What is Euclidean Geometry?
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
The axioms given by Euclid are:
- Things which are equal to the same thing are also equal among themselves.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things which coincide with one another are equal.
- The whole is greater than the part.
The postulates given by Euclid are:
- A line segment can be drawn by joining any two points in any dimension: This means when any two points are joined, they will give the unique line segment.
- An indefinite straight line can be drawn by extending any segment: In easy words, when there is an indefinite extension of any line segment, it will give a unique line.
- A circle can be drawn with any line segment as its radius and one endpoint of that line segment as its center: Consider a straight-line segment of a particular radius and considering its endpoint as its center, draw a circle then the circle will pass through that another endpoint of the line segment.
- All right angles are equal to each other: Since it is obvious that we are considering all the right angles, all these will be 90°, thus all of these will be equal.
- If a straight line intersecting two other straight lines makes the internal angles on either side of it, which when taken together, less than two right angles, if the two straight lines are formed indefinitely, they will meet on that side, at which the sum of the angles is less than two angles. This postulate is called a 'parallel postulate.'
Congruence of Triangles
When two triangles can be superimposed on each other, then the triangles are referred to be congruent. This congruency can be determined by certain theorems which tell about the conditions for congruency.
The first theorem is Side-Angle-Side (SAS):
If any two sides of a triangle and the angle included in between them is equal to the respective two sides and the included angle of another triangle, then the triangles are referred to be congruent.
Similar to this, there are Angle-Side-Angle (ASA) and Side-Side-Side (SSS) congruency theorems. The ASA property is based on the property of an isosceles triangle which states that the two sides of any triangle are equal if and only if the angles opposite to them are also equal.
The above-stated parallel postulate is the foundation of the property that the sum of all the interior angles of a triangle is equal to 180º.
There is no possibility that the triangles aren't congruent when two sides of a triangle and the angle between them are the same as the corresponding sections of another triangle.
All three vertices of the triangle are fixed when two sides and their included angle are fixed. As a result, all it takes to define a triangle is two sides and their included angle; by demonstrating the congruence of these corresponding components, the congruence of each complete triangle follows.
Similarity of Triangles
- As discussed, congruent triangles refer to triangles of the same shape and size. Now for similar triangles, triangles are similar in shape, but the size can be different. The shape is closely related to the proportion of sides. If two triangles are similar, then the corresponding sides are proportional, i.e., a:b:: c:d, where a and b are the sides opposite to a pair of equal angles of the similar triangles, and c and d are the sides opposite to another pair of equal angles of the similar triangles.
- Now the basic fundamental theorem of similarity is the "Thales’ theorem". It was proposed by a mathematician named "Thales".
- According to the Thales’ theorem, when a line is drawn from one side of a triangle intersecting another side such that this line splits the respective sides into the same ratio, this line will be parallel to the third side of the triangle.
- A similar property as above can be restated as the Angle-Angle-Angle (AAA) similarity theorem, which says that in a triangle, two sides are proportional only when their corresponding angles are equal.
The Pythagoras theorem states that the square of the length of the hypotenuse of a right triangle is the same as the sum of the squares of the lengths of the other two sides of the triangle.
The converse of Pythagoras theorem states that, for any given triangle, if the sum of squares of two sides is equal to the third side squared, then that triangle is right-angled.
It is one of the most important theorems in mathematics. It is used to calculate the distances, angles, etc.
- Let’s first know what chord is. A chord is the line segment obtained by joining any two points on the circumference of a circle. When this chord passes through the center of the circle, it is called the diameter of that circle.
- Let’s now consider a semicircle. Its endpoints will be on a diameter. Thales proves that the angle subtended by diameter on any point on the circle will be a right angle.
- Thus, on summarizing the above points, it is concluded that most of the advanced theorems of plane Euclidean geometry are proved with the help of these theorems.
The world around us is three-dimensional, with width, depth, and height. The solid geometry is related to the objects in the space, such as cubes and spheres. In comparison, plain geometry deals with objects that are flat or have only two dimensions, such as triangles and lines, that can be drawn on a flat piece of paper. From plane to solid geometry, some concepts remain the same such as proportions and angles.
In conjunction with Euclidean geometry, there is another geometry referred to as non-Euclidean geometry. In arithmetic, non-Euclidean geometry includes two geometries based totally on axioms carefully associated with the ones that specify Euclidean geometry. As the Euclidean geometry rests on the convergence of the metric and the affine geometry, the non-Euclidean geometry metric makes an appearance with the help of relaxing the requirement or replacing the parallel postulate with an alternative.
Context and Applications
This topic is significant in the professional exams for k-12, undergraduate and postgraduate courses, especially for:
- Class 10
- Class 11
- Class 12
- B.Sc. Mathematics
- M.Sc. Mathematics
- Parabolic geometry
- Euclid geometry
- Solid geometry
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