(i) The terms need to be defined are:
Polygon: A simple closed figure made up of three or more line segments.
Line segment: part of a line with two end points.
line: undefined term.
Point: undefied term.
Angle: A figure formed by two rays with a common initial point.
Ray: Part of a line with one end point.
Right angle: Angle whose measure is 90^o
Undefined terms used are: Line, point
Euclid's fourth postulate says that "all right angles are equal to one another." in a square, all angles are right angles, therefore all angles are equal (From Euclid's fourth postulate)
Three line segments are equal to fourth line segment.
Therefore, all the four sides of a square are equal( by Euclid's first axiom " things which are equal to one another")
(ii) Introduction to Euclid's geometry
(iii) Equality leads to democracy.

AD is part of AF.
As whole is greater than part
Two more axioms
If equals are added two equals, the whole are equal.
e.g., if \(m\angle1=m\angle2,\) then
\(m\angle1+m\angle3=m\angle2+m\angle3\)
if equals are subtracted from equals, the remainders are equal.
e.g., if \(m\angle1=m\angle2\), then
\(m\angle1-m\angle3=m\angle2-m\angle3\) 

Here \(OX=\frac{1}{2}XY\)
\(PX=\frac{1}{2}XZ\)
Also OX = PX
\(\Rightarrow \frac{1}{2}XY=\frac{1}{2}XZ\)
Things equal to half of equals, are equal to one another.
Two other axioms:
Things coincide with one another are equal to one another.
e.g., If \(\overline{AB}\) coincide with \(\overline{XY}\), such that A falls on X and B falls on Y, then \(\overline {AB}=\overline{XY}\)
The whole is greater than the part
e.g., if \(m\angle1=m\angle2+m\angle3,\ then\ m\angle1>m\angle2\ and\ m\angle1>m\angle3\)

Since \(\angle1=\angle3 \ and \ \angle2=\angle4,\) therefore adding both equation
\(\angle1+\angle2=\angle3+\angle4\)
\(\Rightarrow \angle BAD=\angle BCD\)
\(\Rightarrow \angle A=\angle C\)
If equals are added to equal, the wholes are equal
Two more axioms:
Things which are equal to the same thing are equal to one another
e.g., if \(\overline {AB}=\overline{ PQ} \ and\ \overline {PQ}=\overline {XY}, then\ \angle{AB}=\angle{XY}\)
If equals are subtracted from equals, the remainders are equal.
e.g., if \(m\angle1=m\angle2\) then
\(m\angle3=m\angle3\)
\(=m\angle2-m\angle3\)

AC = BC             I Given

⇒ AC + AC =  BC + AC         | If equals are added to equals, the wholes are equal.
⇒ 2AC = AB
AC = \(1\over2\)  AB
(ii) AC = OC              I Given
AC = \(1\over2\)AB                  I Proved in (i) above
\(\therefore\) CB = \(1\over2\)  AB
(iii) AC = BC                I Given
\(\therefore\) AC: BC = 1 : 1
\(\therefore\) Apala is correct. So, the value 'Sharpness' is depicted by her statement.
(iv) The mathematical concept 'Introduction to Euclid's Geometry' has been covered in this problem.
(v) The formulae used in the solution are as follows:
1. If equals are added to equals, the wholes are equal.
2. Concept of ratio.

\(\because\)  C is the midpoint of AB
\(\therefore \)  AC = CB
AC + AC = CB + AC
| If equals are added to equals, then   the wholes are equal (Euclid's Axiom (ii))]
\(\Rightarrow \)  2AC =  AB      I CB + AC coincides with AB 
\(\Rightarrow \)  \(1\over2\)(2AC) = \(1\over2\) AB
| Things which are halves of the same thing are equal (Euclid's Axiom (vii»]
\(\Rightarrow \)  AC =\(1\over2\)AB
\(\Rightarrow \) \(1\over2\)AC  = \(1\over2\)(\(1\over2\)AB)
| Things which are halves of the same thing are equal to one another (Euclid's Axiom (vii))]
\(1\over2\)AC = \(1\over2\)AB
AD = \(1\over4\)AB
\(\because\) D is the mid-point of AC
\(\therefore \)AD = DC =\(1\over2\)AC (as above)

\(\angle \)2 = \(\angle \)4          ,,,,,,,(1) | Given
\(\angle \)1= \(\angle \)4          ........(2) | Given
\(\angle \)3 =\(\angle \)2         .......(3) | Given
From (1), \(\angle \)4 = \(\angle \)2           ......(4)
From (3) and (4),
\(\angle \)3 = \(\angle \)4               .....(5)
I Things which are equal to the same thing are equal to one another
From (2) and (4),
\(\angle \)1 = \(\angle \)3
I Things which are equal to the same thing are equal to one another

AC = XD I Given
2AC = 2XD
\(\therefore \)  Things which are double of the same things are equal to one another
\(\Rightarrow \) AB = XY
\(\therefore \)  C is the midpoint of AB and D is the midpoint of XY

We have, AX=CY
AB=CD
\(\Rightarrow \) AB + BC = CD + BC
| If equals are added to equals, the wholes are equal (Euclid's Axiom (ii))
AC= BD
I AB + BC coincides with AC;
CD + BC coincides with BD
[Things which coincide with one another are equal to one another (Euclid's Axiom (iv))] 

We have,
x - 15 = 25
\(\Rightarrow \)  x - I5 + 15 = 25 + 15  | If equals are added to equals, the wholesare equal (Euclid's Axiom (ii))
\(\Rightarrow \) x = 40