#### Chapters

Chapter 2: Polynomials

Chapter 3: Coordinate Geometry

Chapter 4: Linear Equation In Two Variables

Chapter 5: Introduction To Euclid's Geometry

Chapter 6: Lines & Angles

Chapter 7: Triangles

Chapter 8: Quadrilaterals

Chapter 9: Areas of Parallelograms & Triangles

Chapter 10: Circles

Chapter 11: Construction

Chapter 12: Heron's Formula

Chapter 13: Surface Area & Volumes

Chapter 14: Statistics & Probability

## Chapter 5: Introduction To Euclid's Geometry

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 5 Introduction To Euclid's Geometry Exercise 5.1 [Pages 46 - 47]

#### Choose the correct alternative:

The three steps from solids to points are ______.

Solids - surfaces - lines - points

Solids - lines - surfaces - points

Lines - points - surfaces - solids

Lines - surfaces - points - solids

The number of dimensions, a solid has ______.

1

2

3

0

The number of dimensions, a surface has ______.

1

2

3

0

The number of dimension, a point has ______.

0

1

2

3

Euclid divided his famous treatise “The Elements” into ______.

13 chapters

12 chapters

11 chapters

9 chapters

The total number of propositions in the Elements are ______.

465

460

13

55

Boundaries of solids are ______.

Surfaces

Curves

Lines

Points

Boundaries of surfaces are ______.

Surfaces

Curves

Lines

Points

In Indus Valley Civilisation (about 3000 B.C.), the bricks used for construction work were having dimensions in the ratio ______.

1:3:4

4:2:1

4:4:1

4:3:2

A pyramid is a solid figure, the base of which is ______.

Only a triangle

Only a square

Only a rectangle

Any polygon

The side faces of a pyramid are ______.

Triangles

Squares

Polygons

Trapeziums

It is known that if x + y = 10 then x + y + z = 10 + z. The Euclid’s axiom that illustrates this statement is ______.

First Axiom

Second Axiom

Third Axiom

Fourth Axiom

In ancient India, the shapes of altars used for household rituals were ______.

Squares and circles

Triangles and rectangles

Trapeziums and pyramids

Rectangles and squares

The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is ______.

Seven

Eight

Nine

Eleven

Greek’s emphasised on ______.

Inductive reasoning

Deductive reasoning

Both A and B

Practical use of geometry

In Ancient India, Altars with combination of shapes like rectangles, triangles and trapeziums were used for ______.

Public worship

Household rituals

Both A and B

None of A, B, C

Euclid belongs to the country ______.

Babylonia

Egypt

Greece

India

Thales belongs to the country ______.

Babylonia

Egypt

Greece

Rome

Pythagoras was a student of ______.

Thales

Euclid

Both A and B

Archimedes

Which of the following needs a proof?

Theorems

Axiom

Definition

Postulate

Euclid stated that all right angles are equal to each other in the form of ______.

An axiom

A definition

A postulate

A proof

‘Lines are parallel, if they do not intersect’ is stated in the form of ______.

An axiom

A definition

A postulate

A proof

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 5 Introduction To Euclid's Geometry Exercise 5.2 [Pages 48 - 49]

#### State whether the following statement is True or False:

Euclidean geometry is valid only for curved surfaces.

True

False

The boundaries of the solids are curves.

True

False

The edges of a surface are curves.

True

False

The things which are double of the same thing are equal to one another.

True

False

If a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C.

True

False

The statements that are proved are called axioms.

True

False

“For every line l and for every point P not lying on a given line l, there exists a unique line m passing through P and parallel to l ” is known as Playfair’s axiom.

True

False

Two distinct intersecting lines cannot be parallel to the same line.

True

False

Attempts to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries.

True

False

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 5 Introduction To Euclid's Geometry Exercise 5.3 [Pages 50 - 52]

Solve the following question using appropriate Euclid’s axiom:

Two salesmen make equal sales during the month of August. In September, each salesman doubles his sale of the month of August. Compare their sales in September.

Solve the following question using appropriate Euclid’s axiom:

It is known that x + y = 10 and that x = z. Show that z + y = 10?

Solve the following question using appropriate Euclid’s axiom:

Look at the figure. Show that length AH > sum of lengths of AB + BC + CD.

In the figure, we have AB = BC, BX = BY. Show that AX = CY.

In the figure, we have X and Y are the mid-points of AC and BC and AX = CY. Show that AC = BC

In the figure, we have BX = `1/2` AB, BY = `1/2` BC and AB = BC. Show that BX = BY.

In the figure, we have ∠1 = ∠2, ∠2 = ∠3. show that ∠1 = ∠3.

In the figure, we have ∠1 = ∠3 and ∠2 = ∠4. show that ∠A = ∠C.

In the figure, we have ∠ABC = ∠ACB, ∠3 = ∠4. Show that ∠1 = ∠2.

In the figure, we have AC = DC, CB = CE. Show that AB = DE.

In the figure, if OX = `1/2` XY, PX = `1/2` XZ and OX = PX, show that XY = XZ.

In the figure AB = BC, M is the mid-point of AB and N is the mid-point of BC. Show that AM = NC.

In the figure BM = BN, M is the mid-point of AB and N is the mid-point of BC. Show that AB = BC.

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 5 Introduction To Euclid's Geometry Exercise 5.4 [Pages 52 - 53]

Read the following statement :

An equilateral triangle is a polygon made up of three line segments out of which two line segments are equal to the third one and all its angles are 60° each.

Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in a equilateral triangle.

Study the following statement:

“Two intersecting lines cannot be perpendicular to the same line”. Check whether it is an equivalent version to the Euclid’s fifth postulate.

Read the following statements which are taken as axioms:

(i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.

(ii) If a transversal intersect two parallel lines, then alternate interior angles are equal.

Is this system of axioms consistent? Justify your answer

Read the following two statements which are taken as axioms:

(i) If two lines intersect each other, then the vertically opposite angles are not equal.

(ii) If a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°.

Is this system of axioms consistent? Justify your answer.

Read the following axioms:

(i) Things which are equal to the same thing are equal to one another.

(ii) If equals are added to equals, the wholes are equal.

(iii) Things which are double of the same thing are equal to one another.

Check whether the given system of axioms is consistent or inconsistent.

## Chapter 5: Introduction To Euclid's Geometry

## NCERT solutions for Mathematics Exemplar Class 9 chapter 5 - Introduction To Euclid's Geometry

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Concepts covered in Mathematics Exemplar Class 9 chapter 5 Introduction To Euclid's Geometry are Concept for Euclid’S Geometry, Euclid’S Definitions, Axioms and Postulates, Equivalent Versions of Euclid’S Fifth Postulate.

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