Euclid’s Geometry Class 9 Study Notes

Class 9 Maths syllabus is full of some really tough topics that can make your head spin! Apart from the continuation of old topics being upgraded like Real Numbers, Trigonometry, Polynomials, Linear Equations in Two Variables, you will also be introduced to some unique concepts. One such chapter is on the Introduction to Edulid’s geometry which deals with a range of axioms and theorems. Want to know more about it? Checkout these amazing study notes on Euclid’s Geometry Class 9.

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What is Euclid’s Geometry?

Euclid was a famous mathematician and teacher popularly known as the ‘Father of Geometry’. He was the first one to introduce methods to prove mathematical concepts by using logical reasoning. Euclidean geometry deals with the understanding of geometrical shape and figures on a flat or plain surface using axioms and theorems. The definition of Euclid’s Geometry Class 9 is as follows: 

• A Point has no component or part
• Anything which has length but does not have any breadth is called Line
• The endpoints of a line are known as points, and such a line is called a line segment
• Anything that has length and breadth but no height is called Surface
• When a line has points on itself, such a line is called a straight line
• Edges of the surface are in the form of lines
• A surface with straight lines on itself is referred to as a Plane Surface

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What are Axioms in Euclid’s Geometry?

Some common properties used in mathematics that are not directly related to mathematics are called Axioms. The ones used in Euclid’s geometry class 9 are as follows:

• If two things are equal to the same thing, then they are similar to one another also. If a=b and b=c, then a=c
• If the same things are added to equals, then wholes are also equal. If a is added to b and c where b=c, then a+b=a+c
• If the same things are subtracted to equals, then wholes are also equal. If a is subtracted from b and c where b=c, then b-a=c-a
• If two things coincide with one another, then they are equal
• The whole is always greater than the part
• Things which are double of the same thing are always equal
• Things which are halves of the same thing, then they are equal

Also Read: BODMAS Questions

What are Postulates in Euclid’s Geometry Class 9?

Postulates are assumptions specifically related to Geometry. Euclid gave five postulates, all of which are part of the syllabus for Euclid’s Geometry class 9.

1. A straight line may be drawn from anyone point to any other point. Axiom related to this Postulate states that only a single line can be drawn from 2 unique points.

2. Euclid named a terminated line as a segment stating that it can be drawn indefinitely. So Line Segment is referred to as a line which can be extended from both sides.

3. A circle can be formed with any value of radius and any point as the centre.

4.  All right angles formed are always equal. For Example Angle A = 90° and Angle B = 90°, then Angle A = Angle B.

5.  If there is a line segment passing through two straight lines such that forming two different interior angles on the same side of the line where the sum of angles is less than 180°, then these two lines will intersect with each other if extended on the side where the sum of two interior angles is less than 2 Right angles.

6. If the sum of 2 interior angles on the same side of the line is equal to 2 right angles or 180°, then these two lines will be parallel to each other.

Equivalent Versions of Euclid’s Fifth and Last Postulate

Euclid’s Geometry gives states two equivalent versions of Euclid’s Fifth Postulate which states that sum of 2 interior angles on the same side is equal to 180° means that lines are parallel and If the sum is less than 180° then lines will intersect with each other if extended.

An equivalent version of Euclid’s Fifth Postulate is:

Play fair Axiom: This axiom states that if you have any ‘I’ and any point ‘P’ on some other line except ‘I’, then even if you make infinite lines from Point P still there can be only 1 line parallel to Line I passing from Point P. 

Two distinct intersecting lines can never be parallel to the same line. Still, this version also states that if two lines are intersecting each other, then a line parallel to one of them can never be parallel to the other intersecting line.

Solved Questions for Euclid’s Geometry Class 9

Let’s now understand Euclid’s Geometry class 9 with the help of some solved examples: 

Example 1:
If Point C lies between Line Segment AB such that AC = CB, then prove that AC = ½AB

Solution:
Given: AC = CB
Adding AC to both 
Now, AC + AC = CB + AC
2AC = AB
Hence, AC = ½ AB 

Example 2:
Prove that each and every Line Segment has only one midpoint

Solution:
Let’s assume that Line Segment AB has 2 mid points P and Q
So, AP = PB and AQ = QB
Adding AP and AQ to respective equations 
So, 2AP = AP + PB and 2AQ = QB + AQ
So, Both 2AP and 2AQ are equal to AB
Thus, 2 AP = 2AQ (6^th Axiom)
So, AP = AQ (7^th Axiom)
This, P and Q coincide each other 
So, It is proved that each line segment has only one midpoint

Practice Questions Euclid’s Geometry Class 9 

1. Prove existence of Parallel lines with the help of Euclid’s 5^th Postulate. 
2. If 2 points, B and C between line segment AD such that AC = BD, then prove that AB = CD
3. If Point R lies between Line Segment PQ such that PR = RQ, then prove that PR = ½ PQ
4. What are the five postulates of Euclid’s Geometry?
5. If 2 sales person make the same number of sales in the month of October and their sales double up in the month of November then calculate their sales in November. 

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