Line Segment, Ray, and Line

Three related but different objects

A complete line can't be drawn on paper; what we draw is a segment with arrowheads showing it continues.

Collinear points and concurrent lines

Three or more points on the same line are collinear. Three or more lines through the same single point are concurrent.

Two facts from Euclid

• Given two distinct points, there is one and only one line through them.
• Two distinct lines cannot have more than one point in common.

Worked example: a ray standing on a line

If a ray stands on a line, the two adjacent angles it forms add up to $180^\circ$ (a linear pair). If those angles are $3x$ and $2x + 20^\circ$:

So the angles are $96^\circ$ and $84^\circ$, which do add to $180^\circ$.

Worked example: counting segments

How many line segments are determined by 5 points on a line? Each pair of points gives one segment, so there are $\dbinom{5}{2} = 10$. For 4 points with no three collinear, each pair also determines a different line: $\dbinom{4}{2} = 6$ lines.

Quick true or false

• 'A ray has a definite length.' False. It starts at one endpoint and goes on forever in one direction.
• 'Two distinct points determine exactly one line.' True. This is Euclid's axiom above.
• 'Any three points are collinear.' False. Three points are collinear only if one line passes through all of them.

Common mistakes

• Naming a ray in the wrong order. Ray $AB$ starts at $A$ and passes through $B$; ray $BA$ is different.
• Treating a segment as a line. Two segments can fail to meet even when the lines containing them would.
• Forgetting the 180° rule when angles sit on a straight line.

Where this comes up in exams

The 2025-26 CBSE Class IX syllabus lists Euclid's geometry, including the axiom that two distinct points have one and only one line through them and the theorem that two distinct lines cannot have more than one point in common, and, under Lines and Angles, that if a ray stands on a line the two adjacent angles add up to 180°.

Checked against CBSE Mathematics curriculum, Classes IX-X (2025-26). Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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