# Question on Euclid’s 5 Postulates

Here are the five postulates:

1. Each pair of points can be joined by one and only one straight line segment.
2. Any straight line segment can be indefinitely extended in either direction.
3. There is exactly one circle of any given radius with any given center.
4. All right angles are congruent to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than two right angles.

Questions:

1. These to me sounds more like something that shouldn't require proving... does it?

2. Why is it important to stress things that are obvious? For example, what other answers can you get when extending a line segment other than it can be extended indefinitely in either direction?

3. Similarly, what space can allow two circle of the same radius and center to be not the same?

4. Saying all right angles are congruent ... isn't that the same as saying all 64.506 degree angles are congruent? Isn't it ANY angle are congruent if they are the same degrees measures from the same reference point (say x-axis)?

5. Why do we need the 5th postulate?

• I would like to add, that in a 3D Euclidian space, given a line and a point not on the line, we have more than 1 line that goes through the point and is parallel to the first line. Does this mean the postulates are based on 2D Euclidian space? Perhaps I'm missing something? – ChaoSXDemon Oct 10 '15 at 3:32

The point of Euclid's Elements is to collect statements and constructions concerning lines, points and circles in the two-dimensional plane, all of which are known to be absolutely true. To show the reader that these statements are indeed true, Euclid uses a technique which is even today still the basis of all mathematics, namely mathematical proof. This works as follows: you start with a set of statements, all of which you know to be true beyond doubt, and you show that some other statement is a logical consequence of this statement. It then follows that this last statement must also be true beyond doubt. As an admittedly contrived example, if you already know that the statements $x = 3$ and $y =5$ are both true, then you also know that the statement $x \cdot y = 15$ is true.