Euclidean geometry can be defined as the study of geometry (especially for the shapes of geometrical figures) which is attributed to the Alexandrian mathematician Euclid who has explained in his book on geometry which is known as Euclid’s Elements of Geometry. This geometry can basically universal truths, but they are not proved. Euclid introduced the geometry fundamentals like geometric figures and shapes in his book elements and has also stated 5 main axioms or postulates. We are going to discuss the definition of Euclidean geometry, Euclid’s elements of geometry, Euclidean geometry axioms and the five important postulates of Euclidean Geometry.
What is Euclidean Geometry?
Let’s know what is Euclidean Geometry-
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Euclidean Geometry is an axiomatic system. Here all the theorems are derived from the small number of simple axioms which are known as Euclidean geometry axioms.
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We know that the term “Geometry” basically deals with things like points, line, angles, square, triangle, and other different shapes, the Euclidean Geometry axioms is also known as the “plane geometry”.
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Euclidean Geometry deals with the properties and the relationship between all the things. Euclidean geometry is different from Non-Euclidean. They differ in the nature of parallel lines.
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In Euclid geometry, for the given point and a given line, there is exactly a single line that passes through the given points in the same plane and doesn’t intersect.
Elements of Euclidean Geometry
Euclid’s Elements can generally be defined as a mathematical and geometrical work consisting of thirteen number of books that is written by ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt. Now further, the ‘Elements’ was further divided into thirteen books which had popularized geometry all over the world. As a whole, these Elements are basically a collection of definitions, postulates or axioms, propositions ( that is theorems and constructions), and mathematical proofs of the propositions.
We have already discussed what is Euclidean geometry , now let’s know what are Euclid’s axioms or Euclidean geometry axioms.
Euclid’s Axioms
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Axiom 1 |
Things are equal to one another if only those things are equal to the same thing. |
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Axiom 2 |
The wholes are equal if the equals are added to equals. |
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Axiom 3 |
The remainders are always equal if equals are subtracted from equals. |
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Axiom 4 |
Things are equal to one another only if they coincide with one another. |
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Axiom 5 |
The whole is always greater than the part. |
Euclid Geometry Postulates:
Let us discuss a few terms that are listed by Euclid in his book 1 of the ‘Elements’ before discussing Euclid’s geometry Postulates .The postulated statements of these are as follows:
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Assume that the three steps from solids to points as solids-surface-lines-points. And now in each step, one dimension is lost.
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A solid has generally has three dimensions, the surface has two dimensions, the line has 1 and the point is dimensionless.
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A point is anything that has no part, and a breadth less length is a line and the ends of a line point.
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A surface is something which has only length and breadth.
Let’s get know these Euclid’s geometry postulates in a better way!
Euclid’s Postulate No 1 |
“A straight line can be drawn from any one point to another given point.” The first postulate states that at least one straight line passes through two distinct points but it has not been mentioned that there cannot be more than one such line. Although throughout Euclid’s work he has assumed there exists only a unique line passing through two points. For better understanding see the picture given below: () |
Euclid’s Postulate No 2 |
“A terminated line can always be further produced indefinitely.” In simple words we can say that a line segment was defined by Euclid as a terminated line. Therefore the second postulate says that we can extend a line segment or a terminated line in either direction to form a line. The line segment AB can be extended as shown to form a line in the diagram given below. () |
Euclid’s Postulate No 3 |
“A circle can be drawn with any centre and with any radius.” We can draw any circle 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 No 4 |
The fourth postulate says that “All right angles are equal to one another.” All the right angles (right angles are the angles whose measure is 90°) will always be congruent to each other i.e. they are always equal irrespective of the length of the sides or their orientations. |
Euclid’s Postulate No 5 |
“If a straight line that falls on two other straight lines makes interior angles on the same side of the lines taken together less than two right angles, then these two straight lines, if produced indefinitely, they meet on the side on which the sum of the angles is less than two right angles.”Further, these Postulates and axioms were used by Euclid to prove other different geometrical concepts using deductive reasoning. |
Questions to be Solved:
Question 1): If a point C lies between given two points A and B such that AC is equal to BC, then prove that AC is equal to [frac{1}{2}] AB. Explain your understanding by dra
wing the figure.
Solution)Given, the length of AC = BC
()
Now, you need to add AC on both sides.
L.H.S + AC = R.H.S + AC
Then, AC + AC = BC + AC
We can now write 2AC = BC + AC
Since, we already know,
BC +AC = AB (as it coincides with the given line segment AB, from figure)
Therefore, 2 AC = AB (If equals are added to equals, then the wholes are equal.)
⇒ AC = [frac{1}{2}] AB.