Class 9 — Mathematics (NCERT)
Introduction to Euclid's Geometry: Axioms and Postulates
Class 9
- ✓By the end of this lesson students will be able to understand Euclid's contribution to geometry.
- ✓By the end of this lesson students will be able to define and differentiate between axioms and postulates.
- ✓By the end of this lesson students will be able to state Euclid's seven axioms.
- ✓By the end of this lesson students will be able to state Euclid's five postulates.
- ✓By the end of this lesson students will be able to identify the relevant axiom or postulate in simple geometric statements.
Key concepts
Euclid, a Greek mathematician, is often referred to as the 'Father of Geometry'. Around 300 B.C., he collected and organised all the known work in geometry into his famous treatise called 'The Elements'. This book is divided into 13 chapters, each called a 'book'. Euclid's approach to geometry was revolutionary as he started with a few basic, self-evident truths and then logically deduced a vast system of geometric theorems. This method of reasoning from general statements to specific conclusions is known as deductive reasoning.
Axioms (or Common Notions) are basic facts which are taken for granted without proof. These are general statements that are not specific to geometry but apply to all branches of mathematics. They are self-evident universal truths.
Euclid stated seven axioms in 'The Elements':\n1. Things which are equal to the same thing are equal to one another.\n2. If equals are added to equals, the wholes are equal.\n3. If equals are subtracted from equals, the remainders are equal.\n4. Things which coincide with one another are equal to one another.\n5. The whole is greater than the part.\n6. Things which are double of the same things are equal to one another.\n7. Things which are halves of the same things are equal to one another.
Postulates are also basic facts which are taken for granted without proof. However, unlike axioms, postulates are statements specific to geometry. They are self-evident truths that are fundamental to geometric constructions and reasoning.
Euclid stated five postulates:\n1. A straight line may be drawn from any one point to any other point.\n2. A terminated line can be produced indefinitely.\n3. A circle can be drawn with any centre and any radius.\n4. All right angles are equal to one another.\n5. 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.
Theorems are statements that can be proved using definitions, axioms, postulates, and previously proved statements (theorems). They are derived logically from the foundational truths.
Key facts to remember
- 1Euclid's 'The Elements' is a monumental work that systematised geometry.
- 2Axioms are general, self-evident truths applicable across mathematics.
- 3Postulates are self-evident truths specific to geometry.
- 4Theorems are statements that are proved using definitions, axioms, and postulates.
- 5Euclid stated 7 axioms and 5 postulates.
- 6Euclid's Postulate 5 (Parallel Postulate) is particularly famous and led to the development of non-Euclidean geometries.
Worked examples
Example 1
If x = 5 and y = 5, then x = y. Which of Euclid's axioms supports this statement?
Answer
Euclid's Axiom 1.
This axiom is fundamental for establishing equality between quantities.
Example 2
Can we draw a unique straight line passing through two distinct points P and Q? Which postulate justifies this?
Answer
Yes, a unique straight line can be drawn. This is justified by Euclid's Postulate 1.
This postulate is foundational for defining lines in geometry.
Example 3
A line segment AB has a point C lying on it, such that C is between A and B. Explain why AB > AC. Which axiom is being used here?
Answer
AB > AC. This is supported by Euclid's Axiom 5.
This axiom helps in understanding relationships between parts and wholes in various contexts, not just geometry.
Common mistakes
- ✗Confusing axioms with postulates: Axioms are general, postulates are specific to geometry.
- ✗Trying to prove an axiom or postulate: They are fundamental assumptions, not statements to be proven.
- ✗Misremembering the exact wording of Euclid's axioms and postulates.
- ✗Not understanding the self-evident nature of axioms and postulates.
Exam tips
- ★Memorise the definitions of axiom, postulate, and theorem.
- ★Learn the statements of Euclid's seven axioms and five postulates thoroughly.
- ★Be prepared to identify which axiom or postulate is being used in a given mathematical statement or geometric situation.
- ★Understand the historical context of Euclid's work and its significance in mathematics.
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