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How are a straightedge and a compass used in Euclidean geometry?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

In Euclidean geometry, the straightedge and compass are the fundamental tools used for geometric constructions. As established by the Greek mathematician Euclid in his seminal work, the Elements, these instruments are restricted to specific functions to maintain the rigor of deductive proofs.1

The Straightedge: Used exclusively as an unmarked tool to draw a straight line segment between two points. It cannot be used to measure distances or mark specific lengths.21
The Compass: Used to draw circles or arcs with a given center and radius. It allows for the transfer of distances, as the radius of a circle can be maintained to mark equal lengths elsewhere in a construction.21

This restriction to an unmarked straightedge and compass remains a hallmark of elementary Euclidean geometry. While later mathematicians like Archimedes explored methods involving marked tools—such as neusis—to solve complex problems like angle trisection, the classical Euclidean tradition strictly adheres to the use of these two basic tools.12