CBSE Class 9 Maths Notes Chapter 5 Introduction to Euclids Geometry

These axioms of Euclid establish fundamental principles for reasoning about magnitudes:

1. Axiom 1 : If $x=z$ and $y=z$ , then $x=y$ .
2. Axiom 2 : If $x=y$ , then $x+z=y+z$ .
3. Axiom 3 : If $x=y$ , then $x-z=y-z$ .
4. Axiom 4 : Everything equals itself, justifying the principle of superposition.
5. Axiom 5 : Provides a concept of comparison: If $x$ is a part of $y$ , then there exists a quantity $z$ such that $x=y+z$ or $x>y$ .

These axioms allow for the addition, subtraction, and comparison of magnitudes of the same kind.

Euclid's Postulates

Euclid's postulates are fundamental assumptions unique to geometry:

1. Postulate 1 : A straight line may be drawn from any one point to any other point. In other words, given two distinct points, there is a unique line that passes through them.
2. Postulate 2 : A terminated line can be produced indefinitely.
3. Postulate 3 : A circle can be drawn with any center and any radius.
4. Postulate 4 : All right angles are equal to one another.
5. 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 the two straight lines, if produced indefinitely, will meet on that side where the sum of the angles is less than two right angles.

Postulates 1 to 4 are considered self-evident truths due to their basic and obvious nature. Postulate 5 is more complex and requires further discussion. It essentially describes conditions under which two straight lines intersect.

System of Consistent Axioms

A system of axioms is considered consistent if it does not lead to any contradictions when used to derive statements or propositions.

Proposition or Theorem

Propositions, also known as theorems, are statements or results that have been proven using Euclid's axioms and postulates.

Theorem

According to Euclid's geometry, two distinct lines cannot have more than one point in common.

Proof: Let AB and CD be two lines.

To prove: They intersect at one point or they do not intersect.

Proof: Assume that the lines AB and CD intersect at points P and Q.

Since line AB passes through points P and Q, and line CD also passes through points P and Q, it implies that there are two lines passing through two distinct points, P and Q. However, according to the axiom that only one line can pass through two separate points, this contradicts the assumption that two lines can share more than one point. Hence, the lines AB and CD cannot pass through points P and Q simultaneously.

Equivalent Versions of Euclid's Fifth Postulate

The two different version of fifth postulate For every line l 𝑙 and for every point P 𝑃 not lying on l 𝑙 , there exist a unique line m 𝑚 passing through P 𝑃 and parallel to l.

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

Practice Problems

Solve the following problems of class 9 Maths chapter 5 introduction to Euclid’s geometry.

1. If two circles are equal, then their radii are equal. Is this statement true or false? Also, justify your answer.
2. There is an infinite number of lines that pass through two distinct points. Is this statement true or false? Also, justify your answer.
3. Assume that point C lies between two points, say A and B, such that AC = BC. Prove that AC = ½ AB.

Benefits of CBSE Class 9 Maths Notes Chapter 5 Introduction to Euclids Geometry

1. Conceptual Clarity : These notes provide a clear explanation of the fundamental concepts of Euclidean geometry, enabling students to understand the principles and postulates laid down by Euclid.
2. Systematic Learning : The notes follow a structured approach, presenting Euclid's axioms, postulates, and propositions in a systematic manner. This helps students grasp the logical progression of geometric reasoning.
3. Comprehensive Coverage : The notes cover all the key topics of Chapter 5, including definitions, postulates, and theorems, ensuring that students have a thorough understanding of Euclidean geometry.
4. Easy Accessibility : Students can easily access these notes in PDF format, making it convenient for them to review the concepts anytime, anywhere, using their digital devices.

CBSE Maths Notes For Class 9

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

CBSE Class 9 Maths Notes Chapter 10 - Circles Notes

CBSE Class 9 Maths Notes Chapter 11 - Constructions Notes

CBSE Class 9 Maths Notes Chapter 12 - Herons Formula Notes

CBSE Class 9 Maths Notes Chapter 13 - Surface Areas and Volumes Notes

CBSE Class 9 Maths Notes Chapter 14 - Statistics Notes

CBSE Class 9 Maths Notes Chapter 15 - Probability Notes