Exercise 3.1 — Basic Concepts

Problems based on basic concepts and Euclid's postulates.

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Exercise 3.1 – The Elements of Geometry

Exercise 3.1 from Chapter 3, The Elements of Geometry, of Class 9 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) is where you apply Euclid's definitions, axioms, and postulates to actual problems. This exercise contains a mix of short-answer questions, true/false statements with reasoning, logical proofs using "whole is greater than part," and one geometric construction — the equilateral triangle.

The skills practiced here — reasoning from accepted axioms to logical conclusions — form the basis for every geometric proof you'll encounter in Class 9 and 10, especially in chapters on triangles, parallel lines, and congruence.

Euclid's Axioms Euclid's Postulates Whole > Part Construction of Equilateral Triangle

💡 How to approach this exercise: Most answers here are not calculations — they are short, logical statements supported by one of Euclid's axioms or postulates. Always quote the specific axiom or postulate you are using as your reason.

Question 1 — Short Answer Questions

This question checks your basic recall of facts from the introduction to the chapter — dimensions, Euclid's work, the faces of solids, the angle sum of a triangle, and the undefined terms of geometry.

Part (i)

How many dimensions does a solid have?

Answer: A solid has three dimensions — length, breadth, and height.

Part (ii)

How many books are there in Euclid's "Elements"?

Answer: Euclid's "Elements" consists of 13 books.

Part (iii)

Write the number of faces of a cube and a cuboid.

Answer: Both a cube and a cuboid have 6 faces each.

Part (iv)

What is the sum of interior angles of a triangle?

Answer: The sum of the interior angles of a triangle is always 180° (two right angles).

Part (v)

Write three undefined terms of geometry.

Answer: The three undefined terms of geometry are point, line, and plane.

📌 Why these matter: Each of these facts reappears throughout Class 9 and 10 — the 180° angle sum is used constantly in triangle proofs, while "point, line, plane" form the vocabulary of every geometric statement you'll write.

Question 2 — True or False, with Reasons

This question tests whether you can connect everyday geometric statements to the correct Euclidean axiom or postulate. For each statement, you must decide if it is True or False, and justify your answer using one of Euclid's axioms or postulates.

Statement	True / False	Reason
(a) Only one line can pass through a given point.	False	We can draw infinitely many lines through a single given point — they all pass through that point but go in different directions.
(b) All right angles are equal.	True	This is Euclid's 4th Postulate — all right angles (90°) are equal to one another, regardless of where or how they are drawn.
(c) Circles with the same radii are equal.	True	Two circles drawn with the same radius will always have the same size and area — they are congruent, hence "equal."
(d) A line segment can be extended on both sides endlessly to get a straight line.	True	This is Euclid's 2nd Postulate — a line segment can be extended indefinitely in both directions to form a full straight line.
(e) From the figure (A, C, B on a line with C between A and B), AB > AC.	True	Since AC is only a part of the whole segment AB, and the whole is always greater than a part (Euclid's axiom), AB must be greater than AC.

⚠️ Don't confuse (a) with Postulate 1: Postulate 1 says a unique line passes through two distinct points. Statement (a) talks about one point only — and through a single point, infinitely many lines can pass. The number of points matters!

Question 3 — Proving AH > AB + BC + CD

This question gives a straight line with eight marked points A, B, C, D, E, F, G, H placed in order, and asks you to show that the length AH is greater than the combined length AB + BC + CD.

Points A through H lying on a straight line

Solution

Show that AH > AB + BC + CD

From the figure: AH = AB + BC + CD + DE + EF + FG + GH This means (AB + BC + CD) is only a part of AH. We know: The whole is always greater than a part. ∴ AH > AB + BC + CD

✅ Key takeaway: This proof uses just one axiom — "the whole is greater than the part." Whenever a question asks you to compare a total length with a portion of it, this is the axiom to quote.

Question 4 — If PQ = QR, Prove that PQ = ½ PR

Here, point Q lies between points P and R on a line, such that PQ = QR (Q is the midpoint). You must prove that PQ = ½ PR using Euclid's axioms.

Q lies between P and R, with PQ = QR

Solution

Prove that PQ = ½ PR

From the figure: PR = PQ + QR But given: PQ = QR So, PR = PQ + PQ ← substituting QR with PQ PR = 2PQ PR ÷ 2 = PQ ∴ PQ = ½ PR

💡 Concept used: This is a simple substitution proof — replacing QR with PQ (since they are equal) turns the equation PR = PQ + QR into PR = 2PQ, from which PQ = ½ PR follows directly.

Question 5 — Construct an Equilateral Triangle of Side 5.2 cm

This is the only construction problem in this exercise, directly applying Euclid's 1st and 3rd Postulates — drawing a unique line through two points, and drawing circles of a given radius from a given centre.

Construction of equilateral triangle ABC

Steps of Construction

Draw an equilateral triangle with each side = 5.2 cm

Draw a line segment AB of length 5.2 cm.
Draw a circle of radius 5.2 cm with A as the centre.
Draw another circle of radius 5.2 cm with B as the centre, intersecting the first circle at point C.
Join A to C and B to C.
Notice that AB = BC = AC = 5.2 cm (since each is a radius of one of the equal circles, or the original segment).
Hence, △ABC is an equilateral triangle.

📌 Why this works: Every point on a circle is at the same distance (the radius) from its centre. Since C lies on both circles of radius 5.2 cm centred at A and B, we get AC = 5.2 cm and BC = 5.2 cm automatically — combined with AB = 5.2 cm, all three sides are equal.

Question 6 — What is a Conjecture? Give an Example

Answer

Definition and Example of a Conjecture

A conjecture is a statement which is neither proved nor disproved — it is an educated guess based on patterns observed in many examples, but without a complete logical proof.

Example: The Goldbach Conjecture — every even number greater than 4 can be written as the sum of two prime numbers.

For instance: 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 = 5 + 5, and so on. This pattern holds for every even number checked so far, but no one has proven it true for all even numbers.

Question 7 — How Many Lines Are Parallel to PQ?

Mark two points P and Q, and draw a line through them. The question asks: how many lines can be drawn parallel to this line PQ?

Answer

Number of lines parallel to PQ

We can draw infinitely many lines parallel to PQ. At every possible distance from line PQ (on either side), a new parallel line can be drawn, and since distance can take infinitely many values, there is no limit to the number of parallel lines possible.

💡 Connecting to Playfair's Postulate: While there are infinitely many lines parallel to PQ in general, Playfair's postulate tells us that through any one specific point not on PQ, exactly one such parallel line can be drawn.

Question 8 — When Interior Angles Sum to Less Than 180°

A transversal line n cuts two lines l and m, forming interior angles ∠1 and ∠2 on one side. We're told that ∠1 + ∠2 < 180°, and asked what this tells us about lines l and m.

Answer

Behaviour of lines l and m when ∠1 + ∠2 < 180°

According to Euclid's 5th Postulate, if a straight line falling on two straight lines makes the sum of the interior angles on one side less than 180°, then those two lines, when extended indefinitely, will meet on that same side.

∠1 + ∠2 < 180° ⟹ lines l and m meet on the side where ∠1 and ∠2 lie

✅ This is the foundation of "parallel lines": If ∠1 + ∠2 were exactly 180°, lines l and m would never meet — they would be parallel. The 5th postulate explains precisely when two lines converge versus when they stay parallel forever.

Question 9 — Relation Between ∠1 and ∠2 Using Euclid's Axiom

Given a figure where ∠1 = ∠3, ∠2 = ∠4, and ∠3 = ∠4, this question asks you to find the relationship between ∠1 and ∠2, justified by one of Euclid's axioms.

Solution

Find the relation between ∠1 and ∠2

Given: ∠1 = ∠3, ∠2 = ∠4, and ∠3 = ∠4 Since ∠1 = ∠3 and ∠3 = ∠4, and ∠2 = ∠4... ∴ ∠1 = ∠2

📌 Axiom used: "Things which are equal to the same thing are equal to one another." Since both ∠1 and ∠2 are equal to ∠3 (= ∠4), they must be equal to each other.

Question 10 — Prove BX = BY Using Euclid's Axioms

In triangle ABC, points X and Y lie on sides AB and BC respectively, such that BX = ½ AB, BY = ½ BC, and AB = BC. You must prove that BX = BY.

X on AB, Y on BC, with BX = ½AB and BY = ½BC

Solution

Prove that BX = BY

Given: AB = BC ∴ ½AB = ½BC ← halves of equal things are equal But given: BX = ½AB and BY = ½BC ∴ BX = BY ← things equal to the same thing are equal

💡 Two axioms used together: First, "halves of equal things are equal" gives ½AB = ½BC. Then, "things equal to the same thing are equal to one another" connects BX and BY through their relationship to ½AB and ½BC.

Common Mistakes to Avoid

Quoting the wrong axiom or postulate: Each "true/false" or "prove" question expects a specific named axiom or postulate as the reason — a vague explanation without naming the rule loses marks.
Confusing "one point" with "two points": Infinitely many lines pass through one point, but only one unique line passes through two distinct points (Postulate 1).
Skipping substitution steps: In proofs like Q4 and Q10, write out each substitution clearly (e.g., replacing QR with PQ) rather than jumping straight to the conclusion.
Forgetting construction steps: In Q5, marks are given for each numbered step of construction — don't just draw the final triangle without showing the circles used to locate point C.
Mixing up Postulate 4 and Postulate 5: Postulate 4 is about right angles being equal (a simple fact); Postulate 5 is about when two lines will meet based on interior angles (a more complex condition).

⛔ Most common board exam error: In Q2(a), many students answer "True" because "a line passes through the point" sounds correct — but the statement says only one line, which is false since infinitely many lines can pass through a single point.

Quick Reference — All Answers at a Glance

Question	Topic	Answer
Q1(i)	Dimensions of a solid	3 (length, breadth, height)
Q1(ii)	Books in Euclid's Elements	13
Q1(iii)	Faces of cube/cuboid	6 each
Q1(iv)	Angle sum of triangle	180°
Q1(v)	Undefined terms	Point, Line, Plane
Q2(a)	One line through a point	False
Q2(b)	All right angles equal	True
Q2(c)	Circles with same radii	True
Q2(d)	Extending a line segment	True
Q2(e)	AB > AC (whole > part)	True
Q3	AH > AB+BC+CD	Proved using "whole > part"
Q4	PQ = ½PR	Proved by substitution
Q5	Equilateral triangle (5.2 cm)	Constructed using two circles
Q6	Conjecture example	Goldbach Conjecture
Q7	Lines parallel to PQ	Infinitely many
Q8	∠1+∠2 < 180°	Lines l, m meet on that side
Q9	∠1 vs ∠2	∠1 = ∠2
Q10	BX = BY	Proved using halves & equals axioms

What This Exercise Prepares You For

Exercise 3.1 builds your ability to reason logically using axioms and postulates — a skill that is directly tested in the chapter on Triangles, where congruence proofs depend on combining given equalities exactly as you did in Questions 9 and 10. The construction skills from Question 5 also prepare you for more advanced constructions in Construction of Quadrilaterals.

The reasoning behind Euclid's 5th postulate in Question 8 is especially important — it directly explains why parallel lines never meet, a concept that returns when studying angle pairs formed by a transversal cutting two parallel lines.

For Telangana and Andhra Pradesh board exams, true/false-with-reason questions (like Q2) and short proof questions (like Q3, Q4, Q9, Q10) are extremely common as 1-mark and 2-mark questions. Practicing the exact axiom names used in this exercise will help you answer similar questions confidently.

📐 Board Exam Tip (Telangana & AP): When a question asks you to "give a reason," always state the axiom or postulate in words (e.g., "Things which are halves of the same things are equal to one another") rather than just writing "by Euclid's axiom" — examiners look for the specific statement.