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.

3

(a)

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.

3

(a)

Point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.

3

(a)

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

3

(a)

(i) Why is Axiom 5, in the list of Euclid's axioms, considered a 'universal truth'?  (Note that the question is not about the fifth postulate).
(ii) How would you rewrite Eulid's fifth postulate so that it would be easier to understand?

3

(a)

In the given figure, if \(\angle1=\angle3, \angle2=\angle4\ and\ \angle3=\angle4\), write the relation between \(\angle1\ and\ \angle2\) using Euclid's axiom.

3

(a)

In a triangle ABC, X and Y are the points On AB and BC such that BX = BY and AB = BC. Show that AX = CY. State the Euclid's Axiom used.

3

(a)

In the given figure, we have \(\angle ABC=\angle ACB, \angle3=\angle4\). Show that \(\angle 1=\angle2.\)

3

(a)

In the given figure, we have \(\angle1=\angle3\) and \(\angle2=\angle4\). Show that, \(\angle A=\angle C.\)

3

(a)