Revision Notes for Class 9 Mathematics: Introduction to Euclid's Geometry
Review targeted revision notes for Class 9 Mathematics with the CBSE Class 9 Mathematics Introduction to Euclid's Geometry Notes Set 04. Built according to official educational guidelines for the 2026-27 academic year, these downloadable summaries for Introduction to Euclid's Geometry support daily study and last-minute exam readiness.
Review Introduction to Euclid's Geometry for Class 9 Mathematics
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Points, Lines and Surfaces
1. A solid has shape, size and position and can be moved. Its boundaries are surfaces.
2. The boundaries of a surface are curves or straight lines, and the ends of a line are points.
3. Euclid described a point as that which has no part, a line as length without breadth, and a surface as that which has only length and breadth.
4. A straight line lies evenly with the points on itself, and a plane surface lies evenly with the straight lines on itself.
Axioms and Postulates
Euclid listed the assumptions he used without proof in two groups. Those meant only for geometry are called postulates. The general ones used in all of mathematics are called common notions or axioms.
| Common notion (axiom) | In simple words |
|---|---|
| Things equal to the same thing are equal to one another. | If a = c and b = c, then a = b. |
| If equals are added to equals, the wholes are equal. | a = b gives a + c = b + c. |
| If equals are subtracted from equals, the remainders are equal. | a = b gives a - c = b - c. |
| Things which coincide with one another are equal to one another. | Shapes that fit exactly on each other are equal. |
| The whole is greater than the part. | A segment is longer than its part. |
| Things which are double of the same things are equal to one another. | Doubles of the same thing match. |
| Things which are halves of the same things are equal to one another. | Halves of the same thing match. |
Euclid's Five Postulates
1. A straight line may be drawn from any one point to any other point.
2. A terminated line (segment) can be produced indefinitely.
3. A circle can be drawn with any centre and any radius.
4. All right angles are equal to one another.
5. If a straight line falling on two straight lines makes the interior angles on one side together less than two right angles, then the two lines, if produced indefinitely, meet on that side.
The fifth postulate is long and was the subject of centuries of study. An equivalent form is Playfair's axiom: for every line l and every point P not on l, there is exactly one line through P that is parallel to l.
Results that Follow
1. Two distinct points are joined by exactly one line, so two distinct lines cannot have more than one point in common.
2. Two distinct intersecting lines cannot both be parallel to the same line.
3. An equilateral triangle can be constructed on any given line segment.
4. Every line segment has one and only one mid-point.
Terms to Remember
| Term | Meaning |
|---|---|
| Line segment | Part of a line with two end points |
| Ray | Part of a line with one end point |
| Collinear points | Three or more points on the same line |
| Non-collinear points | Points that do not all lie on one line |
| Angle | Formed by two rays from the same end point |
| Mid-point | Point between P and R with PQ = QR |
| Theorem | A statement proved using definitions, axioms and earlier results |
| Consistent system | A system of axioms that never leads to a contradiction |
Worked Examples
Example 1. If PQ = SR where P, S, Q, R lie on a line in this order, show that PS = QR. Solution: PQ = PS + SQ and SR = SQ + QR. Since PQ = SR, PS + SQ = SQ + QR. Subtracting SQ from both sides (equals subtracted from equals) gives PS = QR.
Example 2. Point C lies between A and B with AC = CB. Show that C is the only mid-point. Solution: Suppose D is another mid-point, so AD = DB. Then AC and AD are both halves of AB, so they are equal, which means C and D are the same point.
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Revision Notes and Key Concepts for Class 9 Mathematics Introduction to Euclid's Geometry
About Introduction to Euclid's Geometry Revision Notes
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You can download the teacher prepared revision notes for CBSE Class 9 Mathematics Introduction to Euclid's Geometry Notes Set 04 from StudiesToday.com. These notes are designed as per 2026-27 academic session to help Class 9 students get the best study material for Mathematics.
Yes, our CBSE Class 9 Mathematics Introduction to Euclid's Geometry Notes Set 04 include 50% competency-based questions with focus on core logic, keyword definitions, and the practical application of Mathematics principles which is important for getting more marks in 2026 CBSE exams.
Yes, our CBSE Class 9 Mathematics Introduction to Euclid's Geometry Notes Set 04 provide a detailed, topic wise breakdown of the chapter. Fundamental definitions, complex numerical formulas and all topics of CBSE syllabus in Class 9 is covered.
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