Class 9 Mathematics Chapter 5 Introduction To Euclid’s Geometry

Key Concept: Limitation of Egyptian Geometry

c) Absence of systematic proofs

[Solution Description]

Egyptian geometry focused on practical results without establishing general axioms or deductive proofs. Greeks later introduced logical rigor through postulates and theorems.

Key Concept: Axiom 7 Application

c) Axiom 7 (Halves Equality)

[Solution Description]

Axiom 7 states “Things which are halves of the same things are equal”. Both AB and EF are halves of CD. Thus AB = EF by axiom 7.

Key Concept: Equal Transitivity

d) Things equal to the same thing are equal to one another

[Solution Description]\

This uses Euclid’s first axiom: Things equal to the same thing are equal to one another. Here Rajesh = Mohan and Mohan = Suresh implies Rajesh = Suresh.

Key Concept: Axioms vs. Postulates

b) Postulates

[Solution Description]

Euclid distinguished postulates (geometry-specific assumptions) from axioms (universal truths). The parallel postulate example confirms this distinction.

Key Concept: Postulate 5

c) They intersect on the same side as the given angles

[Solution Description]

Step 1: Calculate total interior angles: $110^\circ + 65^\circ = 175^\circ$.

Step 2: Compare with two right angles ($180^\circ$). Since $175^\circ < 180^\circ$, Euclid’s fifth postulate applies. Step 3: The postulate states that the two lines will meet on the side where the sum of interior angles is less than $180^\circ$.

Key Concept: Undefined Terms

b) It requires other terms like ‘part’ to define it

[Solution Description] Euclid’s definition states “A point is that which has no part”. However, the word ‘part’ itself requires further definition. Since basic terms like ‘part’, ‘area’, etc., cannot be endlessly defined, geometry takes ‘point’, ‘line’, and ‘plane’ as undefined terms. These are explained intuitively through physical models.

Key Concept: Definition of a Point

b) That which has no part

[Solution Description]

Euclid’s first definition states: “A point is that which has no part.” This means a point has zero dimensions (no length, breadth, or thickness). Other options describe practical representations or modern interpretations but do not match Euclid’s original wording.

Key Concept: Segment Addition Theorem

c) $AB + BC = AC$

[Solution Description] By Theorem 1, when three points are collinear with $B$ between $A$ and $C$, the sum of segments $AB$ and $BC$ equals $AC$. This follows directly from Euclid’s axioms on line segment addition.

Key Concept: Ancient Indian geometry

c) Ancient India

[Solution Description]

The Sulbasutras (800–500 BCE) were Vedic texts from ancient India that provided geometric methods for constructing vedis (altars). These rules predate Greek systematization of geometry. No other option matches this historical context from the syllabus.

Key Concept: Fourth Postulate

d) All right angles are equal to one another

[Solution Description]

Euclid’s Fourth Postulate explicitly states: “All right angles are equal to one another.” This directly answers the question without needing calculations.

Key Concept: Axioms

a) Axiom 1: Equality by same reference

[Solution Description]

According to Euclid’s first axiom: “Things equal to the same thing are equal to one another.” Here, both $a$ and $b$ are equal to $c$. Applying axiom 1, $a$ must equal $b$.

Key Concept: Dimensionality of Surfaces

b) Two dimensions

[Solution Description]

Euclid’s three-step model states that solids have 3 dimensions. When we move from solids to surfaces, we lose one dimension. Thus, surfaces have $3 – 1 = 2$ dimensions. This matches Definition 5 (“A surface…has length and breadth only”).

Key Concept: Meaning of Greek Roots

c) To measure

[Solution Description]

The term “metrein” translates to “to measure” in English, while “geo” means “earth”. This forms the basis of the word “geometry”.

Key Concept: Greek Geometry Approach

c) Deductive reasoning derived from axioms

[Solution Description]

Greek geometry emphasized logical proofs and deductive reasoning rather than practical applications. This focus on formal justification became its defining feature.

Key Concept: Undefined Terms

d) Surface

[Solution Description]

Euclid’s undefined terms are \textit{point}, \textit{line}, and \textit{plane}. A \textit{surface} is explicitly defined in Definition 4 as having “length and breadth only,” making it a defined term.

Key Concept: Fifth Postulate Condition

b) The lines meet on the side with angle sum $115^\circ$

[Solution Description]

Step 1: Refer to Fifth Postulate statement. If sum of interior angles on one side is less than $180^\circ$ (i.e., two right angles), the lines meet on that side.

Step 2: Given sum = $115^\circ < 180^\circ$. Hence, lines will intersect on the side where this sum occurs.

Key Concept: Vedic India Applications

c) Altars (vedis)

[Solution Description]

According to the syllabus, the Sulbasutras contained rules for building altars (vedis) with geometric shapes like squares and triangles. This is stated in the section on Vedic India (800–500 BCE).

Key Concept: Undefined Terms

c) Avoid infinite regression in definitions

[Solution Description]

Per Key Development 4, Euclid recognized leaving terms undefined prevented infinite regression caused by endless definitions. Option c) directly quotes syllabus content.

Key Concept: Axioms

c) The totals are equal

[Solution Description] Euclid\'{}s second axiom states: “If equals are added to equals, the wholes are equal.” If we add two pairs of equal lengths (sticks), the resulting total lengths will be equal. This aligns with the axiom.

Key Concept: Harappan Civilization – Brick Ratio

c) 28 cm

[Solution Description]

Given ratio = $4:2:1$ (length : breadth : thickness). Thickness is 7 cm which corresponds to the third term (1 part). Length corresponds to 4 parts. Calculation: Length = $4 \times$ (Thickness value) = $4 \times 7$ cm = 28 cm.

Key Concept: Axiom 1

d) Things equal to the same thing are equal.

[Solution Description]

Apply Axiom 1: Things equal to the same thing are equal. Since both $M$ and $P$ are equal to $N$, they must be equal to each other.

Key Concept: Indus Valley Geometry

b) $4:2:1$

[Solution Description]

The Indus Valley Civilisation used bricks with a consistent ratio of 4:2:1 (length:breadth:height) for construction. This fact is directly mentioned in the syllabus.