Basics of Geometry

Class 6 Maths Chapter: Basics of Geometry – Introduction

Geometry is one of the most interesting branches of Mathematics because it helps us understand the shapes, patterns, and spaces around us. In Class 6 Maths, the chapter Basics of Geometry introduces students to the fundamental ideas that form the foundation of geometric learning. This chapter explains important concepts such as points, lines, line segments, rays, curves, angles, and different types of geometrical figures. These concepts help students observe and describe objects found in daily life, from roads and buildings to maps and designs.

The chapter begins with the simplest geometrical idea—a point—which represents an exact location. Students then learn about lines, rays, and line segments, understanding how they differ from one another. The study of curves, simple closed figures, polygons, and angles further develops spatial reasoning and visualisation skills. Through practical examples and diagrams, learners discover how geometry is connected to the world around them. Must-read Class 6 Maths Notes and NCERT Solutions for Class 6, NCERT exemplar for class 6. 

A strong understanding of the Basics of Geometry is essential because it serves as the starting point for advanced topics such as triangles, quadrilaterals, circles, symmetry, mensuration, and coordinate geometry. By mastering this chapter, students build confidence in drawing, identifying, and analysing geometric figures, which strengthens their overall mathematical thinking and problem-solving abilities.

Definitions to Learn First in Geometry

Before diving into complex concepts, students must understand these fundamental building blocks of geometry:

1. Point

A point is the most basic element of geometry. It represents a specific location in space and has no length, breadth, or thickness. A dot made with a sharp pencil roughly represents a point.

Main Points:

• Has position but no dimensions
• Denoted by capital letters (A, B, C, D)
• Read as "Point A," "Point B," etc.

Examples: Corners of a square, tip of an ice cone, corner of a book

2. Line

A line is a straight path that extends indefinitely in both directions. It has no endpoints and continues forever on both sides.

Key Characteristics:

• Has no definite length (infinite)
• Cannot be drawn completely on paper
• Represented with arrows on both ends
• Denoted as line AB or ↔AB

3. Ray

A ray is a portion of a line that starts at one point (called the initial point or starting point) and extends endlessly in one direction.

Key Characteristics:

• Has one endpoint (initial point)
• Has no definite length
• Denoted as ray OA or →OA

4. Line Segment

A line segment is a part of a line with two definite endpoints. Unlike a line, it has a measurable length.

Key Characteristics:

• Has two endpoints
• Has a definite length
• Can be drawn on paper
• Denoted as segment AB or AB̅

5. Plane

A plane is a flat surface that extends indefinitely in all directions. Think of it as an infinite flat sheet with no thickness.

Examples: Surface of a table, wall of a room, floor of a classroom

6. Collinear Points

Points that lie on the same line are called collinear points. If points P, Q, R, and S all lie on line l, they are collinear.

7. Concurrent Lines

Three or more lines passing through the same point are called concurrent lines. The common point is called the point of concurrence.

8. Coplanar Lines

Two or more lines that lie in the same plane are called coplanar lines.

Comparison: Line vs Ray vs Line Segment

Types of Lines

Parallel Lines

Lines that lie in the same plane and never intersect are called parallel lines. The distance between them remains constant throughout.

Properties:

• Two lines parallel to the same line are parallel to each other
• If two lines are perpendicular to the same line, they are parallel to each other
• The angle between two parallel lines is zero
• Every line is parallel to itself

Perpendicular Lines

If two lines intersect at a right angle (90°), they are called perpendicular lines.

Symbol: ⊥ (AB ⊥ CD means line AB is perpendicular to line CD)

Examples: Letter 'T', letter 'L', table legs perpendicular to table top

Intersecting Planes

• Parallel planes: Planes that do not intersect (e.g., floor and ceiling of a room)
• Intersecting planes: Planes that intersect along a line (e.g., two adjacent walls of a room)

Transversal and Related Angles

A transversal is a line that intersects two or more lines in a plane at different points.

When a transversal intersects two lines, it creates 8 angles. These angles have special names and relationships:

Interior Angles

Angles that lie between the two lines (have the segment between the lines as one arm).

Exterior Angles

Angles that lie outside the two lines.

Corresponding Angles

A pair of angles where one is interior and one is exterior, both on the same side of the transversal, and they are not adjacent.

Pairs: ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8

Alternate Interior Angles

Interior angles on opposite sides of the transversal that are not adjacent.

Pairs: ∠3 & ∠5, ∠4 & ∠6

Alternate Exterior Angles

Exterior angles on opposite sides of the transversal that are not adjacent.

Pairs: ∠1 & ∠7, ∠2 & ∠8

Properties When Lines are Parallel

When a transversal cuts two parallel lines:

1. Alternate interior angles are equal (∠4 = ∠6, ∠3 = ∠5)
2. Corresponding angles are equal (∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, ∠4 = ∠8)
3. Alternate exterior angles are equal (∠1 = ∠7, ∠2 = ∠8)
4. Sum of interior angles on same side of transversal = 180° (∠3 + ∠6 = 180°, ∠4 + ∠5 = 180°)
5. Sum of exterior angles on same side of transversal = 180° (∠1 + ∠8 = 180°, ∠2 + ∠7 = 180°)

Angles and Their Measurement

What is an Angle?

An angle consists of two different rays with the same initial point. The common initial point is called the vertex, and the rays are called the arms or sides of the angle.

Types of Angles

Special Angle Relationships

Complementary Angles: Two angles whose sum equals 90°

• Examples: (30°, 60°), (50°, 40°), (45°, 45°)

Supplementary Angles: Two angles whose sum equals 180°

• Examples: (60°, 120°), (135°, 45°), (90°, 90°)

Congruent/Equal Angles: Two angles with the same measure

Measuring Line Segments

Method 1: Comparison by Observation

Simply looking at two line segments to determine which is longer. This method is unreliable for segments of similar length.

Method 2: Comparison by Tracing

Trace one segment on transparent paper and place it along the other segment to compare lengths.

Method 3: Using a Graduated Ruler

Place the ruler along the segment with zero mark at one endpoint. Read the mark at the other endpoint.

Method 4: Using a Divider

Place the divider's endpoints on the segment's endpoints. Without disturbing the opening, place one end on the ruler's zero mark and read the other end's position.

Congruent Segments

Two line segments with the same length are called congruent segments.

If AB = CD, then AB ≅ CD

Difference Between Plane Geometry and Solid Geometry

Understanding the distinction between these two branches is essential for Class 6 students:

Plane Geometry (2D Geometry)

Definition: Deals with flat shapes that exist in two dimensions (length and width).

Characteristics:

• Shapes lie on a single plane
• Can be drawn on paper
• Has only length and breadth (area)
• No thickness or depth

Examples:

• Triangle
• Square
• Rectangle
• Circle
• Pentagon
• Hexagon

Measurements:

• Perimeter (boundary length)
• Area (surface coverage)

Solid Geometry (3D Geometry)

Definition: Deals with three-dimensional objects that have length, width, and height.

Characteristics:

• Objects occupy space
• Have volume (capacity)
• Cannot be completely represented on flat paper
• Have thickness/depth

Examples:

• Cube
• Cuboid
• Sphere
• Cylinder
• Cone
• Pyramid

Measurements:

• Surface Area
• Volume
• Lateral Surface Area

Differences Table

How to Calculate Area and Perimeter of Common Shapes

Perimeter

Definition: The total length of the boundary of a closed figure.

Area

Definition: The amount of surface enclosed within a closed figure, measured in square units.

Formulas for Common Shapes

Triangle

• Perimeter: P = a + b + c (sum of all three sides)
• Area: A = ½ × base × height

Example: For a triangle with sides 10m, 40m, and 55m:

• Perimeter = 10 + 40 + 55 = 105m

Square

• Perimeter: P = 4 × side = 4s
• Area: A = side × side = s²

Rectangle

• Perimeter: P = 2(length + breadth) = 2(l + b)
• Area: A = length × breadth = l × b

Circle

• Circumference (Perimeter): C = 2πr = πd
• Area: A = πr²

Where r = radius, d = diameter, π ≈ 22/7 or 3.14

Essential Geometry Formulas to Memorize for Exams

Quick Reference Table

Clock Angle Formula

Angle between consecutive digits on clock = 30°

In one hour, the minute hand turns 360° In one minute, the minute hand turns 6° (360° ÷ 60)

Polygons

What is a Polygon?

A polygon is a simple closed figure formed by three or more line segments. The line segments are called sides, and their meeting points are called vertices.

Types of Polygons by Number of Sides

Convex vs Concave Polygons

Convex Polygon: All interior angles are less than 180°

Concave Polygon: At least one interior angle is greater than 180°

Regular Polygon

A polygon with all sides equal and all angles equal is called a regular polygon.

Diagonals

A line segment joining two non-adjacent vertices is called a diagonal.

Triangles

Definition

A triangle is a polygon with three sides and three angles. It is formed by three non-collinear points connected by line segments.

Classification by Sides

Classification by Angles

Angle Sum Property

The sum of all three angles of a triangle is always 180° (or two right angles).

∠A + ∠B + ∠C = 180°

Exterior Angle Property

An exterior angle of a triangle equals the sum of the two opposite interior angles.

If side BC is extended to point D, then: ∠ACD = ∠A + ∠B

Congruent Triangles

Two triangles are congruent if all corresponding angles and sides are equal.

Simple Practice Problems with Step-by-Step Solutions

Problem 1: Finding Angles in Parallel Lines

Question: If p ∥ q ∥ r and ∠1 = 80°, find ∠2.

Solution:

1. Since p ∥ q, ∠1 and ∠3 are corresponding angles
2. Therefore, ∠1 = ∠3 = 80°
3. Sum of interior angles on the same side of transversal = 180°
4. ∠3 + ∠5 = 180°
5. ∠5 = 180° - 80° = 100°
6. Since p ∥ r, alternate interior angles are equal
7. ∠5 = ∠2 = 100°

Answer: ∠2 = 100°

Problem 2: Finding Exterior Angle of Triangle

Question: In the given figure, if ∠B = 70° and ∠BCA = 130°, find the exterior angle x at vertex A.

Solution:

1. ∠y + 130° = 180° (linear pair)
2. ∠y = 180° - 130° = 50°
3. Exterior angle = sum of opposite interior angles
4. x = 70° + 50° = 120°

Answer: x = 120°

Problem 3: Interior Angles of a Pentagon

Question: Find the sum of interior angles of a pentagon.

Solution:

1. Number of sides (n) = 5
2. Sum of interior angles = (2n - 4) × 90°
3. = (2 × 5 - 4) × 90°
4. = (10 - 4) × 90°
5. = 6 × 90°
6. = 540°

Answer: Sum of interior angles = 540°

Problem 4: Finding Type of Regular Polygon

Question: Each interior angle of a regular polygon is 60°. What type of polygon is it?

Solution:

1. Each interior angle = [(2n - 4) × 90°] ÷ n
2. 60° = [(2n - 4) × 90°] ÷ n
3. 60n = (2n - 4) × 90
4. 60n = 180n - 360
5. 360 = 180n - 60n
6. 360 = 120n
7. n = 3

Answer: Since n = 3 and each angle is 60°, it is an equilateral triangle.

Problem 5: Ratio of Interior to Exterior Angle

Question: The ratio of interior to exterior angle of a regular polygon is 7:2. Find the number of sides.

Solution:

1. Interior angle = [(2n - 4) × 90°] ÷ n
2. Exterior angle = 360° ÷ n
3. Ratio: [(2n - 4) × 90°/n] ÷ [360°/n] = 7/2
4. (2n - 4) × 90° / 360° = 7/2
5. (2n - 4) / 4 = 7/2
6. 2(2n - 4) = 28
7. 4n - 8 = 28
8. 4n = 36
9. n = 9

Answer: The polygon has 9 sides (Nonagon).

Problem 6: Cost of Fencing

Question: Find the cost of fencing a triangular field at ₹30 per meter. The sides measure 10m, 40m, and 55m.

Solution:

1. Perimeter = 10 + 40 + 55 = 105m
2. Cost = 105 × ₹30 = ₹3,150

Answer: Total cost = ₹3,150

Problem 7: Clock Angles

Question: Through how many degrees does the minute hand turn in: (a) One hour? (b) One minute?

Solution: (a) In one hour, minute hand completes one full rotation = 360°

(b) In one minute = 360° ÷ 60 = 6°

Alternatively, the angle between two consecutive digits = 30° In 5 minutes, minute hand moves 30° In 1 minute = 30° ÷ 5 = 6°

Problem 8: Angles of a Triangle

Question: The angles of a triangle are in ratio 1:3:5. Find the measure of each angle.

Solution:

1. Let the angles be x, 3x, and 5x
2. Sum of angles = 180°
3. x + 3x + 5x = 180°
4. 9x = 180°
5. x = 20°

Answer: The angles are 20°, 60°, and 100°

Problem 9: Finding Complements and Supplements

Question: (a) Find the complement of 70° (b) Find the supplement of 150°

Solution: (a) Complement of 70° = 90° - 70° = 20°

(b) Supplement of 150° = 180° - 150° = 30°

Problem 10: Counting Line Segments

Question: How many line segments are in a figure with 4 collinear points A, B, C, D?

Solution: Line segments: AB, BC, CD, AC, BD, AD

Answer: 6 line segments

Basics of Geometry Worksheet

Section A: Fill in the Blanks

1. A point has __________ dimensions.
2. Points lying on the same line are called __________ points.
3. The sum of angles in a triangle is __________.
4. Two angles are complementary if their sum equals __________.
5. A polygon with 6 sides is called a __________.
6. The angle between two parallel lines is __________.
7. An angle of 180° is called a __________ angle.
8. Two segments with equal length are called __________ segments.
9. A line that intersects two or more lines is called a __________.
10. In an equilateral triangle, each angle measures __________.

Section B: True or False

1. A ray has two endpoints.
2. Every line is parallel to itself.
3. The sum of exterior angles of any polygon is 360°.
4. An obtuse angle is greater than 90° but less than 180°.
5. Corresponding angles are always supplementary.

Section C: Multiple Choice Questions

1. The number of points on the boundary of a square is: (a) 4 (b) 2 (c) 8 (d) Infinite
2. A point on a line divides it into ____ rays: (a) 2 (b) 3 (c) 4 (d) 5
3. The arms of an angle are: (a) Segments (b) Lines (c) Rays (d) None of these
4. The distance between two intersecting lines is: (a) 1 (b) 0 (c) Undefined (d) Cannot be determined
5. If a transversal intersects two parallel lines, the sum of exterior angles on the same side is: (a) 90° (b) 360° (c) 180° (d) 0°

Section D: Short Answer Questions

1. Define collinear and concurrent lines with examples.
2. What is the difference between a line segment and a ray?
3. If ∠A and ∠B are supplementary and ∠A = 65°, find ∠B.
4. Find the sum of interior angles of a hexagon.
5. Two angles of a triangle are 45° and 75°. Find the third angle.

Section E: Long Answer Questions

1. Prove that the sum of angles in a triangle is 180°.
2. If two parallel lines are cut by a transversal, prove that alternate interior angles are equal.
3. Find the number of sides of a regular polygon whose each interior angle is 140°.
4. The angles of a quadrilateral are in ratio 2:3:4:6. Find each angle.

Basics of Geometry Test

Time: 45 minutes | Maximum Marks: 40

Section A: Objective Questions (1 mark each = 10 marks)

1. The sum of all angles around a point is: (a) 0° (b) 180° (c) 360° (d) 90°
2. In ΔDEF, if ∠D = 45° and ∠E < 45°, the triangle is: (a) Acute (b) Right (c) Obtuse (d) Cannot be determined
3. With angle measurements 50°, 70°, 60°, can a triangle be constructed? (a) Yes (b) No (c) Sometimes (d) Insufficient data
4. The measures of two angles of a triangle are 75° and 65°. The third angle is: (a) 45° (b) 40° (c) 50° (d) 55°
5. Each interior angle of an equilateral triangle is: (a) 45° (b) 60° (c) 90° (d) 120°
6. Two lines perpendicular to the same line are: (a) Perpendicular to each other (b) Parallel to each other (c) Intersecting (d) Concurrent
7. A reflex angle is: (a) Less than 90° (b) Between 90° and 180° (c) Between 180° and 360° (d) Equal to 360°
8. The exterior angle of a triangle is 110°. One of the opposite interior angles is 40°. The other is: (a) 70° (b) 80° (c) 60° (d) 50°
9. In a concave polygon, at least one interior angle is: (a) Less than 90° (b) Equal to 90° (c) Less than 180° (d) Greater than 180°
10. Through how many degrees does the minute hand turn in 20 minutes? (a) 60° (b) 120° (c) 180° (d) 240°

Section B: Short Answer Questions (2 marks each = 10 marks)

1. Find the complement of 35° and the supplement of 110°.
2. Name all the line segments in a figure with 4 collinear points P, Q, R, S.
3. If angles on a straight line are (2x + 10)° and (3x + 20)°, find x.
4. The angles of a triangle are x°, (x + 20)°, and (x + 40)°. Find all angles.
5. Find each interior angle of a regular octagon.

Section C: Long Answer Questions (4 marks each = 20 marks)

1. In the figure, if PQ ∥ RS and a transversal makes an angle of 65° with PQ, find all eight angles formed.
2. The ratio of the exterior to the interior angle of a regular polygon is 1:4. Find: (a) each interior angle (b) each exterior angle (c) the number of sides
3. A triangular field has sides 120 m, 150 m, and 180m. Find: (a) The perimeter (b) Cost of fencing at ₹25 per meter (c) Cost of three rounds of fencing
4. Prove that the exterior angle of a triangle equals the sum of opposite interior angles.
5. In the given figure, prove that x + y + z = 360° when three lines intersect at different points forming a triangle with exterior angles.

Basics of Geometry PDF

For a downloadable version of these comprehensive notes, including all diagrams, formulas, and practice problems, students can access the complete PDF resource.

What's Included in the PDF:

• All definitions with clear diagrams
• Comparison tables for quick revision
• Step-by-step solved examples
• Practice worksheets with answers
• Formula sheets for exam preparation
• Chapter-wise important questions

Answer Keys

Worksheet Answers

Section A: 1. Zero/No, 2. Collinear, 3. 180°, 4. 90°, 5. Hexagon, 6. 0°/Zero, 7. Straight, 8. Congruent, 9. Transversal, 10. 60°

Section B: 1. False, 2. True, 3. True, 4. True, 5. False

Section C: 1. (d), 2. (a), 3. (c), 4. (b), 5. (c)

Test Answers

Section A: 1. (c), 2. (c), 3. (a), 4. (b), 5. (b), 6. (b), 7. (c), 8. (a), 9. (d), 10. (b)

Section B: 11. Complement = 55°, Supplement = 70° 12. PQ, QR, RS, PR, QS, PS (6 segments) 13. x = 30° 14. 40°, 60°, 80° 15. 135°

Tips for Exam Preparation

1. Memorize key formulas – Create flashcards for angle sum formulas
2. Practice diagrams – Draw accurate figures while solving problems
3. Understand properties – Don't just memorize; understand why rules work
4. Solve previous year questions – Identify frequently asked question types
5. Use the correct notation – Distinguish between line (↔), ray (→), and segment (—)
6. Check your work – Verify angles add up correctly
7. Time management – Practice solving within time limits