# R S AGGARWAL AND V AGGARWAL Solutions for Class 9 Maths Chapter 6 - Introduction to Euclid's Geometry

## Chapter 6 - Introduction to Euclid's Geometry Exercise Ex. 6

What is the difference between a theorem and an axiom?

A theorem is a statement that requires a proof. Whereas, a basic fact which is taken for granted, without proof, is called an axiom.

Example of Theorem: Pythagoras Theorem

Example of axiom: A unique line can be drawn through any two points.

Define the following terms:

(i) Line segment (ii) Ray (iii) Intersecting lines (iv) Parallel lines (v) Half-line (vi) Concurrent lines (vii) Collinear points (viii) Plane

(i) Line segment: The straight path between two points is called a line segment.

(ii) Ray: A line segment when extended indefinitely in one direction is called a ray.

(iii) Intersecting Lines: Two lines meeting at a common point are called intersecting lines, i.e., they have a common point.

(iv) Parallel Lines: Two lines in a plane are said to be parallel, if they have no common point, i.e., they do not meet at all.

(v) Half-line: A ray without its initial point is called a half-line.

(vi) Concurrent lines: Three or more lines are said to be concurrent, if they intersect at the same point.

(vii) Collinear points: Three or more than three points are said to be collinear, if they lie on the same line.

(viii) Plane: A plane is a surface such that every point of the line joining any two points on it, lies on it.

In the adjoining figure, name:

(i) Six points

(ii) Five line segments

(iii) Four rays

(iv) Four lines

(v) Four collinear points

(i) Six points: A,B,C,D,E,F

(ii)
Five line segments: _{}

(iii)
Four rays: _{}

(iv)
Four lines: _{}

(vi) Four collinear points: M,E,G,B

In the adjoining figure, name:

(i) Two pairs of intersecting lines and their corresponding points of intersection

(ii) Three concurrent lines and their points of intersection

(iii) Three rays

(iv) Two line segments

(i) _{} and their corresponding point of intersection is R.

_{} and their corresponding point of intersection is P.

(ii)
_{} and their point of intersection is R.

(iii) Three rays are:

_{}.

(iv) Two line segments are:

_{}.

From the given figure, name the following:

(i) Three lines

(ii) One rectilinear figure

(iii) Four concurrent points

(i) Three lines: Line AB, Line PQ and Line RS

(ii) One rectilinear figure: EFGC

(iii) Four concurrent points: Points A, E, F and B

(i) How many lines can be drawn to pass through a given point?

(ii) How many lines can be drawn to pass through two given points?

(iii) In how many points can the two lines at the most intersect?

(iv) If A, B, C are three collinear points, name all the line segments determined by them.

(i) An infinite number of lines can be drawn to pass through a given point.

(ii) One and only one line can pass through two given points.

(iii) Two given lines can at the most intersect at one and only one point.

(iv)
_{}

Which of the following statements are true?

(i) A line segments has no definite length.

(ii) A ray has no end point.

(iii) A line has a definite length.

(iv) A line _{}is the same as line _{}.

(v) A ray _{}is the same as ray _{}.

(vi) Two distinct points always determine a unique line.

(vii) Three lines are concurrent if they have a common point.

(viii) Two distinct lines cannot have more than one point in common.

(ix) Two intersecting lines cannot be both parallel to the same line.

(x) Open half-line OA is the same thing as ray _{}

(xi) Two lines may intersect in two points.

(xii) Two lines l and m are parallel only when they have no point in common.

(i) False

(ii) False

(iii) False

(iv) True

(v) False

(vi) True

(vii) True

(viii) True

(ix) True

(x) True

(xi) False

(xii) True

In the given figure, L and M are mid-points of AB and BC respectively.

(i) If AB = BC, prove that AL = MC.

(ii) If BL = BM, prove that AB = BC.

(ii) BL = BM

⇒ 2BL = 2BM

⇒ AB = BC

## Chapter 6 - Introduction to Euclid's Geometry Exercise MCQ

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

(a) squares and rectangles

(b) squares and circles

(c) triangles and rectangles

(d) trapeziums and pyramids

Correct option: (b)

Squares and circular altars were used for household rituals.

Whereas altars having shapes as combinations of rectangles, triangles and trapeziums were used for public worship.

'Lines are parallel if they do not intersect' is started in the form of

(a) a definition

(b) an axiom

(c) a postulate

(d) a theorem

Correct option: (a)

'Lines are parallel if they do not intersect' is started in the form of a definition.

Euclid stated that 'All right angles are equal to each other' in the form of

(a) a definition

(b) an axiom

(c) a postulate

(d) a proof

Correct option: (c)

Euclid stated that 'All right angles are equal to each other' in the form of a postulate.

This is Euclid's Postulate 4.

Note: The answer in the book is option (a). But if you have a look at the Euclid's postulate, the answer is **a postulate**.

A pyramid is a solid figure, whose base is

(a) only a triangle

(b) only a square

(c) only a rectangle

(d) any polygon

Correct option: (d)

A pyramid is a solid figure, whose base is any polygon.

The side faces of a pyramid are

(a) triangles

(b) squares

(c) trapeziums

(d) polygons

Correct option: (a)

The side faces of a pyramid are triangles.

The number of dimensions of a solid are

(a) 1

(b) 2

(c) 3

(d) 5

Correct option: (c)

A solid has 3 dimensions.

The number of dimensions of a surface is

(a) 1

(b) 2

(c) 3

(d) 0

Correct option: (b)

A surface has 2 dimensions.

How many dimensions dose a point have

(a) 0 dimension

(b) 1 dimension

(c) 2 dimension

(d) 3 dimension

Correct option: (a)

A point is an exact location. A fine dot represents a point. So, a point has 0 dimensions.

Boundaries of solids are

(a) lines

(b) curves

(c) surfaces

(d) none of these

Correct option: (c)

Boundaries of solids are surfaces.

Boundaries of surfaces are

(a) lines

(b) curves

(c) polygons

(d) none of these

Correct option: (b)

Boundaries of surfaces are curves.

The number of planes passing through three non-collinear points is

(a) 4

(b) 3

(c) 2

(d) 1

Correct option: (d)

The number of planes passing through three non-collinear points is 1.

In ancient India, altars with combination of shapes like rectangles, triangles and trapeziums were used for

(a) household rituals

(b) public rituals

(c) both (a) and (b)

(d) none of (a), (b) and (c)

Correct option: (b)

In ancient India, altars with combination of shapes like rectangles, triangles and trapeziums were used for public rituals.

Axioms are assumed

(a) definitions

(b) theorems

(c) universal truths specific to geometry

(d) universal truths in all branches of mathematics

Correct option: (d)

Axioms are assumed as universal truths in all branches of mathematics because they are taken for granted, without proof.

Which of the following is a true statement?

(a) The floor and a wall of a room are parallel planes

(b) The ceiling and a wall of a room are parallel planes.

(c) The floor and the ceiling of a room are the parallel planes.

(d) Two adjacent walls of a room are the parallel planes.

Correct option: (c)

Two lines are said to be parallel, if they have no point in common.

Options (a), (b) and (d) have a common point, hence they are not parallel.

In option (c), the floor and the ceiling of a room are parallel to each other is a true statement.

Which of the following is true statement?

(a) Only a unique line can be drawn to pass through a given point

(b) Infinitely many lines can be drawn to pass through two given points

(c) If two circles are equal, then their radii are equal

(d)A line has a definite length.

Correct option: (c)

In option (a), infinite number of line can be drawn to pass through a given point. So, it is not a true statement.

In option (b), only one line can be drawn to pass through two given points. So, it is not a true statement.

In option (c),

'If two circles are equal, then their radii are equal' is the true statement.

In option (d), A line has no end points. A line has an indefinite length. So, it is not a true statement.

Which of the following is a false statement?

(a) An infinite number of lines can be drawn to pass through a given point.

(b) A unique line can be drawn to pass through two given points.

(c)

(d)A ray has one end point.

Correct option: (c)

Option (a) is true, since we can pass an infinite number of lines through a given point.

Option (b) is true, since a unique line can be drawn to pass through two given points.

Consider option (c).

As shown in the above diagram, a ray has only one end-point. So, option (d) is true.

Hence, the only false statement is option (c).

A point C is called the midpoint of a line segment , if

(a) C is an interior point of AB

(b) AC = CB

(c) C is an interior point of AB such that =

(d) AC + CB = AB

Correct option: (c)

A point C is called the midpoint of a line segment , if C is an interior point of AB such that =.

A point C is said to lie between the points A and B if

(a) AC = CB

(b) AC + CB = AB

(c) points A, C and B are collinear

(d) options (b) and (c)

* Options modified

Correct option: (d)

Observe the above figure. Clearly, C lies between A and B if AC + CB = AB.

That means, points A, B, C are collinear.

Euclid's which axiom illustrates the statement that when x + y = 15, then x + y + z = 15 + z?

(a) first

(b) second

(c) third

(d) fourth

Correct option: (b)

Euclid's second axiom states that 'If equals are added to equals, the wholes are equal'.

Hence, when x + y = 15, then x + y + z = 15 + z.

A is of the same age as B and C is of the same age as B. Euclid's which axiom illustrates the relative ages of A and C?

(a) First axiom

(b) Second axiom

(c) Third axiom

(d) Fourth axiom

Correct option: (a)

Euclid's first axiom states that 'Things which are equal to the same thing are equal to one another'.

That is,

A's age = B's age and C's age = B' age

⇒ A's age = C's age

The number of interwoven isosceles triangles in Sriyantra is

(a) five

(b) seven

(c) nine

(d) eleven

Correct option: (c)

The Sriyantra consists of nine interwoven isosceles triangles.

In Indus Valley Civilization (about 300 BC) the bricks used for construction work were having dimensions in the ratio

(a) 5:3:2

(b) 4:2:1

(c) 4:3:2

(d) 6:4:2

Correct option: (b)

In Indus Valley Civilization (about 300 BC) the bricks used for construction work were having dimensions in the ratio is 4:2:1.

Into how many chapters was the famous treatise, 'The Elements' divided by Euclid?

(a) 13

(b) 12

(c) 11

(d) 9

Correct option: (a)

The famous treatise 'The Elements' was divided into 13 chapters by Euclid.

Euclid belongs to the country

(a) India

(b) Greece

(c) Japan

(d) Egypt

Correct option: (b)

Euclid belongs to the country, Greece.

Thales belongs to the country

(a) India

(b) Egypt

(c) Greece

(d) Babylonia

Correct option: (c)

Thales belongs to the country, Greece.

Pythagoras was a student of

(a) Euclid

(b) Thales

(c) Archimedas

(d) Bhaskara

Correct option: (b)

Pythagoras was a student of Thales.

Which of the following needs a proof?

(a) axiom

(b) postulate

(c) definition

(d) theorem

Correct option: (d)

A statement that requires a proof is called a theorem.

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