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Euclidean geometry is a mathematical system attributed to the ancient Greek mathematician Euclid, which describes the properties and relations of points, lines, surfaces, and solids in two and three dimensions. It is based on five postulates, including the famous parallel postulate, which forms the foundation for much of classical geometry taught in schools today.
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Non-Euclidean geometry is a branch of mathematics that explores geometries based on axioms fundamentally different from those of Euclidean geometry, allowing for the study of curved spaces. It is crucial for understanding the geometry of the universe in the context of general relativity, where space can be curved by gravity.
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Plane geometry is a branch of mathematics that deals with shapes and figures on a two-dimensional surface, such as lines, circles, and polygons. It is fundamental for understanding spatial relationships and is widely used in fields like engineering, architecture, and computer graphics.
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Solid geometry is the branch of mathematics that studies three-dimensional figures, including their properties, measurements, and relationships. It extends the principles of plane geometry into three dimensions, allowing for the exploration of volume, surface area, and the spatial relationships between different geometric solids.
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Geometric constructions involve creating shapes, angles, and lines using only a compass and straightedge, adhering to the principles of classical geometry. These constructions are fundamental for understanding geometric properties and relationships without relying on measurements or numerical calculations.
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Congruence refers to the idea that two figures or objects are identical in shape and size, meaning they can be perfectly overlapped. In mathematics, congruence is a fundamental concept in geometry and number theory, where it describes figures that are equivalent under rigid transformations or numbers that have the same remainder when divided by a given modulus.
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Similarity is a fundamental concept in mathematics and science, referring to the degree of resemblance between two objects, patterns, or sets of data, often used to identify relationships, patterns, or predict outcomes. It is crucial in fields like geometry, where similar figures have proportional sides and identical angles, and in machine learning, where similarity measures help in clustering and classification tasks.
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The Pythagorean theorem is a fundamental principle in geometry that establishes a relationship between the lengths of the sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, providing a crucial tool for calculating distances and solving various mathematical problems.
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Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric shapes and their properties in a coordinate system. It allows for the precise calculation of distances, angles, and other geometric properties using coordinates, making it an essential tool in fields such as physics, engineering, and computer graphics.
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Geometric proofs are logical arguments presented in a structured format, often involving a series of deductive steps, to demonstrate the truth of geometric propositions or theorems. They rely on axioms, definitions, and previously established theorems to validate geometric relationships and properties.
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Angles are a measure of rotation between two intersecting lines or rays, often measured in degrees or radians. They are fundamental in geometry, trigonometry, and various applications like navigation, physics, and engineering, providing a way to quantify direction and orientation.
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Triangles are three-sided polygons characterized by the sum of their interior angles always equaling 180 degrees. They can be classified based on side lengths (equilateral, isosceles, scalene) or angles (acute, right, obtuse), each with unique properties and applications in geometry and trigonometry.
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A circle is a two-dimensional shape consisting of all points in a plane that are equidistant from a fixed point called the center. It is characterized by its radius, diameter, circumference, and area, and plays a fundamental role in geometry and trigonometry.
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Polygons are two-dimensional geometric figures with straight sides that are fully closed, and they can have any number of sides greater than two. Understanding polygons is crucial for studying more complex geometrical shapes and for applications in various fields such as computer graphics and architecture.
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Transformations refer to operations that alter the form, appearance, or structure of something, often used in mathematics and science to describe changes in position, shape, or size while preserving certain properties. They are fundamental in understanding symmetry, geometry, and the behavior of functions and systems across various domains.
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A straightedge is a tool used in geometry to draw straight lines, typically without any measurement markings, distinguishing it from a ruler. It is essential in classical geometric constructions, where it is used alongside a compass to create precise geometric figures without numerical measurements.
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