# NCERT Solutions For Class 9 Maths Chapter 5 Introduction To Euclid’s Geometry (Ex 5.1) Exercise 5.1

The National Council of Educational Research and Training (NCERT) is a self-governing organisation. In 1961, the Societies Registration Act authorised the formation of this organisation. It is under the administration of the Indian government. Its mission is to promote the literary, artistic, and humanitarian progress of the country. NCERT textbooks and curriculum are available for students in grades 1–12. NCERT textbooks and courses are trusted resources for the CBSE Board Exam. In addition to being applicable for annual exams or CBSE Board Examinations, these textbooks and curricula serve as the foundation for competitive exams such as the National Eligibility Entrance Test (NEET), the Union Public Service Commission (UPSC), the Indian Institute of Technology (IIT), the Provincial Civil Service (PCS), and others.

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For students preparing for examinations, the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, Introduction to Euclid’s Geometry, are available on the Extramarks website. The Extramarks specialists curated the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, that adhere to the CBSE syllabus for Class 9. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, are designed in such a way that students will be able to tackle tough questions relating to Euclid’s Geometry after completing the questions and practising them. To download the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, students can visit the Extramarks website.

The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 facilitate studying and assists students in understanding the fundamentals of Geometry according to the great Mathematician Euclid. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 can be utilised by Class 9 students as a reference tool when practising the NCERT book questions, these solutions can also be downloaded in PDF format.

NCERT Solutions for Class 9 are available for all topics on Extramarks. It is the only location where students may find every NCERT solution they require. Along with the other exercise solutions of the chapter, the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 is provided. Examples and images are used to assist in illustrating the answers. Thanks to the subject-matter specialists at Extramarks, the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 are developed per CBSE rules. Students can practice all the exercise questions by working through the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1.

**NCERT Solutions For Class 9 Maths Chapter 5 Introduction To Euclid’s Geometry (Ex 5.1) Exercise 5.1**

There are several advantages of having NCERT Solutions on hand. To avoid confusion, the Extramarks solutions are organised into exercises for each chapter, such as the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, and other practice solutions make studying easy for Class 9 students. Professional faculty members attentively design the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, along with others. On the Extramarks website, all 15 chapters of the Class 9 NCERT Mathematics book are accessible and may be downloaded in PDF format. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, are available for download on this website.

Along with textbook exercises, examples with answers included in the text, and reference resources, these solutions are essential study aids. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, is a fantastic study tool for students who are solving a variety of test questions. Students can simply obtain the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, by clicking on the supplied link.

**Access NCERT Solutions For Class 9 Mathematics Chapter 5 – Introduction To Euclid’s Geometry**

The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, deals with a variety of geometrical terms and forms. These solutions contain two exercises in which students will discover the relation between theorems, postulates, and axioms. Students must demonstrate relationships between axioms and theorems. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, is essential for the NCERT Class 9 academic examination. Students must acknowledge all theorems and postulates.

Students are introduced to an old mathematical discipline – ‘Euclid’s Geometry,’ in the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 which offer an insight into the genesis of Geometry and how it led to the present-day Geometry.

Euclid’s Elements: An Introduction Geometry is the foundation for fundamental geometry principles and their applications. Based on the axioms and postulates offered by the Egyptian mathematician Euclid, students will study solids, planes, points, lines, curves, angles, circles, and so on. Geometry is an important subject in math, so students must grasp it fully.

**Class 9 Maths Chapter 5 – Exercise 5.1**

Geometry is a fundamental subject in the study of Mathematics. Many technical specialists, like architects, use it to develop buildings that will survive forever. This section provides the finest overview of Euclid’s Geometry exercise 5.1.

The first portion of Class 9 Maths Exercise 5.1 focuses on the fundamentals. It explains what geometry is and where it came from. It explains how the term came to be and how ancient civilizations throughout the world exploited it to their advantage. In this chapter, students will learn about the unsystematic ways of early geometry and how they were fostered by many civilizations. It also tells students when and how the first mathematical proof was stated, as well as how these events influenced the famous Greek mathematician Euclid and his work.

After students have finished studying the theory part, they must answer the seven questions included in Exercise 5.1 Class 9 Maths to assess their comprehension of the chapter. Students must thoroughly read the questions and then write relevant replies. If they take references from the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1

Postulate No. 1

This part initially teaches students about Euclid, the famous Greek mathematician who formulated the first postulate. It explains the precise postulate first, then goes into further depth on what has been stated and what has not been expressed. It discusses how he arrived at this conclusion, the approach he used, and the procedures. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, also include detailed illustrations and sketches to aid comprehension.

Postulate No. 2

This section discusses Euclid’s second postulate. It delves into the meaning of terminated lines as defined by the renowned mathematician. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 also assists the students in comprehending what occurs when a line segment is stretched on either side. Diagrams and sketches are provided to make the notion more understandable. To grasp all the key concepts, learners can refer to the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1.

Postulate No. 2

The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 delves deeper into Euclid’s theories. It establishes numerous circular qualities. It explains whether a circle may be drawn from any given centre or any specified radius.

Postulate No. 4

This portion of the textbook discusses the key geometric principles in mathematics. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, will tell students whether or not all right angles can be equal. This idea is critical in mathematics and has been widely employed in subsequent experiments.

Postulate No 5

In this portion, the notion of straight lines, notably the one advanced by Euclid, is discussed. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, teach the learner if it is feasible for a straight line to fall on another straight line in any situation. If this occurs in a particular situation, where do the two straight lines intersect? With this crucial explanation, the learner can master the subject. To learn more comprehensively students must refer to the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1.

**Exercise 5.1 Class 9Covers The Following Topics:**

Each chapter must be thoroughly studied by students, who are also required to understand the definitions of all the themes and subtopics. Understanding the principles of each chapter is crucial to successfully preparing the chapters. Solving problems is the best way to understand the foundational concepts of mathematics. Students should tackle the easy questions first before moving on to the challenging ones. All students must adequately prepare for the board exams. Students must become familiar with the syllabus before beginning their preparation for the mathematics test. The syllabus is necessary to design a strategy for learning the mathematics chapters.

This chapter covers Euclid’s approach to Geometry and attempts to connect it to modern

Geometry. Introduction to Euclid’s Geometry teaches students how to define

common geometrical forms and terminology.

The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 have all the terms in simplified form to understand efficiently. It can be used directly in the examination.

Experts have provided the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 with accurate solutions and answers.

The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 solves all the sums exactly as they should be in the exam. This ensures that students get the most out of their time and effort. Using the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 assists students in improving their efficiency and earning higher results.

Students would not have to go back and forth to the NCERT textbooks or lecture notes to accomplish last-minute modifications while utilising the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, and other NCERT Solutions. Key points are used to solve all of the problems in NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1. It contains everything a student needs to complete in their zero-hour revision.

The NCERT solutions include all of the relevant questions, as well as detailed reasons for each response, which are critical for comprehension. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 on the Extramarks’ website adhere to the detailed structure. As a result, students preparing for any test should use Exercise 5.1 Class 9Solutions.

**NCERT Solutions For Class 9**

NCERT Class 9 lays the foundation of careers. The syllabus of Class 9 is quite vast and difficult as it involves all the basic concepts that will be useful in further examinations. Hence, students need to prepare well for their exams. To help them with their studies, Extramarks has developed NCERT Solutions for every subject.

A team of expert faculties prepares the NCERT Solutions for Class 9 students under CBSE guidelines to encourage concept-based learning and focus on each topic. As the marks are allocated to each step, Extramarks provides step-by-step solutions to help students from an exam perspective. The questions are provided after each chapter to test the students’ knowledge and assist them in revising for examinations. However, students may be unable to answer some questions since it is impossible to answer every question with the assistance of a teacher. These NCERT Solutions will assist them in locating the solutions. Before looking at NCERT Solutions, students must attempt to solve the problems on their own; if they fail, they should carefully examine how the specialist has solved them. Extramarks strongly advises completely reading all of the chapters and practising the questions in the back.

Extramarks has supplied CBSE Study Materials for Class 9 to assist students with their academics. A group of subject-matter experts produce the CBSE Study Materials for Class 9, and the format is well-structured following NCERT guidelines. It breaks down each answer step by step so that students may comprehend it simply. As points are awarded at each level, students will become more conscious of their test response writing skills. The academic test question papers are based on the NCERT Class 9 book. Thus, resolving concerns with CBSE Study Materials for Class 9 assists students in becoming test-ready.

NCERT Solutions for Class 9 are an important tool for exam preparation. Writing answers has a great influence on the mind; when a student writes about a topic, they fully comprehend it. Students should commit time to self-study in addition to revising. Class 9 students must study courses such as Mathematics, which requires a large number of formulas to be learned and theorems to be proved, science, which requires a large number of ideas from all three subsections, physics, chemistry and biology, and so on. Class 9 lays the groundwork for the Class 10 board examination. Students should start with the NCERT books and then move on to other books to study for the exam.

**CBSE Study Materials For Class 9**

The National Council of Educational Research and Training (NCERT) is an autonomous organisation of the Government of India that creates and develops textbooks, educational materials, and research materials. NCERT books give sufficient content for students to understand the fundamentals of all areas covered. Each chapter or topic concludes with exercises and questions for students to practice. These exercises and questions will be extremely beneficial to students as they prepare for their board and competitive examinations. Extramarks has supplied the CBSE Study Materials for Class 9 to assist students with their academics.

A group of subject specialists develop the CBSE Study Materials for Class 9, and the format is well-structured and under NCERT standards. Each answer is broken down step-by-step so that students may readily comprehend it. Students will become more conscious of their ability to write effective exam-style answers as points are awarded for each level. The academic test question papers are based on the NCERT Class 9 book. Thus, fixing problems with the Class 9 CBSE study materials aids students in getting ready for the exam.

**CBSE Study Materials**

CBSE Study Materials are usually the best way to prepare for a test. It raises the student’s awareness of topics and helps in understanding them. Using CBSE Study Materials improves the student’s learning abilities. Students must only learn from the NCERT textbooks. It is recommended that students in grades 1 to 12 keep separate study materials for each topic. It is sufficient to study from the NCERT book for the specific topic; however, if students have completed the NCERT books for their subjects, they can refer to the CBSE Study Materials.

Students in grades 1 to 12 must focus completely on the ideas. The fundamentals of the concepts should be clear in their brains so that they can comprehend what they are learning. This should be done for all courses, including mathematics, history, science, and others. Having CBSE study materials for essential topics is the greatest technique to retain any concept. The materials are also useful for revising.

The material should be easy to memorise, succinct, and to the point, explaining the solution briefly. Students that use notes may readily memorise the subject, which helps them perform well in their exams. In addition to keeping track of their resources, students should devise a study schedule and approach. It assists students in completing the CBSE syllabus on schedule.

Students may learn and prepare for examinations faster with Extramarks’’ unique, entertaining, and user-friendly CBSE study tools. Subject specialists create the study materials while keeping the most recent CBSE syllabus in mind. The Extramarks NCERT Solutions for Class 9 are created by subject specialists to provide a one-stop solution for all concerns in Mathematics, Physics, Chemistry, and Biology. Students will benefit from the online CBSE study materials, which include the curriculum, books, practice examinations, test questions, NCERT solutions, critical inquiries, and CBSE notes. The CBSE study resources make it easy for students to prepare for their exams. Students are thus advised to study every day. This will assist students in finishing their homework on schedule. They will also have adequate time to study and practice before the yearly exam.

**Q.1** Which of the following statements are true and which are false? Give reasons for your answers.

(i) Only one line can pass through a single point.

(ii) There are an infinite number of lines which pass through two distinct points.

(iii) A terminated line can be produced indefinitely on both the sides.

(iv) If two circles are equal, then their radii are equal.

(v) In the below figure, if AB = PQ and PQ = XY, then AB = XY.

**Ans**

(i) False, many lines can pass through a single point. In the following figure, it can be seen that there are infinite numbers of lines passing through a single point P.

(ii) False, only one line can pass through two distinct points. In the following figure, it can be seen that there is only one single line that can pass through two distinct points P and Q.

(iii) True, a terminated line can be produced indefinitely on both the sides. Let AB be a terminated line. It can be seen that it can be produced indefinitely on both the sides.

(iv) True, If you superimpose the region bounded by one circle on the other, then they coincide. So, their centres and boundaries coincide. Therefore, their radii will coincide. (v) True. It is given that AB and XY are two terminated lines and both are equal to a third line PQ. Euclid’s first axiom states that things which are equal to the same thing are equal to one another. Therefore, the lines AB and XY will be equal to each other.

**Q.2 **Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them? (i) parallel lines (ii) perpendicular lines

(iii) line segment (iv) radius of a circle

(v) square

**Ans**

(i) Parallel lines: If the perpendicular distance between two lines is always constant, then these are called parallel lines. In other words, the lines which never intersect each other are called parallel lines. To define parallel lines, we must know about point, lines, and distance between the lines and the point of intersection.

(ii) Perpendicular lines: If two lines intersect at 90°, then these lines are called perpendicular lines. We should define line and angle before defining perpendicular lines.

(iii) Line segment: A part of a line, whose starting and ending points are known, is called line segment. We should define point and line before defining line segment. AB is a line segment in the figure.

(iv) Radius of a circle: The distance from centre to any point lying on the circumference is called radius of the circle. We should define centre, circle and distance before defining radius of a circle.

(v) Square: A rectangle whose adjacent sides are equal is called a square. In other words, a quadrilateral, whose all sides are equal and each angle is 90°, is called square. We should define quadrilateral, side and agnle before defining the square.

**Q.3 **Consider two ‘postulates’ given below: (i) Given any two distinct points A and B, there exists a third point C which is in between A and B. (ii) There exist at least three points that are not on the same line.

Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.

**Ans**

There are various undefined terms which should be listed.

These postulates are consistent, because they deal with two different situations — (i) says that given two points A and B, there is a point C lying on the line in between them; (ii) says that given A and B, you can take C not lying on the line through A and B.

Both these postulates are not related to Euclid’s postulates, so, these postulates do not follow Euclid’s postulates.

**Q.4 **If a point C lies between two points A and B such that AC = BC, then prove that AC = (1/2)AB. Explain by drawing the figure.

**Ans**

$\begin{array}{l}\mathrm{Since},\text{AC}=\text{BC\hspace{0.33em}}\left[\mathrm{Given}\right]\\ \text{So, \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{AC}+\mathrm{AC}=\mathrm{BC}+\mathrm{AC}\text{\hspace{0.33em}}\left[\mathrm{equals}\text{are added to equals.}\right]\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}2\mathrm{AC}=\mathrm{AB}\text{\hspace{0.33em}}[\because \mathrm{AC}+\mathrm{BC}=\mathrm{AB}]\\ \mathrm{AC}=\frac{1}{2}\mathrm{AB}\end{array}$

**Q.5 **If a point C lies is a mid-point of line segment AB, then prove that every line segment has one and only one mid-point.

**Ans**

Let D be the midpoint of AB other than C, then AD = DB AD + AD = AD + DB [equals are added to equals.] 2AD = AB

AD = (1/2)AB … (i) It is given that C is midpoint of AB, then AC = CB AC + AC = AC + DB [equals are added to equals.] 2AC = AB

AC = (1/2)AB … (ii)

From equation (i) and equation (ii), we have AD = AC

(Things which are double of the same things are equal to one another.) This is possible only when point C and D are representing a single point. Hence, our assumption is wrong that D is a midpoint of AB other than C. Therefore, every line has one and only one midpoint.

**Q.6** In the below figure, if AC = BD, then prove that AB = CD.

**Ans**

Given: AC = BD

To prove: AB = CD

Proof: AC = BD [Given] AB + BC = BC + CD

AB + BC – BC = BC + CD – BC (According to Euclid’s axiom, when equals are subtracted from equals, the resultants are also equal.) AB = CD. Hence proved.

**Q.7** Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)

**Ans**

Axiom 5 states that the whole is greater than the part. This axiom is known as a universal truth because it holds true in any field, and not just in the field of mathematics. Let us take two cases − one in the field of mathematics, and one other than that. Case I Let A represent a whole quantity and only x, y, z are parts of it. A = x + y + z Clearly, A will be greater than all its parts x, y, and z. Therefore, it is rightly said that the whole is greater than the part. Case II Let us consider the country India. Then, let us consider a state Goa which belongs to India.

If we compare the land area of state Goa with that of India, we will find that land area of India is greater than that of Goa.

That is why we can say that the whole is greater than the part. This is true for anything in any part of the world and is thus a universal truth.

## FAQs (Frequently Asked Questions)

### 1. How many chapters are there in the NCERT solutions for Mathematics in Class 9?

There are 15 chapters in total in the NCERT Solutions For Class 9 Mathematics:

- Chapter 1 Number System
- Chapter 2 Polynomials
- Chapter 3 Coordinate Geometry
- Chapter 4 Linear Equations in Two Variables
- Chapter 5 Introduction to Euclid’s Geometry
- Chapter 6 Lines and Angles
- Chapter 7 Triangles
- Chapter 8 Quadrilaterals
- Chapter 9 Areas of Parallelograms and Triangles
- Chapter 10 Circles
- Chapter 11 Constructions
- Chapter 12 Heron’s Formula
- Chapter 13 Surface Areas and Volumes
- Chapter 14 Statistics
- Chapter 15 Probability

### 2. Where can students get the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1?

The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 are available at the Extramarks. The Extramarks is an e-learning site that is well-known for providing authentic NCERT Solutions to assist students to clear their concerns. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 are available online or through Extramakrs’s website. Extramarks offer PDF format of the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1, which include step-by-step answers to the practice problems. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 are offered by subject-matter experts and are created by the most recent NCERT rules and curriculum.

### 3. Why are the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 important?

Extramarks’ experts have created NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 to ensure thorough coverage of the exercise problems. Expert tutors produce the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 in accordance with the most recent NCERT recommendations. Students should download these solutions since it will help them solve the chapter efficiently. To overcome the exercise’s issues, all of the necessary tips and tactics are provided. The NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 and other exercises are an excellent resource for acing the Maths test.

### 4. Why are NCERT Exercise 5.1 Class 9 Solutions recommended?

Subject-matter specialists create the Extramarks’ NCERT Solutions. The Class 9 Maths Exercise 5.1 Solution satisfies students’ curiosity and helps clear any doubts or confusion that may emerge while completing the tasks. This is filled by the extensive explanations and answers provided with each question, leaving no room for doubt in the mind of the learner. Furthermore, these questions are constructed entirely according to the CBSE curriculum pattern, which explains the priority placed on acquiring the most fundamental notions before moving on to tougher topics. To cover all the exercises students can use the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1.

### 5. How many questions are there in the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1?

The questions in the NCERT Solutions For Class 9 Maths Chapter 5 Exercise 5.1 are 7. Students must practice answering many more questions. These are challenging subjects to learn about. When answering questions, students can refer to the Class 9 Maths Exercise 5.1 Solution for guidance. Each solution is worked out step-by-step with precise solutions. Scholars must go through reliable sources for revision.