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What were Euclid's five postulates?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In his foundational work, the Elements, Euclid established five unprovable but intuitive principles, known as postulates or axioms, to serve as the basis for Euclidean geometry. These are:

Given two points, there is a straight line that joins them.
A straight line segment can be prolonged indefinitely.
A circle can be constructed when a point for its center and a distance for its radius are given.
All right angles are equal.
If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles.1

The fifth principle is famously known as the parallel postulate. Because it was less intuitive than the others, it became the subject of centuries of study, eventually leading to the discovery of non-Euclidean geometries in the 19th century.21