Since \(\angle1=\angle3\ and \ \angle2=\angle4\), therefore adding before equations.
\(\angle1+\angle2=\angle3+\angle4\)
\(\Rightarrow \angle BAD= \angle BCD\)
\(\Rightarrow \angle A= \angle C.\)

Given\(\angle ABC=\angle ACB\)
\(\Rightarrow \angle1+\angle4=\angle2+\angle3\)
\(\Rightarrow \angle1+\angle4-\angle4=\angle2+\angle3-\angle3\)
(As, \(\angle3=\angle4\))
\(\Rightarrow \angle1=\angle2\)

AB = BC (given)
BX = BY (given)
If equals are subtracted from equals, then remains are also equal.
AB - BX = BC - BY
\(\Rightarrow\) AX = CY

Here, \(\angle1=\angle3, \angle2=\angle4\ and\ \angle3=\angle4\), Euclid's first axiom says, the things which are equal to same things are equal to one another.

So, \(\angle 1=\angle 2\)

(i) Since this is true for anything in any part of the world, this is a universal truth.
(ii) If the sum of the cointerior angles made by a transversal intersect two straight lines at distinct points is less than 180^o, then the lines cannot be parallel.

We have
AC = BD
\(\Rightarrow \) AC - BC= BD - BC
If equals are subtracted from equals, the remainders are equal (Euclid's Axiom (iii))
\(\Rightarrow \) AB = CD
AC - BC coincides with AB; BD - BC coincides with CD [Things which coincide with one another are equal to one another (Euclid's Axiom (iv))]

Let a line AB have two mid-points, say, C and D. Then,
AC = \(1\over2\) AB     ...(1)
AD = \(1\over2\)AB     ...(2)
From (1) and (2),
AC = AD.
Things which are equal to the same thing are equal to one another

Yes! These postulates contain two undefined terms: Point and Line. Yes! These postulates are consistent because they deal with two different situations
(i) says that given two points A and B, there is a point C lying on the line in between them,
(ii) says that given A and B, we can take C not lying on the line through A and B. These 'postulates' do not follow from Euclid's postulates, however, they follow from Axiom 'Given two distinct lines, there is a unique line that passes through them.

(i) False. This can be seen visually.
(ii) False. This contradicts the Axiom.
[Given two distinct points, there is a unique line that passes through them.]
(iii) True by Euclid's Postulate 
[A terminated line can be produced indefinitely.]
(iv) True. If we superimpose the region bounded by one circle on the other, then they coincide. So, their centres and boundaries coincide, therefore, their radii will coincide.
(v) True by the first Axiom of Euclid.
[Things which are equal to the same thing are equal to one another.]