Quick Revision Notes & Formula Sheet

Class: 9th Mathematics

Chapter: Introduction to Euclid's Geometry

1. Introduction

• Geometry (ज्यामिति): The word 'geometry' is Greek origin. 'Geo' means the 'earth' and 'metrein' means 'to measure'.
• Euclid: Known as the "Father of Geometry". He collected all available mathematical knowledge and built a famous treatise named 'Elements' (divided into 13 books).

2. Basic Terms in Geometry

• Point (बिंदु): A point is that which has no part. It has no length, breadth, or thickness. (Represented by a fine dot).
• Line (रेखा): A line is breadthless length. It extends endlessly in both directions.
• Line Segment (रेखाखंड): A part of a line with two definite end points.
• Ray (किरण): A part of a line with one end point, extending endlessly in one direction.
• Surface / Plane (तल/पृष्ठ): That which has length and breadth only.

3. Euclid’s Definitions

Euclid gave many definitions in Book 1 of 'Elements'. Some important ones are:

1. A point is that which has no part.
2. A line is breadthless 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 the straight lines on itself.

4. Euclid’s Axioms (स्वयंसिद्ध)

Axioms are assumptions that are obvious universal truths throughout mathematics. They are not proved.

1. Things which are equal to the same thing are equal to one another. (If $A = C$ and $B = C$, then $A = B$)
2. If equals are added to equals, the wholes are equal. (If $a = b$, then $a + c = b + c$)
3. If equals are subtracted from equals, the remainders are equal. (If $a = b$, then $a - c = b - c$)
4. Things which coincide with one another are equal to one another. (If two figures overlap completely, they are equal)
5. The whole is greater than the part.
6. Things which are double of the same things are equal to one another.
7. Things which are halves of the same things are equal to one another.

5. Euclid’s Postulates (अभिकल्पनाएँ)

Postulates are assumptions specific to geometry.

• Postulate 1: A straight line may be drawn from any one point to any other point. (One and only one line passes through two distinct points)
• Postulate 2: A terminated line can be produced indefinitely. (A line segment can be extended infinitely on both sides)
• Postulate 3: A circle can be described with any centre and any radius.
• Postulate 4: All right angles are equal to one another (each equal to $90^\circ$).
• Postulate 5 (Parallel Postulate): 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 the angles is less than two right angles.

6. Equivalent Versions of Euclid’s Fifth Postulate

• Playfair's Axiom: "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 $l$."
• Two distinct intersecting lines cannot be parallel to the same line.

7. Important Theorems & Results

• Theorem: There is a unique line that passes through two given distinct points.
• Intersecting lines: Two distinct intersecting lines cannot have more than one point in common.