What is a ray in math?
A 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.
Ray in a picture
Formal definition and notation
Ray AB is the set made of point A and every point P on line AB that lies on the same side of A as B. Point A is its endpoint (or origin). A ray has one endpoint and infinite length. Two rays with the same endpoint that point in exactly opposite directions are opposite rays; together they form a line.
| Symbol | Read it as | Meaning |
|---|---|---|
| ray AB | ray A B | endpoint A, passing through B; in print, AB with a one-headed arrow over it pointing right |
| ray BA | ray B A | a different ray: endpoint B, pointing back through A |
| ∠ QPR | angle Q P R | the angle formed by rays PQ and PR; the vertex P is in the middle |
Key facts
- Order matters: ray AB and ray BA are different rays, but line AB and line BA are the same line.
- A ray has exactly one endpoint, so it has no measurable length.
- Every point on a line splits it into two opposite rays. Opposite rays form a straight angle, 180 degrees.
- The sides of an angle are rays, which is why you can extend the sides of an angle as far as you like without changing its measure.
- If C is on ray AB (and C ≠ A), then ray AC is the same ray as ray AB.
Try it: two rays make an angle
Drag the moving ray around the fixed one. Both rays share one endpoint, the vertex, and the opening between them is the angle.
Ray examples, step by step
Example 1: Same ray or not?
Points A, B and C lie on a line in that order. Are ray AB and ray AC the same ray? Are ray BA and ray BC the same?
- Ray AB and ray AC both start at A and point toward B and C, which are on the same side of A. Same ray.
- Ray BA starts at B and points toward A; ray BC starts at B and points toward C.
- Those directions are opposite, so they are different rays (opposite rays).
Answer: Ray AB and ray AC are the same ray; ray BA and ray BC are opposite rays, not the same.
Example 2: Count the rays
Points A, B and C lie on a line in that order. How many different rays have an endpoint at one of these points and pass through another one of them?
- From A: ray AB and ray AC are the same, so 1 ray.
- From B: ray BA and ray BC point opposite ways, so 2 rays.
- From C: ray CB and ray CA are the same, so 1 ray.
- Total: 1 + 2 + 1 = 4.
Answer: There are 4 different rays.
Example 3: Opposite rays and angles
Rays OA and OB are opposite rays. Ray OC is drawn so that ∠ AOC = 65 degrees. What is ∠ COB?
- Opposite rays form a straight angle: ∠ AOB = 180 degrees.
- Ray OC splits it into two angles: ∠ AOC + ∠ COB = 180.
- ∠ COB = 180 − 65 = 115 degrees.
Answer: ∠ COB = 115 degrees.
Common mistakes
The first letter tells where the ray starts. Swapping the letters names a ray pointing the opposite way.
A ray goes on forever in one direction, so it cannot be measured. Only the part between two endpoints, a segment, has a length.
Arrowheads at both ends mean a line. A ray stops on one side and continues on the other.
Not to be confused with
A line goes on forever in both directions and has no endpoints. A ray stops at one endpoint. Line AB equals line BA, but ray AB does not equal ray BA.
A line segment has two endpoints and a length you can measure. A ray has one endpoint and no length.
A vector also has a direction, but it has a fixed length (magnitude) and can be moved anywhere. A ray is fixed at its endpoint and is infinitely long.
Quick check: 3 questions
How many endpoints does a ray have?
Answer: B. A ray has exactly 1 endpoint and continues forever in the other direction. A line has 0 endpoints and a line segment has 2.Show answer and explanation
In ∠ XYZ, which point is the common endpoint of the two rays?
Answer: C. The vertex is the middle letter, Y. The angle’s sides are ray YX and ray YZ, both starting at Y. X and Z are just points on the two rays.Show answer and explanation
Rays PA and PB are opposite rays, and ∠ APC = 48 degrees. What is ∠ CPB?
Answer: D. Opposite rays form 180 degrees, so ∠ CPB = 180 − 48 = 132 degrees. 42 uses 90 instead of 180, 48 assumes the angles are equal, and 312 uses 360.Show answer and explanation
Ray: frequently asked questions
What is the difference between a ray and a line?
A line goes on forever in both directions and has no endpoints. A ray has one endpoint and goes on forever in only one direction, like sunlight leaving the sun. You can make a ray by picking a point on a line and keeping only one side of it. The two halves are called opposite rays.
How do you name a ray?
Write the endpoint first, then any other point on the ray: ray AB starts at A and goes through B. In print, the two letters get a small arrow above them pointing right. Ray AB and ray BA are different, because they start at different points and point in opposite directions.
Why are the sides of an angle rays?
An angle is formed by two rays that share an endpoint, called the vertex. Because rays never end, the sides of an angle can be drawn as long as you like, and the angle’s measure, the amount of turn between the rays, stays the same. That is why a small angle drawing and a large one can have the same measure.
Practice and tools
Sources
- Common Core State Standards, Grade 4 Geometry (4.G.A.1)
- OpenStax, Contemporary Mathematics, 10.1 Points, Lines, and Planes (LibreTexts edition)
- Weisstein, Eric W. "Ray." MathWorld, Wolfram Research
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 4.G.A.1
Cite this entry
Effortless Math Team. “Ray.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/ray/
Flashcard
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