometric figure. A linear equation, such as y = 2x − 1, describes exactly which points lie on one line.
Quick check: 3 questions
Question 1 of 3
How many endpoints does a line have?
Show answer and explanation
Answer: A. A line goes on forever in both directions, so it has no endpoints. A ray has 1 endpoint and a line segment has 2. A line does contain infinitely many points, but none of them is an end.
Question 2 of 3
How many lines pass through two different points?
Show answer and explanation
Answer: D. Two points determine exactly one line. Infinitely many lines pass through a single point, which is where that wrong answer comes from.
Question 3 of 3
Which three points are collinear?
Show answer and explanation
Answer: B. From (1, 1) to (2, 3) the slope is 2, and from (2, 3) to (4, 7) it is 42 = 2. Same slope through a shared point means one line. Each other set has two different slopes.
Line: frequently asked questions
What is the difference between a line, a line segment and a ray?
All three are straight. A line has no endpoints and extends forever both ways. A line segment has two endpoints, so it has a measurable length. A ray has one endpoint and extends forever in one direction, like a beam of light from a flashlight. Drawings show the difference with arrowheads (it continues) and dots (it stops).
Why is a line called an undefined term?
Geometry has to start with a few words that are not defined using simpler words, or every definition would go in circles. Point, line and plane are those starting words. Instead of a definition, a line is described by its properties: it is straight, has no thickness, extends forever, and exactly one passes through any two points.
Can a line be curved?
Not in geometry. A line is always straight; a curved path is called a curve. Everyday words such as “draw a wavy line” or “line graph” use the word loosely. On a coordinate plane, a straight line is the graph of a linear equation, while curves such as parabolas come from other equations.
UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1Parallel LinesParallel lines are lines in the same plane that never intersect. They stay the same distance apart and point in the same direction, like the rails of a straight railroad track. On a coordinate plane, parallel lines have the same slope but different y-intercepts. The symbol ∥ means is parallel to.Geometry · Grade 4, Grade 5, Grade 6Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4Perpendicular LinesPerpendicular lines are lines that intersect to form right angles: their intersection makes a 90-degree angle. The symbol ⊥ means is perpendicular to. On a coordinate plane, two perpendicular lines that are not vertical have slopes that multiply to −1, such as 2 and −12.Geometry · Grade 4, Grade 5, Grade 6LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3CentimeterA centimeter is a metric unit of length equal to one hundredth of a meter, about the width of a fingernail. Its symbol is cm. There are 100 centimeters in 1 meter and 10 millimeters in 1 centimeter, and 1 inch equals 2.54 centimeters.Measurement · Grade 3, Grade 4, Grade 5Line SegmentA line segment is a straight part of a line with two endpoints. It starts at one point, stops at another, and includes every point between them. A segment with endpoints A and B is written AB. Unlike a line, which goes on forever, a segment has a definite length that you can measure.Geometry · Grade 4, Grade 5, Grade 6RayA ray is part of a line that starts at one endpoint and goes on forever in one direction, like a beam of light from a flashlight. It is named by its endpoint first and then any other point on it, so ray AB starts at A and passes through B. Two rays sharing an endpoint form an angle.Geometry · Grade 4, GeometryDifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7ParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, GeometryPointA point is an exact location in space. It has no length, no width and no thickness, so it has no size at all. You draw a point as a dot and name it with a capital letter, such as point A. On a coordinate plane, an ordered pair such as (3, 2) names a point.Geometry · Grade 4, Grade 5, Grade 6AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6PlaneA plane is a perfectly flat surface that has no thickness and extends forever in every direction, like a tabletop that never ends. It is one of the undefined terms of geometry, along with point and line. Any three points that are not on one line lie in exactly one plane.Geometry · Grade 7, Grade 8, Geometry