Euclid’s Postulates
In Euclidean geometry, postulates are like the basic rules or assumptions that we accept as true without proving them. Think of them as the foundation of a house; before you build the walls and roof, you need a solid base. These postulates give us a starting point to build all the other ideas in geometry. Here are the five key postulates that Euclid came up with, a long time ago:
Postulate 1 -The Straight Line Postulate:: A straight line may be drawn from any one point to any other point. Note that this postulate tells us that at least one straight line passes through two distinct points, but it does not say that there cannot be more than one such line. However, in his work, Euclid has frequently assumed, without mentioning, that there is a unique line joining two distinct points. We state this result in the form of an axiom as follows:
Given two distinct points, there is a unique line that passes through them.
How many lines passing through P also pass through Q ? Only one, that is, the line PQ. How many lines passing through Q also pass through P? Only one, that is, the line PQ. Thus, the statement above is self-evident, and so is taken as an axiom.
Postulate 2-The Line Extension Postulate : A terminated line can be produced infinitely long line. Note that what we call a line segment now-a-days is what Euclid called a terminated line. So, according to the present day terms, the second postulate says that a line segment can be extended on either side to form a line.
Postulate 3-The Circle Postulate : A circle can be drawn with any centre and any radius.Given a point P and a distance r, you can draw a circle with centre P and radius r.
Postulate 4-The All Right Angles Are Equal Postulate: : All right angles are equal to one another.Any two right angles are congruent.
Postulate 5-The Parallel Postulate: 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 sum of angles is less than two right angles.
For example, the line PQ in Fig. 5.6 falls on lines AB and CD such that the sum of the interior angles 1 and 2 is less than 180° on the left side of PQ. Therefore, the lines AB and CD will eventually intersect on the left side of PQ.
These postulates might seem really simple, but they're super powerful. By starting with these basic ideas, mathematicians have been able to figure out all kinds of things about shapes, sizes, and spaces without needing anything else to prove them. It's like having a few LEGO blocks but being able to build a whole castle!