If $A$,$B$ and $C$ are three points on a line,and $B$ lies between $A$ and $C$ (see figure),then prove that $AB + BC = AC$.

How would you rewrite Euclid's fifth postulate so that it would be easier to understand?

Give a definition for the term 'line segment'. Are there other terms that need to be defined first? What are they,and how might you define them?

Give a definition for parallel lines. Are there other terms that need to be defined first? What are they,and how might you define them?

Which of the following statements are true and which are false? Give reasons for your answers.
$(i)$ Only one line can pass through a single point.
$(ii)$ There are an infinite number of lines which pass through two distinct points.
$(iii)$ $A$ terminated line can be produced indefinitely on both the sides.
$(iv)$ If two circles are equal,then their radii are equal.
$(v)$ In the figure,if $AB = PQ$ and $PQ = XY,$ then $AB = XY$.

Consider two 'postulates' given below:
$(i)$ Given any two distinct points $A$ and $B$,there exists a third point $C$ which is in between $A$ and $B$.
$(ii)$ There exist at least three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid's postulates? Explain.