Question

Read the following statement:"A square is a polygon made up of four line segment, out of which , length of three line segments are equal to the length of fourth one and all its angle are right angles ",

Define the term used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all angles and sides of a square are equal?

Answer 1

Hint:
Here we will define all the terms related to geometry like polygon, line segment etc. We will find if there is any undefined term or not. We will then use the concept of Euclid's fourth postulate to prove that all angles and sides of a square are equal.

Complete step by step solution:
Here we will first define the terms which are needed.
Polygon: a polygon is defined as the two dimensional and a simple closed figure which is made up of three or more line segments.
Line segment: A line segment is defined as a line which has two endpoints and the length of the line segment is always fixed and this is equal to the distance between these two end points.
Right angle: An angle whose measure is $90^\circ $ is known as the right angle.
Here the term ‘line’ and the term ‘point’ are undefined.
According to Euclid's fourth postulate, all right angles are always equal to each other and in square, we know that all angles are right angles.
So from Euclid's fourth postulate, we can say that all the angles are equal to each other.
It is also given that the three line segments are equal to fourth line segments. From Euclid's first axiom, we know that the things which are equal to the same thing are also equal to one another.

Therefore, we can say that all the four sides of a square are equal.

Note:
We know that a square is defined as the two dimensional closed geometrical figure whose all sides are equal to each other. Also, all the angles of a square are equal to each other and each of the angles of the square measures $90^\circ $. The diagonals of squares are equal and bisect each other at $90^\circ $. All the opposite sides of the square are parallel to each other, so it is a special case of a parallelogram.