- Euclid was a teacher of mathematics at Alexandria in Egypt, popularly known as ‘Father of Geometry”.
- Axioms or Postulates: Axioms or postulates are the assumptions which are obvious universal truths. They are not proved.
Mainly postulates are used for especially geometry and axioms are used for especially algebra. - An axiom is a statement, which is common and general, and has a lower significance and weight. A postulate is a statement with higher significance and relates to a specific field.
- The word ‘Axiom’ is derived from the Greek word ‘Axioma’ meaning ‘true without needing a proof’.
- Postulate means a fact, or truth of (something) as a basis for reasoning, discussion, or belief. These are the statements assumed to be true without any requirement of proof.
- Theorems are the statements which can be proved.
Euclid gave 5 axioms. Euclid’s 5 axioms are:
1. Things which are equal to the same thing are also equal to one another.
If a = b and b = c, then a = b.
2. If equals be added to equals, the wholes are equal.
This basically means this:
If a = b, then a + c = b + c.
3. If equals be subtracted from equals, the remainders are equal.
This is similar to the second axiom. This tells that:
If a = b, then a – c = b – c.
4. Things which coincide with one another are equal to one another.
Two things which are exactly alike are equal. For e.g. two triangles are congruent, they coincide with each other. There sides, angles, perimeter, area, all are equal.
5. The whole is greater than the part.
This means that x + a > x.
All the other theorems in geometry are derived from these 5 axioms.
Postulates
The assumptions which are very specific in geometry are called Postulates.
There are five postulates by Euclid-
1. A straight line may be drawn from any one point to any other point.
This shows that a line can be drawn from point A to point B, but it doesn’t mean that there could not be other lines from these points.
2. A terminated line can be produced indefinitely.
This shows that a line segment which has two endpoints can be extended indefinitely to form a line.
3. A circle can be drawn with any centre and any radius.
This shows that we can draw a circle with any line segment by taking one of its points as a centre and the length of the line segment as the radius. As we have AB line segment, in which we took A as the centre and the AB as the radius of the circle to form a circle.
4. All right angles are equal to one another.
As we know that a right angle is equal to 90° and all the right angles are congruent because if any angle is not 90° then it is not a right angle.
As in the above figure ∠DCA =∠DCB =∠HE =∠HGF= 90°
5. Parallel Postulate
If there is a line segment which passes through two straight lines while forming two interior angles on the same side whose sum is less than 180°, then these two lines will definitely meet with each other if extended on the side where the sum of two interior angles is less than two right angles.
And if the sum of the two interior angles on the same side is 180° then the two lines will be parallel to each other.
Equivalent Versions of Euclid’s Fifth Postulate
1. Play fair’s Axiom
This says that if you have a line ‘l’ and a point P which doesn’t lie on line ‘l’ then there could be only one line passing through point P which will be parallel to line ‘l’. No other line could be parallel to line ‘l’ and passes through point P.
2. Two distinct intersecting lines cannot be parallel to the same line.
This also states that if two lines are intersecting with each other than a line parallel to one of them could not be parallel to the other intersecting line.