euclid geometry class 9 assignment

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Chapter 5 Class 9 Introduction to Euclid's Geometry

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Updated for new NCERT - 2023-24 Edition.

Get NCERT Solutions to all exercise questions and examples of Chapter 5 Class 9 Introduction to Euclid's Geometry. All questions have been solved in an easy to understand way.

Euclid was a mathematician from Egypt who studied and explained geometry. In this chapter, we will learn Euclid's Geometry. Euclid's Geometry is a bit different from the geometry we study today. We will learn how Euclid explained Geometry and try to relate it with the geometry we study today.

In this chapter, we will study

• Euclid's definitions of Point, Line, Surface
• In these definitions, some terms cannot be explained. So we assume it to be true. These terms are called Axioms.
• Axioms are things which are universally true. We then study some of Euclid's Axioms.
• Postulates are things which are universally true in geometry. 
• We study Euclid's 5 postulates , and also Axiom 5.1 which is a result of Postulate 1
• Using Axioms and Postulates, we prove Theorems. Theorems can always be proved.
• Then, we prove Theorem 5.1 - Two distinct lines cannot have more than one point in common.
• We also different versions of Postulate 5 , and also rewrite it in our own way.

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NCERT Solutions Class 9 Maths Chapter 5 Introduction to Euclid's Geometry

The word geometry finds its origin from the Greek word ‘Geo’ means earth and ‘Metrein’ means to measure. It is an ancient branch of mathematics studied by various civilizations at different times. Euclid’s geometry is the study of solids and planes based on the axioms and postulates given by the Egyptian mathematician Euclid. It mainly deals with points , lines , circles , curves, angles , planes , solids, etc. NCERT solutions for class 9 maths chapter 5 Introduction to Euclid’s Geometry give a brief introduction to the origin of geometry and its link to present-day geometry. This class 9 maths NCERT solutions chapter 5 will help students widen their understanding of basic geometry concepts and their applications.

As geometry forms an important part of the math curriculum required for many fundamental calculations, every student must have a sound knowledge of this foundational topic. NCERT solutions class 9 maths chapter 5 Introduction To Euclid’s Geometry offers a complete reference to all topics enabling students to master all key fundamentals. You can find some of these in the exercises given below.

• NCERT Solutions Class 9 Maths Chapter 5 Ex 5.1
• NCERT Solutions Class 9 Maths Chapter 5 Ex 5.2

NCERT Solutions for Class 9 Maths Chapter 5 PDF

NCERT solutions maths for class 9 chapter 5 are well structured in a downloadable PDF file for students to efficiently cover all the topics as well as subtopics in detail. To prepare the questions provided in these exercises, click on the links of the pdf files as given below.

☛ Download Class 9 Maths NCERT Solutions Chapter 5

NCERT Class 9 Maths Chapter 5   Download PDF

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

Geometry is the basis of understanding measurement and is certainly applied in many practical situations, making it an indispensable part of mathematics studies. Students can study this important topic with the NCERT Solutions Class 9 Maths Chapter 5 Introduction to Euclid’s geometry to broaden their knowledge of basic geometry concepts. To learn and practice with NCERT Solutions Class 9 Maths Chapter 5 Introduction To Euclid’s Geometry, try the detailed exercises-wise questions given below.

• Class 9 Maths Chapter 5 Ex 5.1 - 7 Questions
• Class 9 Maths Chapter 5 Ex 5.2 - 2 Questions

☛ Download Class 9 Maths Chapter 5 NCERT Book

Topics Covered: The important topics covered in class 9 maths NCERT solutions Chapter 5 are a brief history of geometry, introduction to Euclid’s geometry, Euclid’s definitions, axioms, postulates, and equivalent versions of Euclid’s fifth postulate.

Total Questions: Class 9 Maths Chapter 5 Introduction To Euclid’s Geometry has a total of 9 questions, which mainly cover questions based on validation of statements, definitions, and proving sums based on axioms and postulates. These exercises have sub-questions to facilitate the fair practice of key topics.

List of Formulas in NCERT Solutions Class 9 Maths Chapter 5

NCERT solutions class 9 maths Chapter 5 are comprehensive resources designed by experts to provide step-by-step learning of this topic. These solutions are a convenient source of establishing a firm maths foundation for higher grades. NCERT solutions class 9 maths are beneficial for students in many ways. The most important concepts of Euclid’s Geometry covered in the NCERT solutions for class 9 maths Chapter 5 are:

• If A and B are equal to C, it implies A = B.
• If ‘B’ is a part of a quantity ‘A’, then ‘A’ can be represented as the sum of ‘B’ and a third quantity ‘C’. Thus, symbolically, when A is greater than B, there is a C such that A = B + C.

Important Questions for Class 9 Maths NCERT Solutions Chapter 5

Video Solutions for Class 9 Maths NCERT Chapter 5

FAQs on NCERT Solutions Class 9 Maths Chapter 5

What is the importance of ncert solutions for class 9 maths chapter 5 introduction to euclid’s geometry.

NCERT Solutions Class 9 Maths are created by experts after thorough research to provide precise knowledge of Euclid’s Geometry. NCERT solutions are also recommended by CBSE as they offer quality learning. The format of NCERT solutions is quite lucid and apt for understanding complex topics. With multiple examples and sample problems provided in the NCERT solutions, students can easily attain the right approach to excel in board exams. The exceptional quality of NCERT solutions makes them a highly reliable resource for learning and revising for exams.

What are the Important Topics Covered in Class 9 Maths NCERT Solutions Chapter 5?

NCERT Solutions Class 9 Maths Chapter 5 covers a brief history of geometry and its origin, a basic understanding of Euclid’s Geometry, its definitions, axioms, and postulates . NCERT solutions class 9 Maths Chapter 5 Introduction To Euclid’s Geometry is an excellent resource to cover all core topics with suitable examples and sample problems.

Do I Need to Practice all Questions Provided in NCERT Solutions Class 9 Maths Introduction To Euclid’s Geometry?

With a thorough practice of all the questions and sample problems listed in the NCERT Solutions Class 9 Maths Chapter 5, students can quickly gain an understanding of all the key notations. When students possess a sound knowledge of this topic, they can quickly implement the knowledge of the related topics. Practicing complete questions included in these solutions will also provide the necessary guidance and confidence to face various competitive exams.

How Many Questions are there in Class 9 Maths NCERT Solutions Chapter 5 Introduction To Euclid’s Geometry?

NCERT Class 9 Maths Chapter 5 Introduction To Euclid’s Geometry has nine questions in 2 exercises that efficiently cover all the topics in detail. The questions in these exercises are well structured to offer a clear understanding of the fundamental properties of lines, points, planes, and solids. These questions are subcategorized as long answer types and short ones.

What are the Important Formulas in NCERT Solutions Class 9 Maths Chapter 5?

The important concepts covered in the NCERT Solutions Class 9 Maths Chapter 5 are Euclid’s definitions, axioms, postulates, and theorems. Understanding these key notations through regular practice will help students to score better in exams.

Why Should I Practice NCERT Solutions Class 9 Maths Introduction To Euclid’s Geometry Chapter 5?

The CBSE class 9 maths exams are based on NCERT textbooks. By practicing with NCERT Solutions Class 9 Maths Chapter 5 Introduction To Euclid’s Geometry, students can ensure that they don’t miss out on any important topic from an exam point of view. Practicing NCERT solutions also help students to acquire the math skills that are beneficial for academics and practical life.

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Class 9 Mathematics Euclids Geometry Worksheets

We have provided below free printable  Class 9 Mathematics Euclids Geometry Worksheets  for Download in PDF. The worksheets have been designed based on the latest  NCERT Book for Class 9 Mathematics Euclids Geometry . These Worksheets for Grade 9 Mathematics Euclids Geometry  cover all important topics which can come in your standard 9 tests and examinations.  Free printable worksheets for CBSE Class 9 Mathematics Euclids Geometry , school and class assignments, and practice test papers have been designed by our highly experienced class 9 faculty. You can free download CBSE NCERT printable worksheets for Mathematics Euclids Geometry Class 9 with solutions and answers. All worksheets and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Euclids Geometry Class 9. Students can click on the links below and download all Pdf  worksheets for Mathematics Euclids Geometry class 9  for free. All latest Kendriya Vidyalaya  Class 9 Mathematics Euclids Geometry Worksheets  with Answers and test papers are given below.

Mathematics Euclids Geometry Class 9 Worksheets Pdf Download

Here we have the biggest database of free  CBSE NCERT KVS  Worksheets for Class 9  Mathematics Euclids Geometry . You can download all free Mathematics Euclids Geometry worksheets in Pdf for standard 9th. Our teachers have covered Class 9 important questions and answers  for Mathematics Euclids Geometry as per the latest curriculum for the current academic year. All test sheets question banks for Class 9 Mathematics Euclids Geometry and  CBSE Worksheets for Mathematics Euclids Geometry Class 9  will be really useful for Class 9 students to properly prepare for the upcoming tests and examinations. Class 9th students are advised to free download in Pdf all printable workbooks given below.

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Revision Notes on Introduction to Euclid’s Geometry

Introduction to euclid’s geometry.

He introduced the method of proving mathematical results by using deductive logical reasoning and the previously proved result.

He collected all his work in a book called “ Elements ”. This book is divided into thirteen chapters and each chapter is called a book.

Definitions of Euclid’s 

Euclid thought that the geometry is an abstract model of the world which we can see around us. Like the notions of line, plane, surface etc.

He had given these notions in the form of definitions-

1. Anything which has no component is called Point .

2. A length without breadth is called Line .

3. The endpoints of any line are called Points which make it line segment.

4. If a line lies evenly with the points on itself then it is called A Straight Line .

5. Any object which has length and breadth only is called Surface .

6. The edges of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.

Euclid’s Axioms and Postulates

Euclid assumed some properties which were actually ‘obvious universal truth’. He had bifurcated them in two types: Axioms and postulates.

Some common notions which are used in mathematics but not directly related to mathematics are called Axioms .

Some of the Axioms are-

1. If the two things are equal to a common thing then these are equal to one another.

If p = q and s = q, then p = s.

2. If equals are added to equals, the wholes are equal.

If p = q and we add s to both p and q then the result will also be equal.

p + s = q + s

3. If equals are subtracted from equals, the remainders are equal.

This is same as above, if p = q and we subtract the same number from both then the result will be the same.

p – s = q - s

4. Things which coincide with one another are equal to one another.

If two figures fit into each other completely then these must be equal to one another.

5. The whole is greater than the part.

This circle is divided into four parts and each part is smaller than the whole circle. This shows that the whole circle will always be greater than any of its parts.

6. Things which are double of the same things are equal to one another.

This shows that this is the double of the two semicircles, so the two semicircles are equal to each other.

7. Things which are halves of the same things are equal to one another. This is the vice versa of the above axiom.

The assumptions which are very specific in geometry are called Postulates.

There are five postulates by Euclid-

1. A straight line may be drawn from any one point to any other point.

This shows that a line can be drawn from point A to point B, but it doesn’t mean that there could not be other lines from these points.

2. A terminated line can be produced indefinitely.

This shows that a line segment which has two endpoints can be extended indefinitely to form a line.

3. A circle can be drawn with any centre and any radius.

This shows that we can draw a circle with any line segment by taking one of its points as a centre and the length of the line segment as the radius. As we have AB line segment, in which we took A as the centre and the AB as the radius of the circle to form a circle.

4. All right angles are equal to one another.

As we know that a right angle is equal to 90° and all the right angles are congruent because if any angle is not 90° then it is not a right angle.

As in the above figure ∠DCA =∠DCB =∠HE =∠HGF= 90°

5. Parallel Postulate

If there is a line segment which passes through two straight lines while forming two interior angles on the same side whose sum is less than 180°, then these two lines will definitely meet with each other if extended on the side where the sum of two interior angles is less than two right angles.

And if the sum of the two interior angles on the same side is 180° then the two lines will be parallel to each other.

Equivalent Versions of Euclid’s Fifth Postulate

1. Play fair’s Axiom

This says that if you have a line ‘l’ and a point P which doesn’t lie on line ‘l’ then there could be only one line passing through point P which will be parallel to line ‘l’. No other line could be parallel to line ‘l’ and passes through point P.

2. Two distinct intersecting lines cannot be parallel to the same line.

This also states that if two lines are intersecting with each other than a line parallel to one of them could not be parallel to the other intersecting line.

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Introduction to Euclids Geometry Class 9 Notes CBSE Maths Chapter 5 (Free PDF Download)

• Revision Notes
• Chapter 5 Introduction To Euclids Geometry

Class 9 Maths Revision Notes for Introduction to Euclids Geometry of Chapter 5 - Free PDF Download

CBSE Class 9 Maths Notes Chapter 5 Introduction to Euclid’s Geometry by Vedantu help you score better in your upcoming exams. Referring to our Euclid's Geometry Class 9 Notes online, you can learn a step by step and quick approach of solving Mathematical problems. Introduction to Euclid's Geometry is a chapter that needs an understanding of described postulates which can be done with our most preferred study material. Our Class 9 Maths Chapter 5 Notes is prepared by subject experts and comes with a download option.

Download CBSE Class 9 Maths Revision Notes 2024-25 PDF

Also, check CBSE Class 9 Maths revision notes for all chapters:

Access Class IX Mathematics Chapter 5 - Introduction to Euclid Geometry

Introduction to euclid geometry:, geometry's importance has been recognised in various parts of the world since ancient times.  this branch of mathematics sprang from the practical issues that ancient civilizations faced.  let's look at a few examples below; the demarcations of land owners on river-side land were used to wipe out with the floods in the river.  the concept of area was established in order to rediscover the borders. geometry could be used to determine the volume of granaries.  egyptian pyramids show that geometry has been used from ancient times. there was a geometrical construction guidebook called as sulbasutra's throughout the vedic period.  altars of various geometrical shapes were built to fulfil various vedic ceremonies. geometry is derived from the green words 'geo' (earth) and metrein (measurement) (to measure). geometry has been developed and implemented in various parts of the world since ancient times, but it has never been presented in a systematic fashion. later, approximately\[300\text{ }bc\], the egyptian mathematician euclid gathered all known work and organised it into a systematic framework. euclid's 'elements' is a famous treatise on geometry written by him.  this was the book that had most impact.  for several years, the 'element' was utilised as a textbook in western europe. the 'elements' began with 28 definitions, five postulates, and five common conceptions and created the rest of plane and solid geometry in a systematic manner. the euclid method refers to euclid's geometrical approach. the euclid method entails making a small set of assumptions and then using these assumptions to prove a large number of other propositions. the assumptions that were made were self-evident universal truths. axioms and postulates were the two types of assumptions made., euclid's definitions:, euclid listed \[23\] definitions in book \[1\] of the 'elements'.  we list a few of them are as follows: a point is that which has no part a line is a breadth less length the ends of a line are points a straight line is a line which lies evenly with the points on itself. a surface is that which has length and breadth only. the edges of a surface are lines a plane surface is a surface which lies evenly with straight lines on its self. euclid made some assumptions, known as axioms and postulates, based on these definitions., euclid's axioms:, axioms are assumptions that are employed in all areas of mathematics but are not directly related to geometry.  only a few of euclid's axioms are true, they are as follows: things which are equal to the same thing are equal to one another. if equals are added to equals; the wholes are equal. if equals are subtracted from equals, the remainders are equal. things which coincide one another are equal to one another. the whole is greater than the part things which are double of the same thing are equal to one another. things which are half of the same things are equal to one another. all these axioms refer to magnitude of same kind.  axiom - \[1\] can be written as follows: if \[x=z\] and \[y=z\] , then \[x=y\] axiom - \[2\] explains the following: if \[x=y\], then \[x+z=y+z\] according to axiom - \[3\] , if \[x=y\], then \[x-z=y-z\] axiom - \[4\] justifies the principle of superposition that everything equals itself.  axiom - \[5\] , gives us the concept of comparison. if \[x\] is a part of \[y\] , then there is a quantity \[z\] such that \[x=y+z\] or \[x>y\] note that magnitudes of the same kind can be added, subtracted or compared., euclid's postulates:, the term "postulate" was coined by euclid to describe the assumptions that were unique to geometry.  the following are euclid's five postulates: postulate \[1\] :   a straight line may be drawn from any one point to any other point.  same may be stated as axiom \[5.1\] given two distinct points, there is a unique line that passes through them. postulate \[2\] :  a terminated line can be produced indefinitely.  postulate \[3\] :  a circle can be drawn with any centre and any radius.  postulate \[4\] :  all right angles are equal to one another postulate \[5\] :  if a straight line falling on two straight lines makes the interior angle on the same side of it taken together less than two right angles, then two straight lines, if produced indefinitely, meet on that side on which the sum of the angles is less than two right angles. postulates \[1\] to \[4\] are exceedingly basic and obvious, hence they are regarded as "self-evident truths."  postulate \[5\] is complicated and it should be discussed. if the line \[lm\] intersects two lines \[pq\] and \[rs\] at a place where \[\text{sum of angles}=180{}^\circ \] , the lines \[lm\] and \[pq\] will intersect at that point. note: the terms axiom and postulate are often used interchangeably in mathematics, however according to euclid, they have different meanings., system of consistent axioms:, if it is impossible to deduce a statement from these axioms that contradicts any of the given axioms or propositions, the system is said to be consistent., proposition or theorem:, propositions or theorems are statements or results that have been proven using euclid's axioms and postulates., two distinct lines cannot have more than one point in common. proof: given: \[ab\] and \[cd\] are two lines. to prove: they intersect at one point or they do not intersect. proof: assume that the lines \[ab\] and \[cd\] cross at positions \[p\] and \[q\]. the line \[ab\] must therefore pass through the points \[p\]and \[q\]. also, the \[cd\] line also runs through the \[p\] and \[q\] points. this means that there are two lines passing through two distinct points \[p\] and \[q\]. however, we know that only one line can cross through two separate places. this axiom goes against our belief that two separate lines can share more than one point. the lines \[ab\] and \[cd\] are unable to travel via the points \[p\] and \[q\]., equivalent versions of euclid's fifth postulate:, the two different version of fifth postulate for every line \[l\] and for every point \[p\] not lying on \[l\] , there exist a unique line \[m\] passing through \[p\] and parallel to \[l\]. two distinct intersecting lines cannot be parallel to the same line., introduction to euclid's geometry class 9 notes – brief overview of the chapter.

Important Topics Covered in Class 9 Maths Chapter 5

The important topics that are covered under the chapter “Introduction To Euclid’s Geometry are:

Euclid’s Definitions

Axioms and Postulates

Axioms and theorems by a Greek Mathematician Euclid explaining the geometry and analysis of plane and solid figures are the basis of this chapter. You will be well-versed with the concept of a plane and solid geometry by reading and understanding this chapter thoroughly.  You can further learn the geometrical concept with ease by acquiring study help. In this regard, we provide Introduction to Euclid Geometry Notes that explain the related concepts efficiently. Our quality notes can make an excellent resource for revision as each topic is described in a simplified manner, which makes the answers easy-to-learn.  Subsequently, the quality of Euclid’s Geometry Class 9 Notes provided by us is of top-notch as highly qualified professionals and expert teachers prepare these. You can utilise such resources to gain an edge in your exam preparation and eventually perform better in exams.  What makes them the best study aid for students is the ease in accessibility as it can be accessed online or by downloading in their PDF formats. Moreover, you can study our Euclid's Geometry Class 9 Notes while on the go.  

Introduction to euclid's geometry class 9 – revision notes , euclid defined several geometrical terms and derived propositions from the assumptions which are universally accepted. with our 9th class maths chapter 5 notes, you will have clarity of the concept and will be able to learn and practice the advanced concepts effortlessly.  a. euclid geometry class 9 notes – euclid’s definition chapter 5 class 9 maths includes euclid’s definitions of geometrical terms that he listed in book 1 of the ‘elements’ in late 300 bc. 23 definitions were listed in the book that explained different geometrical terms. line  straight line  surface  edges of surface  a plane surface, etc.  the terms defined by euclid have been enlisted in our euclid’s geometry class 9 notes systematically so that you gather all the necessary definitions at a place and can memorise it precisely.  our revision notes put these geometrical concepts in a simplified manner and explain the intricacies of the subject smoothly so that you can learn and memorise them effortlessly.  b. introduction to euclid's geometry notes – euclid’s axioms   axioms aren’t precisely geometrical terms but are assumptions made by mathematicians to deduce a proposition or prove a logical explanation. euclid made use of few such axioms which are known to man for different proposals made by him.  our introduction to euclidean geometry class 9 notes lists a few of the axioms used by euclid for the propositions made.  axiom 1 - follows a basic mathematical assumption, i.e., if a=b and b=c, then a=c  axiom 2 - assumes that if a= b, then a + c = b + c axiom 3 - states that if a= b, then a - c = b – c axiom 4 - is all about the principle of superposition that everything equals itself  axiom 5 - is an assumption made based on the concept of comparison which is, if a is a part of b, then there will be a constant c for which a = b + c or a > b.  you can conveniently get the theoretical explanation of the axioms while retaining the mathematical formula explaining their magnitude by studying from our class 9 chapter 5 maths notes. such quality notes prepared by our expert teachers are aimed to ensure that you make the most of your study material while revising for your exams. with our team of proficient teachers who have years of experience behind them, we are familiar with the challenges that students face during revisions before exams. one of the significant hurdles is strategically going through the entire syllabus from cover to cover the night before exams.  catering to this, we offer our revision notes that highlight the portions of the chapter that are deemed the most essential as far as score divisions are concerned. students can thus rely on our offered material and fall back upon the same as a revision-bible.    c. class 9 maths chapter 5 notes – euclid’s postulates  postulates included in euclid’s geometry class 9 notes are the assumptions made by him which are specific to geometry. his five postulates are covered in chapter 5 of class 9 math books. following are the five postulates  postulate 1 – it states that a straight line can be drawn between two points or a line between two points makes a straight line.  postulate 2 – a terminated line can be extended indefinitely to the infinity. euclid's geometry class 9 notes uses the axioms to reason and say that this geometrical concept holds.  postulate 3 – it states there is no specific value fixed for drawing a circle. you can draw a circle by keeping any point as the centre and for any value of radius. introduction to euclidean geometry class 9 notes pdf explains the postulates briefly while educating students on how these propositions are made. it ensures that they can write accurate answers which can help them secure high marks in the upcoming examination.  postulate 4 – it specifies that all right angles are equal. the right angles in geometry are equal was one of the first speculations made by euclid regarding right angles.  postulate 5 – while the other four postulates were simple and discussed universal truths and can be accepted readily, the fifth postulate requires producing arguments based on assumptions to proof that it holds.  therefore, introduction to euclidean geometry class 9 notes pdf can be useful and resourceful material for students to get well-versed with the postulate and how this can be proved.  euclid termed the proved speculations made from assumptions as postulates or theorems while the premises itself were termed as axioms.  as per the theorem, two different lines will have only one point in common and never more than that. to prove the truthfulness of the theorem, you will have to prove either of the two facts that  two distinct lines intersect one another  two distinct lines never intersect each other  euclid’s geometry class 9 notes help you establish the theorem by assuming that the opposite of theorem holds, i.e., two different lines have more than one point of intersection. then based on the assumptions (axioms), you have to give reasonable arguments to prove that the actual theorem holds.  since proving the geometrical theorems need students to present logical explanation and arguments in favour; therefore, solving the mathematical questions can get challenging or a bit complicated. consequently, acquiring study help from our chapter 5 maths class 9 notes can provide you with a better understanding of the theorem.  you can access the study material online and learn step by step how to prove the theorems accurately. our materials are available online and can be accessed anytime and from anywhere. besides, you can easily download introduction to euclidean geometry class 9 notes pdf and start preparing for upcoming exams immediately without any hassle.  d. class 9 euclid geometry notes – equivalent versions of euclid’s fifth postulate   in this chapter of euclid’s geometry, two different versions of euclid’s fifth postulate has been included. following are the two equivalent versions. the first version states that for any straight line if there is a point ‘p’ which doesn’t lie on the straight line, there will be a unique line which passes through point ‘p’ and is parallel to the former straight line.  second equivalent version defines that two distinct intersecting lines can’t have a single line parallel to them.  both the equivalent theories are forms of the original theorem and support the speculation that there can’t be more than one intersection point between two different lines.   euclid’s geometry class 9 notes prepared by us will help you understand the axioms, propositions, theorem and its equivalents easily as it covers all the key points briefly and accurately. you can rely on the quality notes and utilise them for revision purpose and can write accurate answers that will help you secure higher grades in your academic pursuits.  you can equip our study notes to learn shortcut techniques that will help you write quality answers in upcoming exams. the geometrical concepts provided by euclid have defined many geometrical terms and have provided postulates based on the assumptions made. some of these postulates are easy to understand while some needs to be discussed and proved.  therefore, you can acquire our study material and gain knowledge of such profound concepts effortlessly. also, the standard approach adopted in writing these notes will help you memorise, learn, and understand the topics quickly and therefore, is extremely helpful as revision notes. you can quickly understand a topic while glancing through it and retain the fundamental concept. , key features of ncert solutions for class 9 maths chapter 5: introduction to euclid’s geometry, all the exercise questions of class 9 maths chapter 5 solutions are solved by the experts with a step-by-step procedure. formulas are highlighted wherever used while solving the questions. the solutions are explained in an easy-to-understand language. the questions are solved by expert teachers as per the latest guidelines issued by the cbse board. the questions are solved after a comprehensive analysis of the previous year's question papers. students who refer to the exercise-wise ncert class 9 maths chapter 5 solutions find it easy to complete their homework. to gain detailed knowledge about the class 9 maths chapter 5  introduction to euclid’s geometry, students can explore vedantu’s official website., why is vedantu revision notes a prime choice for students , vedantu offers you a diverse range of study help that can be accessed easily. you can find our revision notes for all subjects and all classes on a chapter-wise basis that can be referred for learning and understanding purpose or can be utilised to brush up your knowledge before the examination.  you can glance through the topics while retaining the easy to understand a concept. euclid's geometry class 9 notes further helps you in acquiring knowledge of the chapter as it explains the postulates and theories briefly and lucidly.  you can make use of the standard notes prepared by our highly qualified teachers and learn the complex topic with convenience.  with exams around the corner, are you still looking out for the best study material with our repute, are there still doubts approach us today and get clear with the quality are potential of the notes.

FAQs on Introduction to Euclids Geometry Class 9 Notes CBSE Maths Chapter 5 (Free PDF Download)

1. What is Euclid's geometry Class 9?

The term geometry consists of geo (earth) and metry (measure). Euclid’s geometry is a brief introduction to the world of solid figures and planes. Euclid, a famous Greek mathematician, is considered the father of geometry. In Euclid’s geometry, there are numerous axioms and postulates. It teaches you about the relationships between various things. It is the study of planes, geometrical shapes and figures. Having a basic understanding of geometry will also help you in your higher classes and various competitive exams. To know more students can refer to the vedantu app.

2. What is a point in geometry Class 9?

A point has no part. It is an element that has no breadth. Various points make a line. A point is characterised by the absence of dimensions and magnitude. It has no thickness. A point only occupies a space. A point can be defined only by geometric properties. A point in the domain of geometry is represented by various coordinates such as x and y in a two-dimensional plane and by x, y and z in a three-dimensional plane.

3. What are Euclid's 5 postulates for geometry?

Euclid’s geometry is the study of various geometrical shapes and figures. He gave 5 postulates for plane geometry. First one among them is that if two or more things are equal to the same thing, then they are also equal to one another. Second is that if equals are added to equals then their whole is also equal. Third postulate states that if equals are subtracted from equals then their remainder will also be equal. Fourth, if things coincide with one another they are equal. Lastly, as parts make up the whole, the whole will be greater than par.

4. Where can I find Class 9 Maths Chapter 5 answers?

Chapter 5 of Class 9 Maths introduces you to Euclid’s geometry. The textbook provides you with all the basic information and all the necessary questions. Vedantu provides NCERT Solutions to all the questions from Chapter 5 of Class 9 Maths. These solutions are prepared by experts. When you have completed solving a set of questions, you can search for the solutions to make sure that your answer is correct and that you have followed all the necessary steps. Make sure to study revision notes before the exam to score good marks. 

5. How many questions of Class 9 Geometry should I practice?

Geometry is a vast area. From a single concept, hundreds of questions can be framed. The most important thing before you start practising geometry questions is to have a strong understanding of the basics. If your basics are clear and strong, you will be able to answer any kind of question that may be asked from this chapter. Practice all your textbook questions. You can find various questions online as well. Write regular tests to check how well prepared you are and work on your shortcomings. To study more and revise the topics students can download the Class 9 maths notes free of cost from the vedantu website (vedantu.com).

STUDY MATERIALS FOR CLASS 9

• Class 9 Maths MCQs
• Chapter 5 Introduction to Euclids Geometry Mcq

Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry MCQs

Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry MCQs are provided here online to solve. These objective type questions are given with correct answers and detailed explanations. Students can solve these chapter-wise problems provided, as per the latest CBSE syllabus and NCERT curriculum. Also, check Important Questions for Class 9 Maths .

Click on the below link to download the PDF for Class 9 Maths Chapter 5 MCQs.

Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry MCQs- Practice Questions

MCQs on Class 9 Maths Chapter 5  Introduction to Euclid’s Geometry

Multiple choice questions for Euclid’s geometry chapter are provided here with four options for each question. Students have to choose the right answer.

1) A solid has __________dimensions.

Explanation: A solid is a three-dimensional object.

2) A point has _______ dimension.

Explanation: A point is always dimensionless.

3) The shape of the base of a Pyramid is:

a. Triangle

c. Rectangle

d. Any polygon

Explanation: A pyramid base could have any polygon shape.

4) The boundaries of solid are called:

a. Surfaces

5) A surface of shape has:

a. Length, breadth and thickness

b. Length and breadth only

c. Length and thickness only

d. Breadth and thickness only

6) The edges of the surface are :

d. None of the above

7) Which of these statements do not satisfy Euclid’s axiom?

a. Things which are equal to the same thing are equal to one another

b. If equals are added to equals, the wholes are equal.

c. If equals are subtracted from equals, the remainders are equal.

d. The whole is lesser than the part.

8) Which of the following statements are true?

a. Only one line can pass through a single point.

b. There is an infinite number of lines that pass through two distinct points.

c. A terminated line can be produced indefinitely on both sides

d. If two circles are equal, then their radii are unequal.

9) The line drawn from the center of the circle to any point on its circumference is called:

b. Diameter

10) There are ________ number of Euclid’s Postulates

11) Euclid’s fifth postulate is

(a) The whole is greater than the part

(b) All right angles are equal to one another.

(c) If a straight line falling on two straight lines makes the interior angles on the same

side of it taken together less than two right angles, then the two straight lines if

produced indefinitely, meet on that side on which the sum of angles is less than two

right angles

(d) A circle may be described with any centre and any radius

Explanation: Euclid’s fifth postulate is If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

12) The three steps from solids to points are: 

(a) Lines – points – surfaces – solids

(b) Lines – surfaces – points – solids

(c) Solids – lines – surfaces – points 

(d) Solids – surfaces – linepoint

Explanation: The three steps from solids to points are Solids to surfaces and surfaces to line points.

13) Axioms are assumed 

(a) Theorems

(b) Definitions

(c) Universal truths specific to geometry

(d) Universal truths in all branches of mathematics

Explanation:  Axioms are assumed universal truths in all branches of mathematics and no mathematical deduction is needed to prove them.

14) Which of the following needs a proof?

(a) Definition

(b) Postulate

(c) Theorem

Explanation: The theorem needs a proof. Whereas definition, axiom and postulates are self-evident and do not require any proof.

15) It is known that if x + y = 10 then x + y + z = 10 + z. Euclid’s axiom that illustrates this statement is

(a) First Axiom

(b) Second Axiom

(c) Third axiom

(d) Fourth Axiom

Explanation: By using Euclid’s second axiom, if equals are added to equals then wholes are equal. Hence, if z has been added to both the sides of equation x + y = 10, then it becomes  x + y + z = 10 + z.

16) ‘Lines are parallel if they do not intersect’ is stated in the form of

(c) Postulate

Explanation:  ‘Lines are parallel if they do not intersect’ is stated in the form of definition. The definition is a statement that gives the exact meaning of the word.

17) The side faces of a pyramid are:

(b) Triangles

(c) Polygons

(d) Trapezium

Explanation: The side faces of a pyramid are triangles. 

18) Euclid stated that all right angles are equal to each other in the form of

Explanation: Euclid stated that all right angles are equal to each other in the form of a postulate. According to Euclid’s fourth postulate, all right angles are equal to each other

19) The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is:

Explanation: The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is nine.

20) The things which are double of the same thing are 

(b) unequal

(c) double of the same thing

(d) halves of the same thing

Explanation: The things which are double of the same thing are equal. According to the Euclidian axiom, the things which are double the same thing are equal to each other. For example, if 2a = 2b, then a =b.

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NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12

NCERT Exemplar Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry

June 28, 2019 by Bhagya

NCERT Exemplar Class 9 Maths Chapter 5 Exercise 5.1

Question 1. The three steps from solids to points are: (A) Solids-surfaces-lines-points (B) Solids-lines-surfaces-points (C) Lines-points-sucfaces-solids (D) Lines-surfaces-points-solids Solution: (A): The three steps from solids to points are solids-surfaces-lines-points.

Question 2. The number of dimensions, a solid has: (A) 1 (B) 2 (C) 3 (D) 0 Solution: (C): A solid e.g., Cuboid has shape, size, and position. So, solid has three dimensions.

Question 3. The number of dimensions, a surface has: (A) 1 (B) 2 (C) 3 (D) 0 Solution: (B): Boundaries of a solid are called surfaces. A surface (plane) has only length and breadth. So, it has two dimensions.

Question 4. The number of dimension, a point has (A) 0 (B) 1 (C) 2 (D) 3 Solution: (A): A point is that which has no part i. e., no length, no breadth and no height. So, it has no dimension.

Question 5. Euclid divided his famous treatise “The Elements” into (A) 13 chapters (B) 12 chapters (C) 11 chapters (D) 9 chapters Solution: (A): Euclid divided his famous treatise ‘The Elements” into 13 chapters.

Question 6. The total number of propositions in the Elements are (A) 465 (B) 460 (C) 13 (D) 55 Solution: (A): The statements that can be proved are called propositions or theorems. Euclid deduced 465 propositions in a logical chain using his axioms, postulates, definitions and theorems.

Question 7. Boundaries of solids are: (A) surfaces (B) curves (C) lines (D) points Solution: (A): Boundaries of solids are surfaces.

Question 8. Boundaries of surfaces are: (A) surfaces (B) curves (C) lines (D) points Solution: (B) : The boundaries of surfaces are curves.

Question 9. In Indus Valley Civilisation (about 3000 B.C.), the bricks used for construction work were having dimensions in the ratio (A) 1 : 3 : 4 (B) 4 : 2 : 1 (C) 4 : 4 : 1 (D) 4 : 3 : 2 Solution: (B) : In Indus Valley Civilisation, the bricks used for construction work were having dimensions in the ratio length : breadth : thickness = 4 : 2 : 1.

Question 10. A pyramid is a solid figure, the base of which is (A) only a triangle (B) only a square (C) only a rectangle (D) any polygon Solution: (D) : A pyramid is a solid figure, the base of which is a triangle or square or some other polygon.

Question 11. The side faces of a pyramid are (A) Triangles (B) Squares (C) Polygons (D) Trapeziums Solution: (A) : The side faces of a pyramid are always triangles.

Question 12. It is known that, if x + y = 10, then x + y + z = 10 + z. The Euclid’s axiom that illustrates this statement is: (A) First Axiom (B) Second Axiom (C) Third Axiom (D) Fourth Axiom Solution: The Euclid’s axiom that illustrates the given statement is second axiom. According to this, if equals are added to equals, the wholes are equal.

Question 13. In ancient India, the shapes of altars used for household rituals were: (A) Squares and circles (B) Triangles and rectangles (C) Trapeziums and pyramids (D) Rectangles and squares Solution: (A) : In ancient India, squares and circular altars were used for household rituals.

Question 14. The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is: (A) Seven (B) Eight (C) Nine (D) Eleven Solution: (C): The Sriyantra (in the Atharvaveda) consists of nine interwoven isosceles triangles.

Question 15. Greek’s emphasised on: (A) Inductive reasoning (B) Deductive reasoning (C) Both (A) and (B) (D) Practical use of geometry Solution: (B) : Greek’s emphasised on deductive reasoning.

Question 16. In ancient India, altars with combination of shapes like rectangles, triangles and trapeziums were used for (A) Public worship (B) Household rituals (C) Both (A) and (B) (D) None of A, B, C Solution: (A): In ancient India altars whose shapes were combinations of rectangles, triangles and trapeziums were used for public worship.

Question 17. Euclid belongs to the country: (A) Babylonia (B) Egypt (C) Greece (D) India Solution: (C) : Euclid belongs to the country Greece.

Question 18. Thales belongs to the country: (A) Babylonia (B) Egypt (C) Greece (D) Rome Solution: (C) : Thales belongs to the country Greece.

Question 19. Pythagoras was a student of: (A) Thales (B) Euclid (C) Both (A) and (B) (D) Archimedes Solution: (A) : Pythagoras was a student of Thales.

Question 20. Which of the following needs a proof ? (A) Theorem (B) Axiom (C) Definition (D) Postulate Solution: (A): The statements that needs a proof are called propositions or theorems.

Question 21. Euclid stated that all right angles are equal to each other in the form of (A) an axiom (B) a definition (C) a postulate (D) a proof Solution: (C) : Euclid stated that all right angles are equal to each other in the form of a postulate.

Question 22. ‘Lines are parallel, if they do not intersect’ is stated in the form of (A) an axiom (B) a definition (C) a postulate (D) a proof Solution: (B) : ‘Lines are parallel, if they do not intersect’ is the definition of parallel lines.

NCERT Exemplar Class 9 Maths Chapter 5 Exercise 5.2

Question 1. Euclidean geometry is valid only for curved surfaces Solution: False Because Euclidean geometry is valid only for the figures in the plane but on the curved surfaces, it fails.

Question 2. The boundaries of the solids are curves. Solution: Because the boundaries of the solids are surfaces.

Question 3. The edges of a surface are curves. Solution: False Because the edges of surfaces are lines.

Question 4. The things which are double of the same thing are equal to one another Solution: True Since, it is one of the Euclid’s axiom.

Question 5. If a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C. Solution: True Since, it is one of the Euclid’s axiom.

Question 6. The statements that are proved are called axioms. Solution: False Because the statements that are proved are called theorems.

Question 7. “For every line l and for every point P not lying on a given line l, there exits a unique line m passing though P and parallel to l is known as Playfair’s axiom. Solution: True Since, it is an equivalent version of Euclid’s fifth postulate and it is known as Playfair’s axiom.

Question 8. Two distinct intersecting lines cannot be parallel to the same line. Solution: True Since, it is an equivalent version of Euclid’s fifth postulate.

Question 9. Attempt to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries. Solution: True All attempts to prove the fifth postulate as a theorem led to a great achievement in the creation of several other geometries. These geometries are quite different from Euclidean geometry and are called non-Euclidean geometry.

NCERT Exemplar Class 9 Maths Chapter 5 Exercise 5.3

Question 1. Two salesmen make equal sales during the month of August. In September, each salesmen doubles his sale of the month of August. Compare their sales in September. Solution: Let the equal sales of two salesmen in August be y. In September, each salesman doubles his sale of August.

Thus, sale of first salesman is 2y and sale of second salesman is 2y.

According to Euclid’s axioms, things which are double of the same things are equal to one another.

So, in September their sales are again equal.

Question 2. It is known that x + y = 10 and that x = z. Show that z + y = 10. Solution: We have, x + y = 10 …(i) and x = z …(ii) On adding y to both sides, we have x + y = z + y …. (iii) [ ∵ If equals are added to equals, the wholes are equal] From (i) and (iii), we get z + y = 10

(ii) Given, BM = BN …(1) M is the mid-point of AB. ∴ AM = BM = \(\frac{1}{2}\)AB ⇒ 2AM = 2BM = AB …(2) and N is the mid-point of BC. ∴ BN = NC = \(\frac{1}{2}\)BC ⇒ 2BN = 2NC = BC …(3) multiplying both sides of (1) by 2, we get 2 BM = 2 BN [ ∵ Things which are double of the same thing are equal to one another] ⇒ AB = BC [Using (2) and (3)]

NCERT Exemplar Class 9 Maths Chapter 5 Exercise 5.4

Question 1. Read the following statement: An equilateral triangle is a polygon made up of three line segments out of which two line segments are equal to the third one and all its angles are 60° each. Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in a equilateral triangle. Solution: The terms need to be defined are

• Polygon : A closed figure bounded by three or more line Segments.
• Line segment : Part of a line with two end points.
• Line: Undefined term.
• Point: Undefined term.
• Angle: A figure formed by two rays with one common initial point.
• Acute angle : Angle whose measure is between 0° to 90°.

Here undefined terms are line and point. All the angles of equilateral triangle are 60° each (given).

Two line segments are equal to the third one (given).

“According to Euclid’s axiom, “things which are equal to the same things are equal to one another”, we conclude that all three sides of an equilateral triangle are equal.

Question 2. Study the following statements: “Two intersecting lines cannot be perpendicular to the same line.” Check whether it is an equivalent version to the Euclid’s fifth postulate. [Hint: Identify the two intersecting lines / and m and the line n in the above statement.] Solution: Two equivalent versions of Euclid’s fifth postulate are as follows: (i) For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l. (ii) Two distinct intersecting lines cannot be parallel to the same line.

From above two statements it is clear that given statement is not an equivalent version to the Euclid’s fifth postulate.

Question 3. Read the following statements which are taken as axioms: (i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal. (ii) If a transversal intersect two parallel lines, then alternate interior angles are equal. Is this system of axioms consistent ? Justify your answer. Solution: As we know that, if a transversal intersects two parallel lines, then each pair of corresponding angles are equal, then first is false, so, not an axiom.

Also, if a transversal intersects two parallel lines, then each pair of alternate interior angles are equal, then second is true so, it is an axiom.

So, in given statements, first is false and second is an axiom.

Thus, given system of axioms is not consistent.

Question 4. Read the following two statements which are taken as axioms: (i) If two lines intersect each other, then the vertically opposite angles are not equal. (ii) If a ray stands on a line, then the sum of two adjacent angles, so formed is equal to 180°. Is this system of axioms consistent ? Justify your answer. Solution: As we know that, if two lines intersect each other, then the vertically opposite angles are equal, then first is false, so, not an axiom. Also, if a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°, then second is true so, it is an axiom.

So, in given statements, first is false and second is an axiom. Thus, given system of axioms is not consistent.

Question 5. Read the following axioms: (i) Things which are equal to the same thing are equal to one another. (ii) If equals are added to equals, the wholes are equal. (iii) Things which are double of the same thing are equal to one another. Check whether the given system of axioms is consistent or inconsistent. Solution: Since, the given three axioms are Euclid’s axioms.

∴ We cannot deduce any statement from the above three axioms which contradicts any axiom. Thus, the given system of axioms is consistent.

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