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

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:

1. A point is that which has no part

2. A line is a breadth less length

3. The ends of a line are points

4. A straight line is a line which lies evenly with the points on itself.

5. A surface is that which has length and breadth only.

6. The edges of a surface are lines

7. 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:

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

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

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

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

5. The whole is greater than the part

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

7. 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.

Theorem:

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

1. 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 \[l\].

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

Introduction to Euclid's Geometry Class 9 Notes – Brief Overview of the Chapter

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. 

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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.

1. Line 

2. Straight line 

3. Surface 

4. Edges of surface 

5. 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. 

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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

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