Q. If a point $C$ lies between two points $A$ and $B$ such that $AC=BC$, then prove that $AC=\frac{1}{2}AB$. Explain by drawing the figure.

Q. Point $C$ is the mid-point of line segment $AB$, prove that every line segment has one and only one mid-point.

Q. Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?
(i) parallel lines
(ii) perpendicular lines
(iii) line segment
(iv) radius of a circle
(v) square

Q. 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 there postulates consistent? Do they follow from Euclid's postulates? Explain.

Q. In the figure, if $AC=BD$, then prove that $AB=CD$.

Q. Why is Axiom $5$, in the list of Euclid's axioms, considered as a 'universal truth'? (Note that the question is not about the fifth postulate.)

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

Q. Does Euclid's fifth postulate imply the existence of parallel lines? Explain.