NCERT Class 9 Mathematics Chapter 5 – Introduction to Euclid’s Geometry:
✅ Example 1:
Statement:
If A, B and C are three points on a line, and B lies between A and C, prove that
AB + BC = AC, using Euclid's reasoning.
Reasoning using Euclid’s Axioms:
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Given: A, B, and C are points on the same line, with B between A and C.
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This means the line segment AC is made up of AB and BC.
Using Euclid’s Axiom 2:
If equals are added to equals, the wholes are equal.
Hence, proved using Euclid’s second axiom.
✅ Example 2:
Statement:
Prove that an equilateral triangle can be constructed on any given line segment.
Let the given line segment be AB.
Construction:
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With A as the center and AB as the radius, draw a circle.
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With B as the center and AB as the radius, draw another circle.
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Let the circles intersect at point C.
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Join AC and BC.
Reasoning:
Conclusion:
Triangle ABC is equilateral.
✅ Theorem 5.1:
Statement:
Two distinct lines cannot have more than one point in common.
Proof using Euclid’s Geometry:
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Let’s assume two distinct lines l₁ and l₂ intersect at two points A and B.
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Then both points A and B lie on both lines.
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From Euclid’s Axiom 5.1:
Given two distinct points, there is a unique line that passes through them.
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But if A and B lie on both lines, then l₁ and l₂ must be the same line, not distinct.
Contradiction arises, hence our assumption is wrong.
Conclusion:
Two distinct lines can intersect at most at one point.
✅ Exercise 5.1 – Q1: True or False with Reasons
(i) Only one line can pass through a single point.
❌ False
(ii) There are an infinite number of lines which pass through two distinct points.
❌ False
(iii) A terminated line can be produced indefinitely on both the sides.
✅ True
(iv) If two circles are equal, then their radii are equal.
✅ True
(v) If AB = PQ and PQ = XY, then AB = XY.
✅ True
Exercise 5.1 of Chapter 5: Introduction to Euclid’s Geometry (Class 9 Mathematics):
2. Definitions and Required Pre-definitions
(i) Parallel Lines:
Two lines in a plane that never meet, no matter how far they are extended, are called parallel lines.
✅ Undefined terms required: Line, plane, point, distance.
To define parallel lines, one must understand what a line is and what it means for lines not to intersect.
(ii) Perpendicular Lines:
Two lines that intersect to form a right angle (90°) are called perpendicular lines.
✅ Undefined terms required: Line, angle, right angle.
A definition of angle and specifically right angle is needed.
(iii) Line Segment:
A part of a line that has two endpoints is called a line segment.
✅ Pre-definition needed: Line, point.
(iv) Radius of a Circle:
The distance from the centre of a circle to any point on the circle is called the radius.
✅ Required terms: Circle, distance, centre.
(v) Square:
A quadrilateral with four equal sides and four right angles is called a square.
✅ Required terms: Line segment, angle, right angle, equality of sides.
3. Postulates Analysis
(i) Postulate: Given any two distinct points A and B, there exists a third point C which lies between A and B.
(ii) Postulate: There exist at least three points that are not on the same line.
(a) Undefined terms involved:
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Point, line, between, and distinct — these are foundational and often taken as primitive terms in geometry.
(b) Are they consistent?
Yes, the two postulates are consistent because:
(c) Do they follow from Euclid’s postulates?
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(i) does not directly follow from Euclid’s postulates but can be accepted as an additional postulate to define betweenness.
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(ii) does not contradict Euclid’s postulates and helps describe configurations beyond collinearity, which is required for planar geometry.
4. If C lies between A and B such that AC = BC, then prove AC = ½ AB
๐️ Proof:
Let points A, C, and B lie on a straight line with C between A and B.
Given:
๐ Figure:
A -------- C -------- B
<--> <-->
AC CB (equal lengths)
5. Prove: Every line segment has one and only one midpoint
๐️ Proof:
Let AB be a line segment.
By definition, a point M is the midpoint of AB if:
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AM = MB
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M lies between A and B
✅ Existence:
We can always find such a point M because a line segment has a measurable length. Halving that gives us a point M such that AM = MB.
✅ Uniqueness:
Suppose there are two midpoints M and N of AB.
Then, AM = MB and AN = NB ⇒ AM = AN
This implies M and N coincide.
Hence, there can be only one such point.
6. If AC = BD, prove that AB = CD (Referring to a straight line A → B → C → D)
๐️ Given:
Points A, B, C, D lie on a line such that:
We want to prove AB = CD.
Let’s assume the line is as follows:
A ---- B ---- C ---- D
Then:
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AC = AB + BC
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BD = BC + CD
Given: AC = BD
So, AB + BC = BC + CD
Subtracting BC from both sides:
⇒ AB = CD
Hence proved.
7. Why is Axiom 5 (The whole is greater than the part) a universal truth?
๐ Explanation:
This is considered a universal truth because:
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It applies not just to geometry, but to all mathematical and physical quantities — length, area, volume, number, etc.
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It is intuitively obvious and doesn’t need proof.
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It holds true across all fields of math and science, not just in Euclidean geometry.
For example:
Hence, it's called a universal axiom — not dependent on any specific geometric figure or construction.
