It looks like you're providing definitions related to right triangles and the Pythagorean theorem. Here’s a concise organization of the terms and their definitions:
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Hypotenuse: The side of a right triangle that is across from the right angle.
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Legs: The sides of a right triangle that form the right angle.
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Proof: A logical sequence of steps used to prove a theorem from other known facts.
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Pythagorean Theorem: A theorem that states if a triangle is a right triangle, then the sum of the squares of the legs is equal to the square of the hypotenuse (usually expressed as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse).
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Right Angle: An angle that measures 90 degrees.
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Right Triangle: A triangle with a right angle.
If you need further information or additional terms, feel free to ask!