The Five Postulates: Where It All Began

Euclid was a Greek mathematician who organized the known mathematical theorems during his time as well as formulated and proved many of his own. These theorems were written in  Elements. In the Elements, Euclid developed the five postulates or axioms, in which all theorems were deduced. The Elements was used by many schools for more than 2000 years and is still the major basis of the high school Geometry that you are learning today. For this achievement, Euclid was considered as the Father of Geometry.

The Five Postulates

  1. A straight line segment maybe can be drawn joining any two points.
  2. A straight line segment maybe extended without bound in a straight line.
  3. Given any line segment, a circle can be drawn having the segment as radius and one point 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. This postulate is equivalent to what is known as the parallel postulate.

All the theorems in Euclidean Geometry was deduced from these give postulates.

Student Trivia

  • The Geometry that you are learning today is called Euclidean Geometry. There are other Geometries that are non-Euclidean, but they are not usually discussed in high school mathematics. As you have guessed, the term Euclidean is in honor of Euclid.
  • There are books that call some theorems postulates; for example, the SAS Postulate. Strictly speaking, they are not really postulates. The only real Euclidean postulates are the five stated above. The reason why some books call theorems postulates so that there is no need for you to prove them.


Weisstein, Eric W. “Euclid’s Postulates.” From MathWorld — A Wolfram Web Resource.

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