---
source: ncert
topic: geometry
class_level: class_9
chapter: ch5
difficulty: beginner
---

Class 9 | Ch5: Introduction to Euclid's Geometry
EUCLID'S DEFINITIONS (important ones):
A point is that which has no part.
A line is breadthless length.
A surface has length and breadth only.
Ends of a line are points. Ends of a surface are lines.
EUCLID'S POSTULATES:
1. A straight line segment can be drawn from any point to any other point.
2. A terminated line (line segment) can be extended indefinitely.
3. A circle can be drawn with any centre and any radius.
4. All right angles are equal to each other.
5. Parallel postulate: If a line falls on two lines such that interior angles on one side sum less than two right angles, those two lines if extended will meet on that side.
EUCLID'S AXIOMS (Common Notions):
1. Things equal to the same thing are equal to each other.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal.
5. The whole is greater than the part.
UNDEFINED TERMS: point, line, plane (defined informally).
THEOREMS FROM POSTULATES:
Theorem 1: Two distinct lines cannot have more than one point in common.
Theorem 2: A unique line passes through two distinct points.
EQUIVALENT VERSIONS OF 5TH POSTULATE:
Given a line and a point not on it, exactly one parallel line passes through that point.
This is Playfair's axiom (equivalent to Euclid's 5th postulate).
SOLVED EXAMPLES:
Example 1: If AB=CD and EF=GH and AB=EF, prove CD=GH.
AB=CD (given). EF=GH (given). AB=EF (given).
By Axiom 1: CD=EF. By Axiom 1: CD=GH.
Example 2: How many lines through 2 distinct points? Exactly 1 (from Postulate 1).
COMMON MISTAKES:
Axioms are assumptions (self-evident truths). Theorems are proved from axioms and postulates.
The 5th postulate is NOT self-evident — this led to non-Euclidean geometry.