Euclid’s Postulates

Euclid's five postulates are foundational assumptions upon which the entire Euclidean geometry is built. Euclid's postulates were proposed by the ancient Greek mathematician Euclid of Alexandria. These five postulates explain how points, lines, and shapes behave in space. In this guide, you will learn about the five postulates along with examples to help understand their real-life applications.

Table of Contents

• What is a Postulate?
• What are Euclid's Five Postulates?
• Solved Examples on Euclid's Five Postulates

What is a Postulate 

Postulates in mathematics are statements in geometry that are valid without being tested or proved. They are basic structures from which lemmas and theorems are derived. They are statements that are accepted as true without proof.

What are Euclid's Five Postulates

A postulate is a statement accepted as true without proof, serving as a starting point for further reasoning and arguments in geometry.

Euclid's five postulates are the following:

• Postulate 1: A straight line can be drawn from any one point to any other point.

This postulate tells us that at least one straight line passes through two distinct points, but does not specify that there cannot be more than one such line. Euclid, throughout his work, assumed that there is a unique line joining two distinct points.

• Postulate 2: A terminated line (line segment) can be further produced indefinitely on either side to form a line.
  
  

• This postulate says that a line segment can be extended on either side to form a line.

• Postulate 3: A circle can be drawn with any centre and any radius.

• Postulate 4: All right angles are equal to one another.
  
  
• Postulate 5 (Parallel Postulate): If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
  
  

In the above figure, we can observe that line PQ intersects lines AB and CD, forming interior angles. The sum of ∠1 and ∠2 is less than 180° on the left side of PQ. ∴ Lines AB and CD will eventually intersect on the left side of PQ.

Solved Examples on Euclid's Five Postulates

Example 1: A line segment AB has length 9 cm. Can it be extended to a line of length 120 cm?
Solution: Yes. A line segment AB of length 9 cm can be extended to a line of length 120 cm using Euclid's second postulate

Example 2: Two lines are cut by a transversal. The co-interior angles on the left side are 35° and 86°. Do the lines meet on the left?
Solution: 35° + 86° = 121° < 180°.
∴ by Euclid's 5th postulate, the lines will meet on the left side.

Example 3: A transversal cuts two lines, making co-interior angles of 70° and 110° on one side. Are the lines parallel?
Solution: 70° + 110° = 180°. ∴ By Euclid's 5th postulate, the lines will be parallel to each other.