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FUNDAMENTAL CONCEPT OF GEOMETRY

__FUNDAMENTAL CONCEPT OF GEOMETRY -__

**Geometry is the branch of mathematics that deals with the properties and relationships of points, lines, shapes, surfaces, and solids. It originated from the study of shapes and spatial relationships in the natural world and has since evolved into a field with a wide range of practical and theoretical applications. Below are the fundamental concepts of geometry:**

**1. Points:-**

**A point is a precise location or position in space. It has no dimensions, no length, width, or height, and is usually represented as a dot.**

**2. Lines:-**

**A line is a straight one-dimensional figure extending infinitely in both directions. It has length but no thickness. Two points determine a line.**

**3. Line Segments:-**

**A line segment is a portion of a line defined by two endpoints. Unlike a line, it has a definite length.**

**4. Rays:-**

**A ray is a part of a line that starts at a point and extends infinitely in one direction.**

**5. Angles:-**

**An angle is formed by two rays (or line segments) that share a common endpoint, called the vertex. Angles are measured in degrees or radians.**__Acute angle:-__Less than 90°__Right angle:-__Exactly 90°__Obtuse angle:-__Greater than 90° but less than 180°__Straight angle:-__Exactly 180°.

**6. Planes:-**

**A plane is a flat, two-dimensional surface that extends infinitely in all directions. It has no thickness.**

**7. Shapes:-**

** 2D Shapes:- These include figures such as triangles, rectangles, circles, and polygons, which lie in a plane.**

__Triangles:-__Three-sided polygons (e.g., equilateral, isosceles, scalene).__Quadrilaterals:-__Four-sided polygons (e.g., squares, rectangles, trapezoids).__Polygons:-__Multi-sided shapes (e.g., pentagons, hexagons).

**8. Solids:-**

__3D Shapes:-__Objects with three dimensions, such as cubes, spheres, pyramids, and cylinders.__Polyhedra:-__Solids with flat polygonal faces (e.g., cubes, pyramids).__Curved solids:-__Objects like spheres and cylinders with curved surfaces.

**9. Congruence and Similarity:-**

__Congruence:-__Two shapes are congruent if they have the same size and shape, meaning they can be superimposed.__Similarity:-__Two shapes are similar if they have the same shape but possibly different sizes, meaning their corresponding angles are equal, and their sides are proportional.

**10. Perimeter, Area, and Volume:-**

__Perimeter:-__The total distance around a 2D shape.__Area:-__The amount of space enclosed within a 2D shape.__Volume:-__The amount of space enclosed within a 3D object.

**11. Pythagorean Theorem:-**

**In right-angled triangles, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Expressed as a^2 + b^2 = c^2.**

**12. Transformations:-**

**Movements of shapes in a plane that include:-**__Translation:-__Sliding a shape without rotating or flipping it.__Rotation:-__Turning a shape around a fixed point.__Reflection:-__Flipping a shape over a line to produce a mirror image.__Scaling (Dilation):-__Resizing a shape larger or smaller while maintaining its proportions.

**13. Coordinate Geometry:-**

**A system where geometric shapes are analyzed using a coordinate plane, typically with x and y-axes. Points are represented by coordinates (x, y), and equations describe lines and curves.**

**14. Basic Theorems:-**

__Euclid's Postulates:-__The foundational rules for classical geometry, formulated by the Greek mathematician Euclid.__Thales' Theorem:-__In a circle, if a triangle is drawn with its base as the diameter, the angle opposite the base is a right angle.

**These concepts form the building blocks of geometry and allow mathematicians and scientists to model, analyze, and understand the physical world.**