Euclidean Geometry

Euclidean geometry is the study of geometrical shapes (plane and solid) and figures based on different axioms and theorems. It is basically introduced for flat surfaces or plane surfaces. Geometry is derived from the Greek words ‘geo’ which means earth and ‘metrein’ which means ‘to measure’. 

Euclidean geometry is better explained especially for the shapes of geometrical figures and planes. This part of geometry was employed by the Greek mathematician Euclid, who has also described it in his book, Elements. Therefore, this geometry is also called Euclid geometry. 

The axioms or postulates are the assumptions that are obvious universal truths, they are not proved. Euclid has introduced the geometry fundamentals like geometric shapes and figures in his book elements and has stated 5 main axioms or postulates. Here, we are going to discuss the definition of euclidean geometry, its elements, axioms and five important postulates.

History of Euclid Geometry

The excavations at Harappa and Mohenjo-Daro depict the extremely well-planned towns of Indus Valley Civilization (about 3300-1300 BC). The flawless construction of Pyramids by the Egyptians is yet another example of extensive use of geometrical techniques used by the people back then. In India, the Sulba Sutras, textbooks on Geometry depict that the Indian Vedic Period had a tradition of Geometry.

The development of geometry was taking place gradually, when Euclid, a teacher of mathematics, at Alexandria in Egypt, collected most of these evolutions in geometry and compiled it into his famous treatise, which he named ‘Elements’.

What is Euclidean Geometry?

Euclidean Geometry is considered an axiomatic system, where all the theorems are derived from a small number of simple axioms. Since the term “Geometry” deals with things like points, lines, angles, squares, triangles, and other shapes, Euclidean Geometry is also known as “plane geometry”. It deals with the properties and relationships between all things.

Examples of Euclidean Geometry

The two common examples of Euclidean geometry are angles and circles. Angles are said as the inclination of two straight lines. A circle is a plane figure, that has all the points at a constant distance (called the radius) from the center.

Euclidean and Non-Euclidean Geometry

There is a difference between Euclidean and non-Euclidean geometry in the nature of parallel lines. In Euclidean geometry, for the given point and line, there is exactly a single line that passes through the given points in the same plane and it never intersects.

Non-Euclidean is different from Euclidean geometry. The spherical geometry is an example of non-Euclidean geometry because lines are not straight here.

Properties of Euclidean Geometry

• It is the study of plane geometry and solid geometry
• It defined point, line and a plane
• A solid has shape, size, position, and can be moved from one place to another.
• The interior angles of a triangle add up to 180 degrees
• Two parallel lines never cross each other
• The shortest distance between two points is always a straight line

Elements in Euclidean Geometry

In Euclidean geometry, Euclid’s Elements is a mathematical and geometrical work consisting of 13 books written by ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt. Further, the ‘Elements’ was divided into thirteen books that popularized geometry all over the world. As a whole, these Elements is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions.

Book 1 to 4th and 6th discuss plane geometry. He gave five postulates for plane geometry known as Euclid’s Postulates and the geometry is known as Euclidean geometry. It was through his works, we have a collective source for learning geometry; it lays the foundation for geometry as we know it now.

What were Euclidean Axioms?

Here are the seven axioms are given by Euclid for geometry.

1. Things which are equal to the same thing are equal to one another. 
2. If equals are added to equals, the wholes are equal. 
3. If equals are subtracted from equals, the remainders are equal. 
4. Things which coincide with one another are equal to one another. 
5. The whole is greater than the part. 
6. Things which are double of the same things are equal to one another. 
7. Things which are halves of the same things are equal to one another

What were Euclid’s Five Postulates?

Before discussing Postulates in Euclidean geometry, let us discuss a few terms as listed by Euclid in his book 1 of the ‘Elements’. The postulated statements of these are:

• Assume the three steps from solids to points as solids-surface-lines-points. In each step, one dimension is lost.
• A solid has 3 dimensions, the surface has 2, the line has 1 and the point is dimensionless.
• A point is anything that has no part, a breadthless length is a line and the ends of a line point.
• A surface is something that has length and breadth only.

It can be seen that the definition of a few terms needs extra specification. Now let us discuss these Postulates in detail.

Euclid’s Postulate 1

“A straight line can be drawn from any one point to another point.”

This postulate states that at least one straight line passes through two distinct points but he did not mention that there cannot be more than one such line. Although throughout his work he has assumed there exists only a unique line passing through two points.

Euclid’s Postulate 2

“A terminated line can be further produced indefinitely.”

In simple words what we call a line segment was defined as a terminated line by Euclid. Therefore this postulate means that we can extend a terminated line or a line segment in either direction to form a line. In the figure given below, the line segment AB can be extended as shown to form a line.

Euclid’s Postulate 3

“A circle can be drawn with any centre and any radius.”

Any circle can be drawn from the end or start point of a circle and the diameter of the circle will be the length of the line segment.

Euclid’s Postulate 4

“All right angles are equal to one another.”

All the right angles (i.e. angles whose measure is 90°) are always congruent to each other i.e. they are equal irrespective of the length of the sides or their orientations.

Euclid’s Postulate 5

“If a straight line falling on two other straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on the side on which the sum of angles is less than two right angles.”

To learn More on 5th postulate, read: Euclid’s 5th Postulate

Further, these Postulates and axioms were used by him to prove other geometrical concepts using deductive reasoning. No doubt the foundation of present-day geometry was laid by him and his book the ‘Elements’.

Euclid Geometry Worksheet

1. How many dimensions do solids, points and surfaces have?
2. What is the shape of a pyramid’s base?
3. If a + b =10 and a = c, then prove that c + b =10.
4. Can two distinct intersecting lines be parallel to each other at the same time? Justify.
5. Read the following sentence and mention which of Euclid’s axiom is followed: “X’s salary is equal to Y’s salary. Due to the recession, the salaries of X and y are reduced to half. Now the final salary of X will still be equal to Y.”

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• Co-ordinate Geometry
• Euclid Geometry

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