1
The brick ratio found in the Indus valley was —
💡 Hint:
Length, breadth, thickness.
Explanation:
4 : 2 : 1.
Class 9 · Maths · Chapter 5 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Introduction to Euclid's Geometry
Introduction to Euclid's Geometry
How to use this page:
1. Read — History · definitions · seven axioms · five postulates · Theorem 5.1 · Exercise 5.1 · forms of the fifth · Exercise 5.2, diagram, worked example, board tip
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- 1 Geometry began in measuring, then asked for reasons
- 2 The definitions the book still prints
- 3 The seven common notions
- 4 The five postulates and the unique line
- 5 Theorem 5.1 and the sum of segments
- 6 Equivalent forms of the fifth postulate
- Chapter winner — every lesson at mastery ★
🎬 Shorts
This chapter has no Short on the channel yet, so ten empty cards are not shown. The notes, diagrams and checks are complete.
📄 PDF download — notes + answer key
Geometry began in measuring, then asked for reasons
ज्यामिति माप से निकली, फिर कारण माँगा · Textbook 5.1 · India and Greece
The work came first, the rules were settled laterNotes
“Geometry” comes from Greek words for earth and for measuring. The Nile floods wiped field boundaries, so Egypt built rules for area. In India the streets of the Indus valley were parallel and the brick ratio was found to be 4 : 2 : 1. The Sulbasutras are manuals for altars and fireplaces. The sriyantra weaves nine isosceles triangles into 43 smaller ones.
There the results were stated, not the reason for every step. In Greece, Thales gave the first known proof that a diameter bisects a circle. Around 300 BCE Euclid arranged the known work in the thirteen books of the Elements.
Worked exampleExample
Question: A brick is in the ratio 4 : 2 : 1 and the length is 20 cm. Find the breadth and the thickness.
Ratio: length : breadth : thickness = 4 : 2 : 1
Substitution: 4 parts = 20 cm, so 1 part = 5 cm. Breadth = 2 × 5 = 10 cm. Thickness = 1 × 5 = 5 cm.
10-second revision
- Geometry is tied to measuring the earth
- In India, the Sulbasutras and the 4 : 2 : 1 brick
- The Elements of Euclid has thirteen books
Board tip · BSEBBoard tip
In a history answer do not write only Greece. The Indus valley and the Sulbasutras are in this chapter too.
Board tip · CBSEBoard tip
The proof of Thales is about a diameter and a circle. Pythagoras is named as his pupil. Do not swap the names.
Check your understandingall correct = mastery ★
2
The Sulbasutras are only a Greek book.
Explanation:
False — they are geometrical construction manuals of ancient India.
3
The Elements of Euclid has ______ books.
💡 Hint:
Each one is called a book.
Explanation:
Thirteen.
4
The first known proof of Thales is about —
💡 Hint:
A circle and its diameter.
Explanation:
A diameter cuts a circle into two equal parts.
5
Write two examples of geometry in ancient India. Also write one different point about Greece.
Model answer (board format)
The Indus valley had parallel streets and bricks in the ratio 4 : 2 : 1. The Sulbasutras describe the construction of altars. In Greece, Thales and Euclid pressed for the reason and the proof of a statement.
Check your answer:
The definitions the book still prints
जो परिभाषाएँ पुस्तक अभी भी छापती है · Textbook 5.2 · point · line · plane
The old lines, and the stand taken todayNotes
The book still prints these statements of Euclid. A point is that which has no part. A line is length without breadth. The ends of a line are points. A straight line lies evenly with the points on it. A surface has only length and breadth. The edges of a surface are lines. A plane surface lies evenly with the straight lines on it.
In these sentences, part, breadth and “evenly” themselves ask for definitions. So a point, a line and a plane are undefined today. A solid has three dimensions, a surface two, a line one, and a point none. In a figure a point is a dot, even though a dot has size.
Worked exampleExample
Question: Why is “a line is length without breadth” not accepted as a definition today?
Answer: Length and breadth are not defined before this sentence. Explaining one word needs a definition of another, and the chain does not stop. So a line is left undefined and the work is done with its picture and the axioms.
10-second revision
- The book still prints seven old definitions
- A point, a line and a plane are now undefined
- Dimensions: solid 3, surface 2, line 1, point 0
Board tip · BSEBBoard tip
If the question says “give the definition”, write the line from the book, and in the next line say that today these are taken as undefined. Do not leave one side out.
Board tip · CBSEBoard tip
Do not merge a ray and a line in one figure. A ray has one end. A line runs without an end both ways.
Check your understandingall correct = mastery ★
1
Today this is left undefined —
💡 Hint:
Three geometric words.
Explanation:
A point, a line and a plane.
2
The book no longer prints the old line about a point.
Explanation:
False — “that which has no part” is still in the chapter. The stand today is that it is undefined.
3
The number of dimensions of a line is ______.
💡 Hint:
A surface has two, a point has zero.
Explanation:
1.
4
The edges of a surface are —
💡 Hint:
The old printed line.
Explanation:
Lines.
5
The dot in a figure is truly a point with no size.
Explanation:
False — a dot has some size. It is only a picture of a point.
6
Write the lines about a point and a line that the book still prints. Then say in one sentence what they are taken to be today.
Model answer (board format)
A point is that which has no part. A line is length without breadth. Today a point and a line are taken as undefined, because the words in these lines themselves ask for definitions.
Check your answer:
The seven common notions
सात सामान्य धारणाएँ · Textbook 5.2 · axioms
These are not private to geometryNotes
The book gives the axioms of Euclid, not in his order, in this form. Things equal to the same thing are equal to one another. If equals are added to equals, the wholes are equal. If equals are subtracted from equals, the remainders are equal. Things that coincide are equal. The whole is greater than the part. Doubles of the same thing are equal. Halves of the same thing are equal.
The fourth axiom is the base of superposition. The fifth gives the meaning of “greater”: A > B means there is some C with A = B + C. Magnitudes of different kinds are not added. A line is not added to a rectangle, and an angle is not compared with a pentagon.
Worked exampleExample
Question: AB = PQ and PQ = XY. What is the relation of AB and XY? Which axiom?
Answer: Both are equal to the same length PQ, so AB = XY. This is the first axiom. The unit stays the same on both sides: if both are 5 cm, then AB is also 5 cm.
10-second revision
- Seven common notions, not proved
- Things that coincide are equal — superposition
- Magnitudes of different kinds are not added
Board tip · BSEBBoard tip
“The whole is greater than the part” is the fifth common notion, not the fifth postulate. When a number is asked, write the name too.
Board tip · CBSEBoard tip
Do not put a line and an area in one sum. Only magnitudes of the same kind are added.
Check your understandingall correct = mastery ★
1
Things which are equal to the same thing are —
💡 Hint:
The first common notion.
Explanation:
Equal to one another.
2
The whole is smaller than its part.
Explanation:
False — the whole is greater than the part. That is the fifth common notion.
3
Things that coincide with each other are ______.
💡 Hint:
Superposition rests on this.
Explanation:
Equal.
4
Which addition is not allowed?
💡 Hint:
The kinds differ.
Explanation:
A line and a rectangle are magnitudes of different kinds.
5
A = B and C = D. Which axiom gives the relation of A + C and B + D? Give one numerical example.
Model answer (board format)
If equals are added to equals, the wholes are equal, so A + C = B + D. Example: 3 cm + 2 cm = 5 cm, and 3 cm + 2 cm on the other side is also 5 cm.
Check your answer:
The five postulates and the unique line
पाँच अभिधारणाएँ और अद्वितीय रेखा · Textbook 5.2 · postulates · Axiom 5.1
Five assumptions made for geometryNotes
First: a straight line may be drawn from any point to any other point. This says “at least one”. The book states Axiom 5.1 separately: exactly one line passes through two distinct points. Second: a terminated segment can be extended both ways. What Euclid called a terminated line is called a line segment today. Third: a circle may be drawn with any centre and any radius. Fourth: all right angles are equal.
The fifth is long. If a straight line falls on two straight lines so that the interior angles on the same side add to less than two right angles, then the lines, if extended, meet on that side. Beside the first four, this one is neither short nor immediately plain. It still could not be proved from the rest, so it stayed a postulate.
Worked exampleExample
Question: P and Q are two distinct points. How many straight lines pass through them?
Answer: By Axiom 5.1, exactly one, the line PQ. The first postulate alone does not say “only one”. The same line is the only one from Q toward P as well.
10-second revision
- Postulate 1 says at least one line
- Axiom 5.1 says exactly one line
- A terminated line is today’s line segment
Board tip · BSEBBoard tip
In the answer “two points, one line”, write the number Axiom 5.1. The first postulate alone stays incomplete.
Board tip · CBSEBoard tip
All right angles are equal. That is the fourth postulate. The measure 90° is a later agreement. The postulate is not a table of measures.
Check your understandingall correct = mastery ★
1
Exactly one line through two distinct points is the statement of —
💡 Hint:
The book adds it separately.
Explanation:
Axiom 5.1. Postulate 1 only says a line may be drawn.
2
The terminated line of Euclid is today’s line segment.
Explanation:
True — extending it both ways makes a line. That is the second postulate.
3
A ______ can be drawn with any centre and any radius.
💡 Hint:
The third postulate.
Explanation:
A circle.
4
All right angles are equal. This is —
💡 Hint:
The circle is the third.
Explanation:
The fourth postulate.
5
The fifth postulate is as short and immediately plain as the first four.
Explanation:
False — the book says it is far more complex than they are.
6
Write the difference between the first postulate and Axiom 5.1 in two lines.
Model answer (board format)
The first postulate says a straight line may be drawn through two points. Axiom 5.1 adds that through two distinct points such a line is exactly one.
Check your answer:
Theorem 5.1 and the sum of segments
प्रमेय 5.1 और रेखाखंड का योग · Textbook 5.2 · Theorem 5.1 · Exercise 5.1
A proved statement and an assumed statement are differentNotes
A statement proved from the axioms is a theorem. Euclid deduced 465 theorems in his chain. Theorem 5.1: two distinct lines cannot have more than one point in common. If two points were common, two lines would pass through those same two points, against Axiom 5.1.
If B lies between A and C, then AC coincides with AB together with BC. Things that coincide are equal, so AB + BC = AC. An equilateral triangle is constructed from the third postulate: two circles with centres A and B and radius AB, and the meeting point C.
Worked exampleExample
Question: B lies between A and C. AB = 4 cm and BC = 5 cm. Find AC.
Relation: AB + BC = AC, because AC coincides with the two segments together.
Substitution: AC = 4 cm + 5 cm = 9 cm.
10-second revision
- A theorem is proved, an axiom is assumed
- Two lines meet in at most one point
- If the point is between, AB + BC = AC
Board tip · BSEBBoard tip
In Theorem 5.1 write “at most one”. “Exactly one” can be wrong, because parallel lines do not meet.
Board tip · CBSEBoard tip
AB + BC = AC only when B is between. Mark the between-point on the figure.
Check your understandingall correct = mastery ★
1
Theorem 5.1 says —
💡 Hint:
It clashes with the unique line.
Explanation:
Not more than one point in common.
2
If B is between, then AB + BC = AC.
Explanation:
True — from the common notion about coincidence.
3
The number of theorems in the chain of Euclid is given as ______.
💡 Hint:
Four hundred and sixty-five.
Explanation:
465.
4
AB = 6 cm and B lies between A and C. BC = 3 cm. Find AC and name the fact used.
Model answer (board format)
AC = AB + BC = 6 cm + 3 cm = 9 cm. This comes from the common notion of coincidence and the result about a point lying between.
Check your answer:
Equivalent forms of the fifth postulate
पाँचवीं अभिधारणा के तुल्य रूप · Textbook 5.3 · Exercise 5.2
It was not proved, but it opened other geometriesNotes
Attempts to prove the fifth from the first four failed. The failure produced several other geometries, called non-Euclidean. The book gives two equivalent forms. The first, Playfair’s: for every line l and every point P not on it, exactly one line m passes through P and is parallel to l. The second: two distinct intersecting lines cannot both be parallel to the same line.
A system of axioms is consistent when no statement deduced from it contradicts an axiom or a statement already proved. Today the words axiom and postulate are also used in the same sense.
Worked exampleExample
Question: Point P is not on line l. How many lines through P can be drawn parallel to l?
Answer: By the Playfair form, exactly one. This is equivalent to the fifth postulate, not a separate new rule. Draw two, and they will meet each other or both will not stay parallel to l.
10-second revision
- The fifth was not proved from the first four
- Playfair: exactly one parallel from an outside point
- Failed attempts led to non-Euclidean geometry
Board tip · BSEBBoard tip
Do not call Playfair a new postulate. Write that it is one of the forms equivalent to the fifth.
Board tip · CBSEBoard tip
A consistent system does not mean “an easy system”. It means no inner contradiction is deduced.
Check your understandingall correct = mastery ★
1
From a point outside a line, the parallel lines are —
💡 Hint:
Playfair.
Explanation:
Exactly one. This is equivalent to the fifth.
2
The fifth postulate has been proved from the first four.
Explanation:
False — the attempts failed. Non-Euclidean geometries opened from that.
3
If no inner contradiction is deduced, the axiom system is called ______.
💡 Hint:
No contradiction.
Explanation:
Consistent.
4
Two distinct intersecting lines —
💡 Hint:
The second equivalent form.
Explanation:
They cannot both be parallel to the same line.
5
Write two forms equivalent to the fifth postulate. Also write one result of the failed proofs.
Model answer (board format)
First: exactly one parallel through a point outside a line. Second: two distinct intersecting lines cannot both be parallel to the same line. Non-Euclidean geometries grew from the failed proofs.
Check your answer:
❓ Full question bank — with answers and explanations — 65 questions
Pick a type. The 31 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ130 tried · 0 correctTrue / falseTrue / false80 tried · 0 correctFill in the blanksFill80 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason50 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA110 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/13Pick one option. A wrong try brings a hint.
1
The books of the Elements are —
💡 Hint:
465 is the count of theorems.
Explanation:
13.
2
The brick ratio was —
💡 Hint:
Length is the largest.
Explanation:
4 : 2 : 1.
3
Undefined today is —
💡 Hint:
The old lines stay printed.
Explanation:
A point, a line and a plane.
4
The dimensions of a line are —
💡 Hint:
A point has zero, a surface two.
Explanation:
1.
5
The statement about the whole and the part is —
💡 Hint:
A postulate belongs to geometry.
Explanation:
The fifth common notion.
6
Drawing a circle is the statement of —
💡 Hint:
Any centre, any radius.
Explanation:
The third postulate.
7
Through two distinct points the lines that pass are —
💡 Hint:
Axiom 5.1.
Explanation:
Exactly one.
8
Theorem 5.1 forbids —
💡 Hint:
Meeting at one point is allowed.
Explanation:
Not more than one common point.
9
From outside a line, the parallel is —
💡 Hint:
The Playfair form.
Explanation:
Exactly one.
10
Non-Euclidean geometry came from —
💡 Hint:
The first four were not enough.
Explanation:
From the failed proofs.
11
A consistent system means —
💡 Hint:
No clash.
Explanation:
No deduced contradiction.
12
Superposition rests on which axiom?
💡 Hint:
The fourth common notion.
Explanation:
Things that coincide are equal.
13
If B is between, AC equals —
💡 Hint:
The two pieces make the whole.
Explanation:
AB + BC.
True or false
0/81
Thales gave a proof that a diameter bisects a circle.
Explanation:
True — it is taken as the first known proof.
2
Today the old line about a point is accepted as a complete definition.
Explanation:
False — a point is undefined. The line is printed in the book as the old wording.
3
Magnitudes of different kinds can be added.
Explanation:
False — a line and a rectangle do not go into one sum.
4
The first postulate alone proves that the line is unique.
Explanation:
False — uniqueness is stated separately in Axiom 5.1.
5
All right angles are equal.
Explanation:
True — the fourth postulate.
6
Two distinct lines can meet at three points.
Explanation:
False — Theorem 5.1 forbids more than one common point.
7
The Playfair form is equivalent to the fifth postulate.
Explanation:
True — exactly one parallel from an outside point.
8
An axiom has to be proved before it is accepted.
Explanation:
False — an axiom is accepted without proof. A theorem is proved.
Fill in the blanks
0/81
The number of books in the Elements is ______.
💡 Hint:
13.
Explanation:
The theorems are 465.
2
The dimensions of a point are ______.
💡 Hint:
0.
Explanation:
A line has one.
3
The third postulate is about drawing a ______.
💡 Hint:
A circle.
Explanation:
Any centre, any radius.
4
The number of lines through two distinct points is ______.
💡 Hint:
1.
Explanation:
Axiom 5.1.
5
If AB = 4 cm and BC = 5 cm, with B between, then AC = ______ cm.
💡 Hint:
9.
Explanation:
4 + 5.
6
The number of parallels from a point outside a line is ______.
💡 Hint:
1.
Explanation:
Playfair.
7
A system with no inner contradiction is called ______.
💡 Hint:
Consistent.
Explanation:
Not a synonym of easy.
8
The chain of Euclid has about ______ theorems.
💡 Hint:
465.
Explanation:
The books are 13.
Match
0/21
Match the statement with its class.
Column B: A. Fourth postulate · B. A common notion · C. Theorem 5.1 · D. An old printed line
1. All right angles are equal
2. The whole is greater than the part
3. Two lines share at most one point
4. A point has no part
Explanation:
Right angles are a postulate, the whole is a common notion, the common point is the theorem, and the point line is the old definition.
2
Match the number with the right thing.
Column B: A. Books of the Elements · B. Theorems · C. The unique line · D. A surface
1. 13
2. 465
3. Axiom 5.1
4. Dimension 2
Explanation:
13 books, 465 theorems, 5.1 the unique line, and a surface has two dimensions.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): Exactly one line passes through two distinct points.
Reason (R): Axiom 5.1 states this uniqueness separately.
Explanation:
Both are true and R is the correct reason.
2
Assertion (A): A point is undefined today.
Reason (R): All right angles are equal.
Explanation:
Both are true, but R does not explain why a point is undefined.
3
Assertion (A): AB + BC = AC when B is between A and C.
Reason (R): The length of a line and the area of a rectangle are put in the same sum.
Explanation:
A is true. R is false — magnitudes of different kinds are not added.
4
Assertion (A): The fifth postulate has been proved from the first four.
Reason (R): Non-Euclidean geometries opened from the failed attempts.
Explanation:
A is false. R is true.
5
Assertion (A): Two distinct lines cannot share three points.
Reason (R): Even two shared points would clash with Axiom 5.1.
Explanation:
Both are true and R explains Theorem 5.1.
Coefficient practice
0/4These are not results of this maths chapter. They are only coefficient practice for H2 + O2 → H2O.
1
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). This is not a geometry theorem.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). Keep the hydrogen atoms equal.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
3
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). Keep the oxygen atoms equal.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
4
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). A blank means coefficient 1.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
Classify
0/21
Place each statement in its class.
The whole is greater than the part
All right angles are equal
Two lines share at most one point
A circle may be drawn from any centre
Explanation:
The whole is an axiom. Right angles and the circle are postulates. The common point is Theorem 5.1.
2
Place each fact in the right slot.
4 : 2 : 1
Thirteen books
One parallel from outside
Sulbasutras
Explanation:
The brick and the Sulbasutras belong to India. Thirteen books belong to the Elements. One parallel is Playfair.
Very short answer
0/51
What is a theorem?
Model answer (board format)
A statement proved from definitions, axioms and earlier proved statements is a theorem.
Check your answer:
2
Write Axiom 5.1.
Model answer (board format)
Exactly one line passes through two distinct points.
Check your answer:
3
Write Theorem 5.1.
Model answer (board format)
Two distinct lines cannot have more than one point in common.
Check your answer:
4
What is the fourth postulate?
Model answer (board format)
All right angles are equal to one another.
Check your answer:
5
Write the Playfair form in one line.
Model answer (board format)
Exactly one line parallel to a given line passes through a point outside it.
Check your answer:
Short answer
0/31
AB = 7 cm, BC = 2 cm and B is between. Find AC. Name the notion used.
Model answer (board format)
AC = 7 cm + 2 cm = 9 cm. Things that coincide are equal, and the between-point makes the two segments into the whole.
Check your answer:
2
Write the old line about a point and say why it is not accepted as a definition.
Model answer (board format)
A point is that which has no part. “Part” itself asks for a definition, so the chain does not stop. Today a point is undefined.
Check your answer:
3
Write the first four postulates in one line each.
Model answer (board format)
A line may be drawn through two points. A segment may be extended both ways. A circle may be drawn with any centre and radius. All right angles are equal.
Check your answer:
Long answer
0/31
Write the first, fourth and fifth of the seven common notions. Say why magnitudes of different kinds are not added. Apply the first axiom to AB = PQ = 5 cm.
Model answer (board format)
Things equal to the same thing are equal. Things that coincide are equal. The whole is greater than the part. A line and a rectangle are different kinds, so they are not added. AB and PQ are both 5 cm, so AB = PQ.
Check your answer:
2
Write the difference between Postulate 1 and Axiom 5.1. Then give the idea of the proof of Theorem 5.1.
Model answer (board format)
Postulate 1 says a line may be drawn. Axiom 5.1 says it is exactly one. If two lines shared two points, two lines would pass through those two points, against 5.1. So there is not more than one common point.
Check your answer:
3
Write the fifth postulate in your own words. Add two equivalent forms and one line on non-Euclidean geometry.
Model answer (board format)
If the interior angles on one side of a transversal add to less than 180°, the lines meet on that side. Equivalent forms: exactly one parallel from an outside point, and two intersecting lines are not both parallel to the same line. When it could not be proved from the first four, non-Euclidean geometries opened.
Check your answer:
BSEB model paper · practice
0/6This model set is for practice. It is not a question from any year’s annual examination. Annual questions will be added only when a source page is available.
1
The number of the unique-line statement is —
💡 Hint:
Postulate 1 is not enough.
Explanation:
Axiom 5.1.
2
The dimensions of a surface are —
💡 Hint:
A line has one, a solid three.
Explanation:
2.
3
In AC = AB + BC, B lies ______.
💡 Hint:
Between.
Explanation:
At an end the sum is not this.
4
Two intersecting lines can both be parallel to the same line.
Explanation:
False — this is against the second form equivalent to the fifth.
5
A 4 : 2 : 1 brick has length 12 cm. Find the breadth and the thickness.
Model answer (board format)
4 parts = 12 cm, so 1 part = 3 cm. The breadth is 6 cm and the thickness is 3 cm.
Check your answer:
6
This is coefficient practice, not a result of Euclid geometry. Balance H2 + O2 making H2O by filling coefficients (blank = 1).
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
CBSE-style questions
0/6These are competency-based practice questions. They are not copies of a CBSE paper.
1
A student says two different straight lines can be drawn through two points. The mistake is —
💡 Hint:
Uniqueness is a separate axiom.
Explanation:
Axiom 5.1 says exactly one line.
2
On a map two roads are shown meeting at two crossings. According to the geometry —
💡 Hint:
Theorem 5.1.
Explanation:
Two distinct lines do not have more than one point in common.
3
Assertion (A): Extending a segment both ways gives a line.
Reason (R): This extension is Theorem 5.1.
Explanation:
A is true, the second postulate. R is false — 5.1 is about a common point.
4
A consistent system means every statement is taken as true without a check.
Explanation:
False — consistent means no contradiction is deduced from the system.
5
A field boundary was wiped out in a flood. How will the straight boundary through two pegs be unique? Which statement will you use?
Model answer (board format)
The two pegs are two distinct points. Axiom 5.1 says exactly one line passes through them, so the straight boundary is drawn in only one way. The first postulate only says that a line may be drawn.
Check your answer:
6
In class a student added an angle to a length. Which caution was broken? Give one example of a correct sum.
Model answer (board format)
Magnitudes of different kinds are not added. An angle and a length are not one sum. A correct sum is 4 cm + 5 cm = 9 cm, or 30° + 60° = 90°.
Check your answer:
🏛️ Board exam corner — Bihar Board (BSEB)— CBSE
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These are case and assertion-reason practice items. Do not treat them as past CBSE questions.
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🧠 What you learned + equation sheet
What you learned
- A point, a line and a plane are undefined today, though the book still prints the old definitions.
- Axioms and postulates are accepted without proof.
- The seven common notions include addition, subtraction, and “the whole is greater than the part”.
- The five postulates cover a line, extension, a circle, right angles, and the meeting in the fifth.
- Axiom 5.1: exactly one line passes through two distinct points.
- Theorem 5.1: two distinct lines do not meet in more than one point.
| What | Keep this |
|---|---|
| Undefined | point, line, plane |
| Axioms | seven common notions |
| Postulates 1–4 | line, extension, circle, right angles |
| Postulate 5 | lines meet if interior sum < 180° |
| Axiom 5.1 | unique line through two points |
| Theorem 5.1 | at most one common point |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.