Postulate I: It is possible to draw a straight line from any point to any other point.

As per this postulate, if we have two points P and Q on a plane, then we can draw at least one line that simultaneously passes through the two points. Euclid does not mention that only one line can pass through two points, but he assumes the same. The fact that only one line can pass through any two points is illustrated in the following figure.

Postulate II: A terminated line can be produced indefinitely.

This postulate can be considered an extension of the first postulate. According to this postulate, we can make a straight line that is different from a given line by extending its points on both sides of the plane.

In the following figure, MN is the original line, while M'N' is the new line formed by extending the original line in either direction.

Postulates III and IV

Postulate III: It is possible to describe a circle with any centre and radius.

According to Euclid, acircleis a plane figure consisting of a set of points that are equidistant from a reference point. It can be drawn with the knowledge of its centre and radius.

Circles with different radii have different sizes but the same shape.

Postulate IV: All right angles are equal to one another.

A right angle is unique in the sense that it measures exactly 90. Hence, all right angles are of the measure 90, irrespective of the lengths of their arms. Thus, all right angles are equal to one another. For example, in the following figure, ABC = GHI = DEF = 90.

Explanation of All Four Postulates

Following rules were observed while measuring the lengths of line segments and angles. These rules were not stated separately but these were assumed by Euclid in the derivation of new postulates. These can be taken as additional postulates.

Rule 1.Every line segment has a positive length.

Rule 2.If a point R lies on the line segment PQ, then the length of PQ is equal to the sum of the lengths of PR and RQ. That is, PQ = PR + RQ

Rule 3.Every angle has a certain magnitude. A straight angle measures 180.

Rule 4.If raysare such thatlies between,then POQ = POR + ROQ

Rule 5.If the angle between two rays is zero then they coincide. Conversely, if two rays coincide, then angle between them is either zero or an integral multiple of 360.

Explanation of All Four Postulates

Example 1:

Prove that an isosceles triangle can be constructed on any given line segment.

Solution:

Say we have a line segment AB of any length.

Let us extend AB to the points X and Y in either direction, such that AX = BY.

Now, as per Euclids second axiom, we have:

AX + AB = BY + AB

⇒BX = AY (1)

Using Euclids third postulate, let us draw a circle with A as the centre and AY as the radius. Similarly, let us draw another circle with B as the centre and BX as the radius. Let the circles intersect at a point C.

Now, let us join A and B to C to get the line segments AC and BC respectively. We thus obtain ΔACB.

Now, we have to prove that ΔACB is isosceles, i.e., AC = BC.

AY and AC are the radii of the circle with centre A; BX and BY are the radii of the circle with centre B.

∴AY = AC (2)

Similarly, BX = BC (3)

From equations 1, 2 and 3 and Euclids axiom that things which are equal to the same thing are equal to one another, we can conclude that AC = BC.

So, ΔACB is isosceles.

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