# NCERT Solutions for Class 9 Maths Chapter 5 – Introduction to Euclid’s Geometry

NCERT Solutions for Class 9 Maths Chapter 5 will help you learn the concepts of Euclid’s Geometry that are practiced since ancient times. This Chapter of NCERT Solutions for Class 9 Maths helps you differentiate between axioms or postulates and theorems while studying the definitions of the Euclidean Geometry. You will also be amazed to know how the four postulates of Euclid failed to prove his fifth postulate yet opened doors to new concepts of Non-Euclidean Geometry.

NCERT Class 9 Maths Chapter 5 has 9 questions in 2 exercises which include topics like drawing a line and defining perpendicular lines, the radius of a circle, square, parallel lines and a line segment. The Chapter includes short answer type questions as well. This Chapter is easy to understand with mostly theoretical solutions and can help you enhance your performance in the Maths exam.

Instasolv has prepared up to date NCERT Solutions for Chapter 5 of Class 9 Maths book adhering to the guidelines provided by the CBSE board. You can use the language written by Instasolv experts in your exams as well and get good marks for writing on-point answers. Preparing NCERT Maths Solutions will save you from the dangers of rote learning by helping you grasp the core gist of Chapter 5 without much hustle.

## Major Topics for NCERT Solutions for Class 9 Maths Chapter 5

**Introduction to Geometry **

- The word ‘geometry’ is an amalgamation of two Greek words, ‘geo’ meaning ‘the earth’, and ‘metrein’ meaning ‘to measure’. This branch of Maths has been explored by various civilizations such as in India, Egypt, Babylonia, China, Greece, the Incas etc. to find the solutions to various practical problems since ancient times.
- There has been much archaeological evidence in support of the study of Geometry in the Indus Valley civilization.
- Extensive and practical use of Geometry has also been found in the later Vedic period in the Sulbasutras. The Sri Yantra mentioned in the Atharva Veda contains nine interwoven isosceles triangles which are of astrological importance even today.
- While in the Indian subcontinent, evidence of practical geometry has been found since time immemorial, the Greeks seemed to be more interested in deriving the truth using deductive reasoning. Thales and his pupil, Pythagoras are some of the most prominent mathematicians who devised many theories of geometry around 572 B.C. which we study and apply even today.
- Euclid was a Mathematics teacher at Alexandria in Egypt. He compiled all the known work in Geometry of his era and arranged it in his famous thesis called ‘Elements’. ‘Elements’ is divided into 13 chapters. Each Chapter constitutes a book. These books were a revolution in this branch of Maths in the coming times.

**Euclid’s Definitions**

**Surfaces:** The boundaries of the solids that have a defined shape, size, position and have mobility. These solids with negligible thickness are known as surfaces. The boundaries of these surfaces separate different spaces and are known as ‘bends’ or ‘straight lines’. These lines have endpoints called points. Points have no dimensions or sizes.

**Line:** A line does not have a breadth and has only one dimension that is the length. In a straight line, the points lie equitably on the line itself. A surface comprises of only length and breadth, and no thickness. The straight lines together on a surface, make up a plane surface

The criticism of these definitions is that some terms such as breadth and length remain undefined. Similarly, in modern geometry, we take a point, a line and a plane, as undefined.

**Axioms, Postulates and Theorems**

- According to Euclid, Axioms were assumptions that did not have any proof and were not specifically related to geometry but had relevance in maths.
- Euclid used the term ‘postulates’ for the assumptions that he used explicitly for geometry.
- An arrangement of Axioms is known as ‘consistent’ in Euclid’s Geometry.
- After establishing the axioms and postulates, Euclid used them to prove further results using the method of deductive reasoning. These new proofs were called ‘theorems’ or propositions.

**Euclid’s Axioms**

- The axiom says that if two objects are equal to the same object, then they will be equal to one another as well.
- If we will add something equivalent to two equal things respectively then their sums will be equal too.
- If we will subtract two equivalent values from two equal values, then their remainder will be equal too.
- A part will always be smaller than the whole.
- Things that are double of, or are half of the equal things are equal to one another.

**Euclid’s Postulates**

- There can be one and only one straight line between two points.
- A line that terminates can be extended indefinitely.
- A circle might have a radius and a centre of any value.
- All the angles that constitute the set of right angles are equal to each other.
- In the event that a straight line falls on two straight lines and makes the interior angles on the same side of it taken together less than two right angles, then if these two straight lines are extended indefinitely, they will meet on this line.

## Highlights of CBSE NCERT Class 9 Maths Chapter 5

**Theorem: Two different lines can have one and only one point in common.**

**Equivalent versions of Euclid’s 5th postulate:**

(i) There is always a line m parallel to any line l passing through a point P that does not lie on l.

(ii) If two lines intersect each other, they cannot be parallel to each other.

**Exercise-wise discussion of NCERT Solutions for Class 9 Maths Chapter 5**

- Exercise 5.1 of NCERT Solutions for Class 9 Maths Chapter 5 includes State True/False Questions in addition to the identification and applications of Euclid’s definitions.
- In the exercise solutions for this Chapter, you will also be required to write an explanation about your understanding of some concepts such as explaining the proof of a theorem using Euclid’s Postulates.
- Exercise 5.2 has only two questions and both of them require a proper understanding of Euclid’s Fifth Postulate.
- Our subject matter experts have provided all the NCERT Solutions for Chapter 5 of Class 9 Maths in a simple manner with an objective to not only help you score good marks but also assist you to build a strong base in geometry.
- The NCERT exercise solutions are updated according to the latest CBSE exam pattern. We make sure that we leave you with no doubts but an opportunity to polish Class 9 maths concepts.