MathsCBSE Class 9 · Introduction to Euclid's Geometry: Axioms and Postulates

Euclid's Definitions and Axioms (Common Notions): Class 9

Updated 17 October 2026 · Fact-checked

Euclid began with definitions such as 'a point is that which has no part', but these use undefined words, so modern geometry treats point, line and plane as undefined terms. His axioms (common notions) are general truths used in all mathematics, such as 'things equal to the same thing are equal to one another' and 'the whole is greater than the part'. A proof in geometry names the axiom used for each step.

Each common notion works on quantities of the same kind: equal bars stay equal after adding or removing equal pieces, and a whole exceeds its part.

Illustration drawn for QuestionBuddy.

Understand Euclid's Definitions and Axioms (Common Notions)

Euclid of Alexandria organised the geometry of his time into a chain of reasoning: definitions, then axioms (common notions), then postulates, then propositions (theorems) proved from them. This page deals with the first two links.

Definitions and undefined terms. Euclid defined a point as "that which has no part", a line as "breadthless length" and the ends of a line as points. But "part" and "breadth" are themselves words that need definitions, which need further words, and so on. A chain of definitions cannot go on for ever, so modern geometry accepts point, line and plane as undefined terms whose meaning is understood intuitively, and defines everything else (segment, ray, angle, triangle) from them.

Axioms (common notions). An axiom is a statement accepted without proof as obviously true, and used across all of mathematics, not only geometry. The common notions in Euclid's list are:
1. Things equal to the same thing are equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things that coincide with one another are equal to one another.
5. The whole is greater than the part.
6. Things which are double of the same things are equal to one another.
7. Things which are halves of the same things are equal to one another.

They speak of magnitudes of the same kind: lengths with lengths, angles with angles, areas with areas. A length of 12 cm and an area of 12 cm $^{2}$ cannot be compared, even though the numbers match.

Using axioms in a proof. Suppose points $A, B, C, D$ lie on a line in that order and $A C = B D$ . Then $A C = A B + B C$ and $B D = B C + C D$ (the whole is the sum of its parts). Since $A C = B D$ , subtracting the common $B C$ from both (axiom 3) gives $A B = C D$ . Each line of such a proof carries a reason: a fact from the figure, an earlier result, or an axiom by name. If $M$ is the midpoint of $A B$ then $A M = M B = \frac{1}{2} A B$ , and by axiom 7 halves of equal segments are equal.

Axiom, postulate, theorem. An axiom is general, a postulate (next page) is specific to geometry, and a theorem is a statement proved from definitions, axioms, postulates and earlier theorems. Modern mathematics uses the word "axiom" for both axioms and postulates. A set of axioms must be consistent: it should never be possible to prove a statement and its opposite from it.

Exam pattern. Expect 1-mark questions that ask you to classify a statement, 2- or 3-mark proofs on a line with points in order, and short reasoning questions on why point, line and plane are not defined.

Euclid's Definitions and Axioms (Common Notions): formulas to remember

Axiom 1: $a = c, b = c \Rightarrow a = b$; Things equal to the same thing are equal to one another.
Axioms 2 and 3: $a = b \Rightarrow a + c = b + c, a - c = b - c$; Equals added to or subtracted from equals.
Axiom 5: $whole > part$; If $B$ lies between $A$ and $C$ then $A C > A B$ .
Axioms 6 and 7: $a = b \Rightarrow 2 a = 2 b and \frac{a}{2} = \frac{b}{2}$; Doubles and halves of equals are equal.
Point between two points: $B between A, C \Rightarrow A B + B C = A C$; Used in nearly every segment proof.
Midpoint: $M midpoint of A B \Rightarrow A M = M B = \frac{1}{2} A B$; Halves of the same thing are equal.

How to solve Euclid's Definitions and Axioms (Common Notions) questions

A segment proof is a chain of reasons; write each reason next to its line.

1Draw a line and place the points in the order given.
2Write the whole in terms of its parts, such as $A C = A B + B C$ .
3Write what is given as an equation between segments.
4Choose the axiom that links the lines: add equals (2), subtract equals (3), same thing (1), doubles (6) or halves (7).
5Write the conclusion and state the axiom beside each step.
6Check that every comparison is between magnitudes of the same kind.

Euclid's Definitions and Axioms (Common Notions) in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Subtracting equals

Suppose four fence posts $A, B, C, D$ stand in a straight line in that order, and the distance from $A$ to $C$ is exactly the same as the distance from $B$ to $D$ .

Show that the gap between the first two posts equals the gap between the last two.

Try it first, then show the solutionSolution

$A C = A B + B C$ and $B D = B C + C D$ .
$A C = B D$ is given, so $A B + B C = B C + C D$ .
Subtract the common gap $B C$ from both sides (equals subtracted from equals): $A B = C D$ .

Answer: The first and last gaps are equal: $A B = C D$ .

Memory hook: Cancel the shared piece.

Tricks to remember Euclid's Definitions and Axioms (Common Notions)

Short memory aids. Say each one aloud once, then use it on a question the same day.

Add, subtract, same, double, half
Axioms 2, 3, 1, 6 and 7 are about adding, subtracting, equal to the same thing, doubles and halves of equals.
Use it when: Naming the axiom in a proof.
Whole is bigger
Axiom 5 is the only inequality: the whole is greater than the part.
Use it when: Proofs that one segment is longer than another.
Same kind only
Compare lengths with lengths, areas with areas.
Use it when: Spotting invalid reasoning.

Euclid's Definitions and Axioms (Common Notions): solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Midpoint and half · 3 marks · Short answerBoard level

In the figure $C$ lies between $A$ and $B$ and $A C = B C$ . Show that $A C = \frac{1}{2} A B$ , and name the axioms used.

Try it first, then show the solutionSolution

$C$ lies between $A$ and $B$ , so $A B = A C + C B$ (the whole equals the sum of its parts). (1)
$C B = A C$ (given), so $A B = A C + A C = 2 A C$ (equals added to equals). (1)
Halves of equals are equal (axiom 7): $A C = \frac{1}{2} A B$ . (1)

Answer: $A C = \frac{1}{2} A B$ .

Example 2: Subtracting equals with numbers · 3 marks · Short answerBoard level

Points $P, Q, R, S$ lie on a line in this order with $P R = QS = 17$ cm and $QR = 5$ cm. Find $P Q$ , $R S$ and $P S$ , and name the axiom that shows $P Q = R S$ .

Try it first, then show the solutionSolution

$P R = P Q + QR$ , so $P Q = 17 - 5 = 12$ cm. (1)
$QS = QR + R S$ , so $R S = 17 - 5 = 12$ cm. Thus $P Q = R S$ because equals ( $P R$ , $QS$ ) with the same $QR$ taken away leave equal remainders (axiom 3). (1)
$P S = P Q + QR + R S = 12 + 5 + 12 = 29$ cm. (1)

Answer: $P Q = R S = 12$ cm and $P S = 29$ cm.

Example 3: Chain of midpoints · 5 marks · Long answerHarder board level

On a line, $B$ is the midpoint of $A C$ and $C$ is the midpoint of $B D$ . Prove that $A B = C D$ , and find $A D$ if $B C = 6$ cm.

Try it first, then show the solutionSolution

$B$ is the midpoint of $A C$ , so $A B = B C$ . (1)
$C$ is the midpoint of $B D$ , so $B C = C D$ . (1)
Things equal to the same thing are equal: $A B = C D$ (axiom 1). (1)
The points lie in the order $A, B, C, D$ , so $A D = A B + B C + C D$ . (1)
With $B C = 6$ : $A B = C D = 6$ and $A D = 18$ cm. (1)

Answer: $A B = C D$ and $A D = 18$ cm.

Example 4: Why not define a point · 3 marks · Short answerHarder board level

Euclid defined a point as "that which has no part". Explain why geometry today treats point, line and plane as undefined terms, using this definition as the example.

Try it first, then show the solutionSolution

The definition uses the word "part", which itself would need a definition. (1)
Defining "part" needs further words, which need definitions again, and so on; the chain can never end or it becomes circular. (1)
So a few terms (point, line, plane) are accepted as undefined and understood by intuition, and all other terms are defined using them. (1)

Answer: Every definition uses other words, so some terms must be left undefined to avoid a never-ending or circular chain.

Example 5: Is the reasoning valid? · 4 marks · Case-basedHOTS / competency

Case: Anu reasons as follows. (A) " $A B = 6$ cm and $C D = 6$ cm, so $A B = C D$ ." (B) "A rectangle has area 12 cm $^{2}$ and a segment is 12 cm long, so they are equal." (C) " $B$ lies between $A$ and $C$ , so $A C > A B$ ." (D) "If $A B = 10$ cm and $B C = 4$ cm, with $B$ between $A$ and $C$ , then $A C = 14$ cm." (i) Which of A and B are valid, and why? (ii) Which axiom justifies C? (iii) Check D.

Try it first, then show the solutionSolution

(i) A is valid: both are equal to 6 cm, and things equal to the same thing are equal (axiom 1). B is invalid: an area and a length are different kinds of magnitude, and the axioms compare only magnitudes of the same kind. (1½)
(ii) C follows from axiom 5: the whole is greater than the part. (1)
(iii) $A C = A B + B C = 10 + 4 = 14$ cm, so D is correct, and also consistent with C because $14 > 10$ . (1½)

Answer: (i) A valid, B invalid; (ii) axiom 5; (iii) D is correct, $A C = 14$ cm.

Where students lose marks in Euclid's Definitions and Axioms (Common Notions)

The slips examiners see most often, and what to write instead.

Mistake: No axiom named
The steps are right, but the reason is missing.
Wrong working
$A B + B C = B C + C D$ , so $A B = C D$ .
Right working
$A B + B C = B C + C D$ ; subtracting the common $B C$ (equals subtracted from equals are equal), $A B = C D$ .
Do this: Write the axiom beside the step.
Usually costs: 1 mark in a 3-mark proof.
Mistake: Comparing unlike magnitudes
Treating equal numbers as equal quantities.
Wrong working
Area $= 16$ cm $^{2}$ and perimeter $= 16$ cm, so they are equal.
Right working
An area and a length are different kinds of magnitude and cannot be equal.
Do this: Check units before using an axiom.
Usually costs: The conclusion.
Mistake: Calling a postulate an axiom, or a theorem a postulate
Mixing the three terms.
Wrong working
"The angles of a triangle add to $18 0^{\circ}$ " is a postulate.
Right working
It is a theorem: it is proved from earlier results.
Do this: Ask: is it assumed (axiom or postulate) or proved (theorem)?
Usually costs: 1 mark.

Exam tips

Draw the line with points in the stated order before writing anything.
Quote the axiom by name or by its meaning (for example "equals subtracted from equals").
Use " $A C = A B + B C$ " only when $B$ lies between $A$ and $C$ ; check the order.
For classification questions, give a one-line reason.

Practice questions on Euclid's Definitions and Axioms (Common Notions)

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
A statement that is accepted without proof and is used in every branch of mathematics is called
(a) a theorem
(b) a common notion (axiom)
(c) a corollary
(d) a construction
Hint
Think of what Euclid called "common notions".
Answer
(b). Axioms or common notions are accepted without proof and used throughout mathematics; a theorem is proved, a corollary follows from a theorem, and a construction is a procedure.
Q2 · 3 marks · Short answerBoard level
Classify each statement as an axiom (common notion), a postulate or a theorem: (i) things which coincide with one another are equal to one another; (ii) a circle can be drawn with any centre and any radius; (iii) the angles of a triangle add up to $18 0^{\circ}$ ; (iv) halves of the same things are equal to one another.
Hint
Axioms are general; postulates are geometric assumptions; theorems are proved.
Answer
(i) Axiom (common notion 4). (ii) Postulate (about constructions in geometry). (iii) Theorem, since it is proved from earlier results. (iv) Axiom (common notion 7).
Q3 · 3 marks · Short answerHarder board level
Points $A, B, C, D$ lie on a line in this order with $A B = B C = C D$ . $M$ is the midpoint of $A B$ and $N$ is the midpoint of $C D$ . Show that $M N = 2 A B$ , and find $M N$ if $A D = 36$ cm.
Hint
Write $M N = M B + B C + C N$ and use halves.
Answer
$M N = M B + B C + C N$ . Here $M B = \frac{1}{2} A B$ and $C N = \frac{1}{2} C D = \frac{1}{2} A B$ (halves of equals are equal), and $B C = A B$ . So $M N = \frac{1}{2} A B + A B + \frac{1}{2} A B = 2 A B$ . With $A D = A B + B C + C D = 3 A B = 36$ , $A B = 12$ cm and $M N = 24$ cm.
Q4 · 5 marks · Long answerHarder board level
Points $A, B, C, D, E$ lie on a line in this order with $A B = D E$ and $B C = C D$ . (i) Prove that $A C = C E$ . (ii) If $A E = 30$ cm and $B C = 6$ cm, find $A B$ and $A C$ . State the axioms you use.
Hint
Add equals to equals: $A B + B C$ and $D E + C D$ .
Answer
(i) $A C = A B + B C$ and $C E = C D + D E$ . Since $A B = D E$ and $B C = C D$ , adding equals to equals gives $A C = C E$ (axiom 2). (ii) $A E = A B + B C + C D + D E = 2 A B + 2 B C$ , so $30 = 2 A B + 12$ and $A B = 9$ cm. Then $A C = 9 + 6 = 15$ cm (and $C E = 6 + 9 = 15$ cm, as proved).
Q5 · 4 marks · Case-basedHOTS / competency
Case: a 36 m rope $A B$ has midpoint $M$ . Another rope $C D$ is also 36 m long and has midpoint $N$ . A third rope $P Q$ is exactly double the length of $A M$ . (i) Find $A M$ . (ii) Show that $A M = C N$ and name the axiom. (iii) Show that $P Q = A B$ and name the axiom. (iv) Which axiom guarantees that $A B > A M$ ?
Hint
Use axioms 7, 6 and 5 in turn.
Answer
(i) $A M = \frac{36}{2} = 18$ m. (ii) $A B = C D$ (both 36 m), and $A M$ , $C N$ are halves of them, so $A M = C N$ (axiom 7: halves of equals are equal). (iii) $P Q = 2 A M$ and $A B = 2 A M$ (since $M$ is the midpoint), so $P Q = A B$ (axiom 6: doubles of the same thing are equal, or axiom 1). (iv) Axiom 5: the whole is greater than the part, as $M$ lies between $A$ and $B$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Is "point" defined in modern geometry?Show answer
No, point, line and plane are undefined terms.
2.How many common notions did Euclid list?Show answer
Seven, in the list used in this course.
3.Which axiom says $A C > A B$ when $B$ lies between $A$ and $C$ ?Show answer
The whole is greater than the part.
4.What is a theorem?Show answer
A statement proved from definitions, axioms, postulates and earlier theorems.
5.Do the axioms compare a length with an area?Show answer
No, only magnitudes of the same kind.

Euclid's Definitions and Axioms (Common Notions): frequently asked questions

What is the difference between an axiom and a postulate?

In Euclid's scheme, axioms (common notions) are general truths for all mathematics, while postulates are assumptions about geometry. Today both are called axioms.

Why can't we prove an axiom?

Because it is a starting point. Proof needs earlier statements to start from, so some statements must be accepted without proof.

Are Euclid's definitions still used?

Their spirit is. We still use points, lines and planes, but we leave them undefined and define the other terms from them.