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Intuitive geometry examples include visualizing a triangle's area by seeing it as half of a rectangle formed by duplicating the triangle. Another example is understanding the concept of symmetry by folding a butterfly image in half and noticing that both sides match perfectly. The notion of parallel lines can be illustrated by train tracks that never meet, while the Pythagorean theorem can be grasped by imagining a right triangle inscribed in a square, where the areas of the squares on the legs add up to the area of the square on the hypotenuse.

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