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