Plane Geometry, Euclid's Postulates

Euclid's Postulates

Here are Euclid's 5 postulates, the foundation for plane geometry. They aren't as rigorous as the axioms of modern mathematics, but it's interesting to see where he was coming from. These ideas will be expanded in subsequent sections.

  1. A straight line segment can be drawn joining any two points.

  2. Any straight line segment can be extended indefinitely, to form a straight line.

  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

  4. All right angles are congruent.

  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The converse is also assumed, although not stated directly. If the inner angles are 180° or more the lines do not meet on that side. Parallel lines never meet, hence the angles on either side sum to 180°. More on this later.

lines almost parallel, but intersecting at the right