Chapter 5 – Introduction to Euclid’s Geometry:

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:

• Given: A, B, and C are points on the same line, with B between A and C.

• 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.

• Segment AB and segment BC are "parts" of AC, so:

  $$AB + BC = AC$$

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:

1. With A as the center and AB as the radius, draw a circle.

2. With B as the center and AB as the radius, draw another circle.

3. Let the circles intersect at point C.

4. Join AC and BC.

Reasoning:

• All radii of the same circle are equal (Postulate 3).

• So, AC = AB and BC = AB.

• Hence, AB = BC = CA.

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:

• Let’s assume two distinct lines l₁ and l₂ intersect at two points A and B.

• Then both points A and B lie on both lines.

• From Euclid’s Axiom 5.1:

  Given two distinct points, there is a unique line that passes through them.

• 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

• Reason: Infinite number of lines can pass through a single point.

(ii) There are an infinite number of lines which pass through two distinct points.
❌ False

• Reason: Euclid’s Axiom 5.1 states: "Given two distinct points, there is a unique line that passes through them."

(iii) A terminated line can be produced indefinitely on both the sides.
✅ True

• Reason: Euclid’s Postulate 2 states: A terminated line (line segment) can be produced indefinitely.

(iv) If two circles are equal, then their radii are equal.
✅ True

• Reason: By definition, equal circles have equal radii.

(v) If AB = PQ and PQ = XY, then AB = XY.
✅ True

• Reason: Euclid’s Axiom 1: Things which are equal to the same thing are equal to one another.

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:

• 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:

• They do not contradict each other.

• They can both be true in the same geometrical space.

(c) Do they follow from Euclid’s postulates?

• (i) does not directly follow from Euclid’s postulates but can be accepted as an additional postulate to define betweenness.

• (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

🖊️ Proof:

Let points A, C, and B lie on a straight line with C between A and B.

Given:

• AC = BC

• AB = AC + CB
  But AC = BC ⇒ AB = AC + AC = 2AC
  Therefore, AC = (1/2) AB

🔍 Figure:

A -------- C -------- B  
      <-->    <-->  
      AC      CB (equal lengths)

5. Prove: Every line segment has one and only one midpoint

🖊️ Proof:

Let AB be a line segment.
By definition, a point M is the midpoint of AB if:

• AM = MB

• 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)

🖊️ Given:

Points A, B, C, D lie on a line such that:

• A → B → C → D

• AC = BD

We want to prove AB = CD.

Let’s assume the line is as follows:

A ---- B ---- C ---- D

Then:

• AC = AB + BC

• 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?

🔍 Explanation:

• Axiom 5: The whole is greater than the part.

This is considered a universal truth because:

• It applies not just to geometry, but to all mathematical and physical quantities — length, area, volume, number, etc.

• It is intuitively obvious and doesn’t need proof.

• It holds true across all fields of math and science, not just in Euclidean geometry.

For example:

• If a rope is cut into two parts, clearly the entire rope is longer than either of the parts.

Hence, it's called a universal axiom — not dependent on any specific geometric figure or construction.