CBSE Class 9 Mathematics Introduction to Euclid's Geometry Notes Set 02

Euclid and the Axiomatic Method

Euclid organised the geometry known in his time into a logical system. He started from a few basic definitions and statements assumed true, and then proved further results (propositions or theorems) step by step.
A corollary is a result that follows easily from a theorem. A system of axioms is consistent if no statement can be deduced from it that contradicts the axioms.

Definitions

A point has no part; a line has length but no breadth; the ends of a line are points. A surface has length and breadth only and its edges are lines.
A straight line lies evenly with the points on itself and a plane surface lies evenly with the straight lines on itself.

Axioms

Things equal to the same thing are equal to one another. Adding equals to equals gives equals; subtracting equals from equals gives equals. Things that coincide are equal. The whole is greater than the part.
Doubles of the same thing are equal to one another, and so are halves of the same thing.

The Five Postulates

1. A straight line can be drawn from any point to any other point (so two distinct points lie on exactly one line). 2. A terminated line can be extended indefinitely. 3. A circle can be drawn with any centre and any radius. 4. All right angles are equal.
5. If a line falling on two lines makes the interior angles on one side together less than two right angles, the two lines meet on that side when produced. If the angles total exactly 180 degrees, the lines are parallel.
An equivalent form (Playfair's axiom): through a point not on a given line there is exactly one line parallel to it.

Lines, Rays and Planes

Two distinct lines have at most one common point (Theorem 5.1, proved by contradiction using the two-point postulate). Lines with one common point are intersecting; lines with none are parallel; three or more lines through one point are concurrent.
A line segment has two end points; a ray has one. A point on a line splits it into two half lines and the point. A line splits a plane into two half planes and the line.
Three non-collinear points determine exactly one plane; three collinear points lie on infinitely many planes.

Worked Examples

Counting lines: four points with no three collinear determine 6 lines; with exactly three collinear, 4 lines; with all four collinear, 1 line.
If \( 2x=2y \), then \( x=y \) because halves of equal things are equal.
Triangle with sides 12, 5 and 13 satisfies \( 12^2+5^2=13^2 \).