Introduction to Euclid's Geometry

Statement: A sentence which is either true of false is called a statement.

Line: A straight path having no end point is called a line.

Ray: A straight path having one end point is called a ray.

Line Segment: A straight path having two end point is called a line segment.

Theorem: A statement which requires proof is called a theorem.

Corollary: A statement which can be derived from a theorem is called a corollary.

Axioms: The basic facts which are taken for granted, without any proof and which are used throughout in the mathematics are called axioms.

Postulates: The basic facts which are taken for granted, without any proof and which are specific to geometry are called postulates.

Great circle: Circles obtained by the intersection of a sphere and planes passing through the centre of the sphere is called a great circle.

Euclid’s axioms:

i).            Things which are equal to the same thing are equal to one another.
Example : For example, if an area of a triangle equals the area of a rectangle and the area of the rectangle equals that of a square, then the area of the triangle also equals the area of the square.

ii).          If equals are added to equals, the wholes are equal.
Example : if x = y, then x +k = y +k

iii).        If equals are subtracted from equals, the remainders are equal.
Example : if x = y, then x - k = y - k

iv).        Things which coincide with one another are equal to one another.
Example : Segment AB = Segment BA ; ∠A = ∠A .

v).          The whole is greater than the part.
Example : If a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C. Symbolically, A > B means that there is some C such that A = B + C.

vi).        Things which are double of the same things are equal to one another.
Example : If 2x = 2y then x = y.

vii).      Things which are halves of the same things are equal to one another.
Example : If ½ x = ½y then x = y.

Parallel Line Axiom / Playfair’s Axiom: For a given line and a given point not on the line, a unique line can be drawn passing through the given point and parallel to the given line.

Euclid’s Five Postulates:

1.   A straight line may be drawn from any one point to any other point.
2.  A terminated line 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 the same side of it, taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

• A system of axioms is called consistent, if it is impossible to deduce from these axioms a statement that contradicts any axioms or previously proved statement.

Theorem: Two distinct lines cannot meet at more than one point.

Proof: If possible let two lines l and m meet at two distinct points, say P and Q.

So, P lies on l as well as m and Q lies on l as well as m.

But we know that using two distinct point a unique line can be drawn.

Therefore, the above statements are true only when the lines l and m coincides with each other, which contradicts that the lines l and m are distinct lines.

So our assumption was wrong and hence two distinct lines cannot meet at more than one point.