What is the difference between a line and a line segment?

A line extends infinitely in both directions and has no endpoints. A line segment has two fixed endpoints.

An angle is formed when two rays share a common endpoint (vertex). It is measured in degrees.

NCERT Solutions for Class 6 Maths Chapter 4 — Basic Geometrical Ideas

Q: How many sides does a triangle have?

A triangle has 3 sides and 3 vertices.

Geometry begins with the simplest ideas — a point, a line, a ray, an angle. These ideas may seem small, but everything in geometry is built on them. In this chapter, you learn the vocabulary of geometry and how to identify, name, and classify the basic shapes. Every solution below follows a step-by-step approach so you understand the reasoning behind each answer.

Key Concepts

Concept	What It Means	Example
Point	A location in space with no size (no length, breadth, or thickness). Represented by a dot and named with a capital letter.	Point A, Point P
Line	A straight path that extends infinitely in both directions. It has no endpoints. Named using any two points on it or a single lowercase letter.	Line AB (written as AB with arrows on both ends)
Line Segment	A part of a line with two fixed endpoints. It has a definite length.	Segment PQ where P and Q are endpoints
Ray	A part of a line that starts at one endpoint and extends infinitely in one direction.	Ray AB starts at A and goes through B forever
Intersecting Lines	Two lines that cross each other at a single point. The point where they meet is called the point of intersection.	Lines l and m crossing at point O
Parallel Lines	Two lines in the same plane that never meet, no matter how far they are extended. The distance between them stays constant.	Railway tracks
Angle	Formed when two rays share a common endpoint (vertex). Measured in degrees.	Angle ABC with vertex at B
Triangle	A closed figure formed by three line segments. Has 3 sides, 3 vertices, and 3 angles.	Triangle ABC with sides AB, BC, CA
Quadrilateral	A closed figure formed by four line segments. Has 4 sides, 4 vertices, and 4 angles.	Rectangle, square, parallelogram
Circle	A closed curve where every point is at the same distance from a fixed point called the centre.	A round plate, a coin
Polygon	A closed figure made entirely of line segments. The line segments are called sides, and their endpoints are called vertices.	Triangle (3 sides), Quadrilateral (4 sides), Pentagon (5 sides)

Exercise 4.1 — Points

Question 1: How many points are marked in the figure? Name them.

Solution

Step 1: Look at the figure and identify each dot that has been marked.
Step 2: Each dot represents a point and is labelled with a capital letter.
Step 3: Count the dots carefully.

AnswerThe points marked are A, B, C, D, E — so there are 5 points.

Question 2: How many line segments are there in the figure? Name them.

Solution

Step 1: A line segment connects two points.
Step 2: List every pair of connected points: AB, BC, CD, DE, AC, AD, AE, BD, BE, CE.
Step 3: Count them.

AnswerThere are 10 line segments — AB, BC, CD, DE, AC, AD, AE, BD, BE, CE.

Question 3: Use the figure to name: (a) Five points (b) A line (c) Four rays (d) Five line segments

Solution

(a) Five points:

Step 1: Identify dots in the figure labelled with capital letters.

AnswerPoints B, O, E, D, C (any five distinct points from the figure).

(b) A line:

Step 1: A line is represented by two points with arrows on both ends.

AnswerLine BD (or line DE — any line that extends infinitely in both directions).

(c) Four rays:

Step 1: A ray starts at one endpoint and extends infinitely.

AnswerRay OB, Ray OC, Ray OD, Ray OE.

(d) Five line segments:

Step 1: A line segment has two fixed endpoints.

AnswerSegment OB, Segment OC, Segment OD, Segment OE, Segment DE.

Exercise 4.2 — Lines and Line Segments

Question 1: Name the line segments in the figure.

Solution

Step 1: Identify all pairs of connected points that form segments with two fixed endpoints.
Step 2: A line segment is named by its two endpoints.

AnswerThe line segments are: AB, BC, CD, DA, AC, BD.

Question 2: Is the following statement true or false? Justify your answer. (a) A line segment has one endpoint only. (b) Two lines are parallel if they do not intersect, however far they are extended. (c) A line has a beginning point but no end point.

Solution

(a) A line segment has one endpoint only — FALSE.

Step 1: Recall the definition of a line segment.
Step 2: A line segment has two fixed endpoints, not one.

AnswerFalse. A line segment has two endpoints.

(b) Two lines are parallel if they do not intersect, however far they are extended — TRUE.

Step 1: This matches the definition of parallel lines.
Step 2: Parallel lines never meet, regardless of how far they are extended.

AnswerTrue.

(c) A line has a beginning point but no end point — FALSE.

Step 1: A line extends infinitely in both directions.
Step 2: It has no beginning and no end.

AnswerFalse. A line has no endpoints at all.

Question 3: From the figure, name: (a) All pairs of parallel lines (b) All pairs of intersecting lines (c) Lines whose point of intersection is L

Solution

(a) All pairs of parallel lines:

Step 1: Identify lines that never meet.

AnswerLine m is parallel to line n (written as m || n).

(b) All pairs of intersecting lines:

Step 1: Identify lines that cross each other.

AnswerLine l intersects line m, and line l intersects line n.

(c) Lines whose point of intersection is L:

Step 1: Find the point L on the figure and see which lines pass through it.

AnswerLine l and line m intersect at point L.

Exercise 4.3 — Rays and Angles

Question 1: Name the angles in the given figure.

Solution

Step 1: An angle is formed at a vertex where two rays meet.
Step 2: Identify all vertices where rays meet and name each angle using three letters (or one letter if only one angle is at that vertex).

AnswerThe angles are: angle AOB, angle BOC, angle COD, and angle AOD.

Question 2: In the given figure, which of the following are angles? (a) AOB (b) BOC (c) COD (d) DOE (e) EOF (f) FOA

Solution

Step 1: Check each pair of rays — they must share a common endpoint (vertex) and the region between them must form an opening.
Step 2: Verify each option against the figure.

AnswerAll of the listed pairs — AOB, BOC, COD, DOE, EOF, FOA — are angles in the figure, because each pair of rays shares a common vertex and forms an opening.

Question 3: Write the names of: (a) All rays starting from point O (b) Angle between which two rays is angle AOB?

Solution

(a) All rays starting from point O:

Step 1: A ray starts at a point and goes in one direction infinitely.
Step 2: List every ray that originates at O.

AnswerRay OA, Ray OB, Ray OC, Ray OD, Ray OE, Ray OF.

(b) Angle between which two rays is angle AOB?

Step 1: Angle AOB has vertex O.
Step 2: The two rays forming this angle are OA and OB.

AnswerAngle AOB is the angle between Ray OA and Ray OB.

Exercise 4.4 — Triangles

Question 1: How many sides does a triangle have? What are they called?

Solution

Step 1: A triangle is formed by three line segments.
Step 2: Each line segment is a side of the triangle.

AnswerA triangle has 3 sides. In triangle ABC, the sides are AB, BC, and CA. They are called the sides of the triangle.

Question 2: How many vertices does a triangle have? Name them from the figure.

Solution

Step 1: A vertex is a point where two sides of a triangle meet.
Step 2: Since a triangle has 3 sides, it has 3 vertices.

AnswerA triangle has 3 vertices. In triangle PQR, the vertices are P, Q, and R.

Question 3: How many angles does a triangle have? Name them from the figure.

Solution

Step 1: An angle in a triangle is formed at each vertex by two sides.
Step 2: Since there are 3 vertices, there are 3 angles.

AnswerA triangle has 3 angles. In triangle ABC, the angles are angle A (BAC), angle B (ABC), and angle C (BCA).

Question 4: Write the names of: (a) Three sides (b) Three angles (c) Three vertices of triangle LMN

Solution

(a) Three sides of triangle LMN:

AnswerSide LM, Side MN, Side NL.

(b) Three angles of triangle LMN:

AnswerAngle L (angle MLN), Angle M (angle LMN), Angle N (angle LNM).

(c) Three vertices of triangle LMN:

AnswerVertex L, Vertex M, Vertex N.

Question 5: Can a triangle have: (a) Two right angles? (b) All angles less than 60 degrees? (c) All angles greater than 60 degrees?

Solution

(a) Can a triangle have two right angles?

Step 1: The sum of angles in a triangle is 180 degrees.
Step 2: If two angles are 90 degrees each, their sum = 90 + 90 = 180 degrees.
Step 3: The third angle would be 180 - 180 = 0 degrees, which is not possible.

AnswerNo, a triangle cannot have two right angles.

(b) Can all angles be less than 60 degrees?

Step 1: If all three angles are less than 60 degrees, their maximum total would be less than 60 + 60 + 60 = 180 degrees.
Step 2: But the sum must be exactly 180 degrees.

AnswerNo, all angles cannot be less than 60 degrees.

(c) Can all angles be greater than 60 degrees?

Step 1: If all three angles are greater than 60 degrees, their minimum total would be greater than 60 + 60 + 60 = 180 degrees.
Step 2: But the sum must be exactly 180 degrees.

AnswerNo, all angles cannot be greater than 60 degrees.

Exercise 4.5 — Circles

Question 1: From the figure, identify: (a) The centre of the circle (b) Three radii (c) A diameter (d) A chord (e) Two points in the interior (f) A point in the exterior

Solution

(a) The centre of the circle:

Step 1: The centre is the fixed point from which every point on the circle is at the same distance.

AnswerPoint O is the centre of the circle.

(b) Three radii:

Step 1: A radius is a line segment from the centre to any point on the circle.

AnswerRadius OA, Radius OB, Radius OC.

(c) A diameter:

Step 1: A diameter passes through the centre and connects two points on the circle.
Step 2: Its length equals twice the radius.

AnswerDiameter AC (it passes through centre O and has endpoints on the circle).

(d) A chord:

Step 1: A chord is any line segment whose both endpoints lie on the circle.

AnswerChord ED (any segment connecting two points on the circle; a diameter is also a chord).

(e) Two points in the interior:

Step 1: Points inside the circle (closer to the centre than the radius).

AnswerPoints P and Q (any points inside the circle).

(f) A point in the exterior:

Step 1: Points outside the circle (farther from the centre than the radius).

AnswerPoint R (any point outside the circle).

Question 2: Is every diameter of a circle also a chord? Is every chord also a diameter?

Solution

Step 1: A chord connects any two points on the circle.
Step 2: A diameter is a chord that passes through the centre.

AnswerYes, every diameter is a chord because its endpoints lie on the circle. But not every chord is a diameter — only chords that pass through the centre are diameters.

Question 3: What is the difference between a chord and a diameter?

Solution

Step 1: A chord is any line segment joining two points on the circle.
Step 2: A diameter is a special chord that passes through the centre.

AnswerA diameter is the longest chord of a circle and it always passes through the centre. A chord can be any size, but a diameter is always twice the radius.

Exercise 4.6 — Polygons and Quadrilaterals

Question 1: How many sides does a quadrilateral have? What is a quadrilateral?

Solution

Step 1: A polygon with four sides is called a quadrilateral.
Step 2: The word "quadrilateral" comes from "quad" (four) and "lateral" (sides).

AnswerA quadrilateral has 4 sides. It is a closed figure formed by four line segments.

Question 2: Name the sides, vertices, and diagonals of quadrilateral PQRS.

Solution

Step 1: A quadrilateral has 4 sides and 4 vertices.
Step 2: A diagonal connects opposite vertices.

Answer

Sides: PQ, QR, RS, SP
Vertices: P, Q, R, S
Diagonals: PR and QS

Question 3: How many diagonals does a quadrilateral have?

Solution

Step 1: A diagonal connects two non-adjacent vertices.
Step 2: In a quadrilateral with vertices P, Q, R, S, the diagonals are PR and QS.

AnswerA quadrilateral has 2 diagonals.

Question 4: Match the following: (a) Triangle (b) Quadrilateral (c) Pentagon (d) Hexagon — with (i) 4 sides (ii) 6 sides (iii) 3 sides (iv) 5 sides

Solution

Step 1: Identify the number of sides for each polygon.
Step 2: Triangle = 3 sides, Quadrilateral = 4 sides, Pentagon = 5 sides, Hexagon = 6 sides.

Answer

(a) Triangle matches with (iii) 3 sides
(b) Quadrilateral matches with (i) 4 sides
(c) Pentagon matches with (iv) 5 sides
(d) Hexagon matches with (ii) 6 sides

Question 5: What is a polygon? Is a circle a polygon?

Solution

Step 1: A polygon is a closed figure made entirely of line segments (straight sides).
Step 2: A circle is made of a curved line, not straight line segments.

AnswerA polygon is a closed figure formed by line segments only. A circle is not a polygon because it is made of a curved line, not straight line segments.

Common Mistakes

1. Confusing line, line segment, and ray. A line has no endpoints (infinite both ways), a line segment has two endpoints, and a ray has one endpoint. Always check: how many endpoints does it have?

2. Saying "a point has some size." A point has no length, breadth, or thickness — it is only a location. Never measure a point.

3. Calling a diameter "just a chord" without qualification. A diameter IS a chord, but it is the longest chord. Not every chord is a diameter — only those passing through the centre.

4. Forgetting that a triangle has exactly 3 sides, 3 angles, and 3 vertices. Do not count more or fewer. A triangle is always formed by exactly three line segments.

5. Thinking parallel lines can meet "eventually." Parallel lines never intersect, no matter how far you extend them. If lines meet, they are not parallel.

6. Confusing interior and exterior of a circle with the circle itself. The circle is the boundary (the curved line). Points inside the boundary are in the interior; points outside are in the exterior.

Exam Checklist

[ ] I can define and identify a point, line, line segment, and ray with correct notation.
[ ] I can distinguish between intersecting lines and parallel lines.
[ ] I can name an angle using three letters with the vertex in the middle (e.g., angle ABC).
[ ] I can identify and name the sides, vertices, and angles of a triangle.
[ ] I can explain why a triangle cannot have two right angles or all angles less than 60 degrees.
[ ] I can identify the centre, radius, diameter, and chord of a circle.
[ ] I understand the difference between a chord and a diameter.
[ ] I can define a polygon and distinguish polygons from non-polygons (like circles).
[ ] I can name the sides, vertices, and diagonals of a quadrilateral.
[ ] I know the number of sides in a triangle (3), quadrilateral (4), pentagon (5), and hexagon (6).

Frequently Asked Questions

A point is a location in space that has no size — it is represented by a dot and named with a capital letter.

NCERT Solutions for Class 6 Maths Chapter 4 Basic Geometrical Ideas | KalviGavani

NCERT Solutions for Class 6 Maths Chapter 4 — Basic Geometrical Ideas

Key Concepts

Exercise 4.1 — Points

Exercise 4.2 — Lines and Line Segments

Exercise 4.3 — Rays and Angles

Exercise 4.4 — Triangles

Exercise 4.5 — Circles

Exercise 4.6 — Polygons and Quadrilaterals

Common Mistakes

Exam Checklist

Frequently Asked Questions

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