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A compass and a straightedge
The angle-bisector construction is proven effective by demonstrating that the two angles formed by the bisector are congruent. This is achieved using the properties of isosceles triangles, where the lengths of the sides opposite the equal angles are shown to be proportional to the lengths of the adjacent sides of the original angle. Additionally, the use of geometric tools like a compass and straightedge allows for the accurate replication of distances and angles, ensuring that the bisector divides the angle into two equal parts. Thus, the congruence of the resulting angles confirms that the construction reliably bisects any angle.
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Plato believed that the only tools needed for geometric calculations was a compass and a ruler. The type of compass being the type with a pencil attached to a pointing device.
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