Question

“An equilateral triangle is a polygon made up of three line-segments out of which two line segments are equal to the third one and all its angles are 60° each.” Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in an equilateral triangle?

Answer 1

Hint: We have to read the given statement carefully and understand it. We have to define the terms which we feel should be defined like polygon, triangle etc. we have to use Euclidean axioms to prove that all sides and all angles are equal to each other.
The axiom is: “Things which are equal to the same thing are equal to one another.”

Complete step by step solution:
Let’s read the statement given to us once again.
“An equilateral triangle is a polygon made up of three line-segments out of which two line segments are equal to the third one and all its angles are 60° each.”
The terms needed to be defined are:
> Equilateral: By definition, equilateral means having all sides or lengths equal. It is used to categorize a shape with all sides equal.
> Triangle: Any shape formed by three straight lines is called a triangle. The shape should be a closed figure. A triangle can be equilateral (all sides equal), isosceles (only two sides equal) and scalene (all three sides are different).
> Polygon: A closed figure formed by straight lines is called a polygon. We know that to form a closed shape, we need at least 3 sides. Therefore, the basic polygon is a triangle. Other polygons are rectangles, pentagons, hexagons etc.
> Line-segment: A line-segment joins any two points. It can also be defined as a part of a line with two endpoints and can not be extended. It has a fixed length.
> Angles: An angle is formed when two rays meet at a common point. It can be acute, right, obtuse etc.
The undefined terms are lines, points and rays.
The given statement tells us that: two line-segments are equal to the third one and all its angles are equal to 60°.
Using the Euclidean axiom: “Things which are equal to the same thing are equal to one another.”
When two line-segments are equal to the third line-segment, this means that all the three line-segments are equal. All the angles are equal to 60°, this means that all angles are equal.
Therefore, all three sides of an equilateral triangle are equal. And all angles are also equal.

Note: A clear understanding of the given statement is very important. Knowing the definitions of basic terms like angles, triangle, polygon etc. will be very helpful in solving the equation. Having a thorough knowledge about Euclidean axioms will help us to prove that all sides and angles are equal in an equilateral triangle.