Axioms or postulates are the assumptions which are obvious universal truths. They are not proved.
Axiom 1: Things which are equal to the same thing are equal to one another.
Draw a line segment AB of length 10cm. Draw a second line CD having length equal to that of AB, using a compass. Measure the length of CD. We see that, CD = 10cm.
We can write it as, CD = AB and AB = 10cm implies CD = 10cm.
Axiom 2: If equals are added to equals, the wholes are equal.
Suppose we have two line segments AB and DE of equal length. Add BC to AB and add EF to DE. If BC = EF, then AC = DF.
Axiom 3: If equals are subtracted from equals, then the remainders are equal.
Suppose we have two line segments of equal length. Remove BC from AC and EF from DE respectively. If BC = EF, then AB = DE.
Axiom 4: Things which coincide with one another must be equal to one another.
This means that if two geometric figures can fit completely one in to another, they are essentially the same.
Axiom 5: The whole is greater than the part.
Take a container of water. Remove some water from it. Will the remaining volume of water the same as the original volume?