It is one thing to assert axioms such as 'parallel lines never meet', an assertion for which there was no obvious falsification for over 2,000 years, as did Euclid.

• There is at most one line that can be drawn parallel to another given one through an external point. (Playfair's axiom)
  
• The sum of the angles in every triangle is 180° (triangle postulate).
  
• There exists a triangle whose angles add up to 180°.
  
• The sum of the angles is the same for every triangle.
  
• There exists a pair of similar, but not congruent, triangles.
  
• Every triangle can be circumscribed.
  
• If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.
  
• There exists a quadrilateral in which all angles are right angles.
  
• There exists a pair of straight lines that are at constant distance from each other.
  
• Two lines that are parallel to the same line are also parallel to each other.
  
• In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).
  
• There is no upper limit to the area of a triangle. (Wallis axiom)
  
• The summit angles of the Saccheri quadrilateral are 90°.
  
• If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. (Proclus' axiom)