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#### Page No 46:

#### Question 1:

The three steps from solids to points are :

(A) Solids - surfaces - lines - points

(B) Solids - lines - surfaces - points

(C) Lines - points - surfaces - solids

(D) Lines - surfaces - points - solids

#### Answer:

The steps from solids to point are solids-surfaces-lines-points.

Hence, the correct answer is option (A).

#### Page No 46:

#### Question 2:

The number of dimensions, a solid has :

(A) 1

(B) 2

(C) 3

(D) 0

#### Answer:

A solid shape has shape, size, position and can be moved from one place to another. So, solid has three dimensions.

Hence, the correct answer is option (C).

#### Page No 46:

#### Question 3:

The number of dimensions, a surface has :

(A) 1

(B) 2

(C) 3

(D) 0

#### Answer:

Boundaries of a solid are called surfaces. A surface(plane) has only length and breadth. So, it has two dimensions.

Hence, the correct answer is option (B).

#### Page No 46:

#### Question 4:

The number of dimension, a point has :

(A) 0

(B) 1

(C) 2

(D) 3

#### Answer:

A point is that which has no part i.e., no length, no breadth and no height. So, it has no dimension.

Hence, the correct answer is option (A).

#### Page No 46:

#### Question 5:

Euclid divided his famous treatise “The Elements” into :

(A) 13 chapters

(B) 12 chapters

(C) 11 chapters

(D) 9 chapters

#### Answer:

Euclid divided his famous treatise The Elements’ into 13 chapters.

Hence, the correct answer is option (A).

#### Page No 46:

#### Question 6:

The total number of propositions in the Elements are :

(A) 465

(B) 460

(C) 13

(D) 55

#### Answer:

Euclid deduced 465 proportions in a logical chain using his axioms, postulates, definitions and theorems.

Hence, the correct answer is option (A).

#### Page No 46:

#### Question 7:

Boundaries of solids are :

(A) surfaces

(B) curves

(C) lines

(D) points

#### Answer:

The boundaries of solids are surfaces.

Hence, the correct answer is option (A).

#### Page No 46:

#### Question 8:

Boundaries of surfaces are :

(A) surfaces

(B) curves

(C) lines

(D) points

#### Answer:

The boundaries of surfaces are curves.

Hence, the correct answer is option (B)

#### Page No 46:

#### Question 9:

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

(A) 1 : 3 : 4

(B) 4 : 2 : 1

(C) 4 : 4 : 1

(D) 4 : 3 : 2

#### Answer:

In Indus valley civilization, the bricks used for construction work were having dimensions in the ratio 4 : 2 : 1.

Hence, the correct answer is option (B).

#### Page No 46:

#### Question 10:

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

(A) only a triangle

(B) only a square

(C) only a rectangle

(D) any polygon

#### Answer:

A pyramid is solid figure, the base of which is a triangle or square or some other polygon.

Hence, the correct answer is option (D).

#### Page No 46:

#### Question 11:

The side faces of a pyramid are :

(A) Triangles

(B) Squares

(C) Polygons

(D) Trapeziums

#### Answer:

The side faces of a pyramid are always triangles.

Hence, the correct answer is option (A).

#### Page No 47:

#### Question 12:

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

(A) First Axiom

(B) Second Axiom

(C) Third Axiom

(D) Fourth Axiom

#### Answer:

The Euclid’s axiom that illustrate the given statement is second axiom, according to which, if equals are added to equals, the wholes are equals.

Hence, the correct answer is option (B).

#### Page No 47:

#### Question 13:

In ancient India, the shapes of altars used for house hold rituals were :

(A) Squares and circles

(B) Triangles and rectangles

(C) Trapeziums and pyramids

(D) Rectangles and squares

#### Answer:

In ancient India, squares and circular altars were used for household rituals.

Hence, the correct answer is option (A).

#### Page No 47:

#### Question 14:

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

(A) Seven

(B) Eight

(C) Nine

(D) Eleven

#### Answer:

The Syriyantra (in the Atharvaveda) consists of nine interwoven isosceles triangles.

Hence, the correct answer is option (C).

#### Page No 47:

#### Question 15:

Greek’s emphasised on :

(A) Inductive reasoning

(B) Deductive reasoning

(C) Both A and B

(D) Practical use of geometry

#### Answer:

Greek’s emphasized on deductive reasoning.

Hence, the correct answer is option (B).

#### Page No 47:

#### Question 16:

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

(A) Public worship

(B) Household rituals

(C) Both A and B

(D) None of A, B, C

#### Answer:

In ancient India altars whose shapes were combinations of rectangles, triangles and trapezium were used for public worship.

Hence, the correct answer is option (A).

#### Page No 47:

#### Question 17:

Euclid belongs to the country :

(A) Babylonia

(B) Egypt

(C) Greece

(D) India

#### Answer:

Euclid belongs to the country Greece.

Hence, the correct answer is option (C).

#### Page No 47:

#### Question 18:

Thales belongs to the country :

(A) Babylonia

(B) Egypt

(C) Greece

(D) Rome

#### Answer:

Thales belongs to Greece.

Hence, the correct answer is option (C).

#### Page No 47:

#### Question 19:

Pythagoras was a student of :

(A) Thales

(B) Euclid

(C) Both A and B

(D) Archimedes

#### Answer:

Pythagoras was a student of Thales.

Hence, the correct answer is option (A).

#### Page No 47:

#### Question 20:

Which of the following needs a proof ?

(A) Theorem

(B) Axiom

(C) Definition

(D) Postulate

#### Answer:

The statements that were proved are called propositions or theorem.

Hence, the correct answer is option (A).

#### Page No 47:

#### Question 21:

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

(A) an axiom

(B) a definition

(C) a postulate

(D) a proof

#### Answer:

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

Hence, the correct answer is option (C).

#### Page No 47:

#### Question 22:

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

(A) an axiom

(B) a definition

(C) a postulate

(D) a proof

#### Answer:

‘Line as parallel, if they do no intersect’ is the definition of parallel lines.

Hence, the correct answer is option (B).

#### Page No 48:

#### Question 1:

Write whether the following statement are **True **or **False**? Justify your answer :

Euclidean geometry is valid only for curved surfaces.

#### Answer:

False

Reason: Because Euclidean geometry is valid only for the figures in the plane but on the curved surfaces it fails.

#### Page No 48:

#### Question 2:

Write whether the following statement are **True **or **False**? Justify your answer :

The boundaries of the solids are curves.

#### Answer:

False

Reason: Because the boundaries of the solids are surfaces.

#### Page No 48:

#### Question 3:

Write whether the following statement are **True **or **False**? Justify your answer :

The edges of a surface are curves.

#### Answer:

False

Reason: Because the edges of surfaces are lines.

#### Page No 48:

#### Question 4:

Write whether the following statement are **True **or **False**? Justify your answer :

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

#### Answer:

True

Reason: Since, it is one of the Euclid’s axiom.

#### Page No 48:

#### Question 5:

Write whether the following statement are **True **or **False**? Justify your answer :

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.

#### Answer:

True, because quantity A is greater than quantity B. So definitely there will be some quantity C by which A is greater than B.

#### Page No 48:

#### Question 6:

Write whether the following statement are **True **or **False**? Justify your answer :

The statements that are proved are called axioms.

#### Answer:

False, because axioms are statements which are self-evident and are accepted without any proof.

#### Page No 49:

#### Question 7:

Write whether the following statement are **True **or **False**? Justify your answer :

“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.

#### Answer:

True, because “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.

#### Page No 49:

#### Question 8:

Write whether the following statement are **True **or **False**? Justify your answer :

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

#### Answer:

True, because two intersecting lines can never be parallel to each other. As they are intersecting so it is not possible that both are parallel to same line.

#### Page No 49:

#### Question 9:

Write whether the following statement are **True **or **False**? Justify your answer :

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

#### Answer:

True, because these geometries are different from Euclidean geometry called non-Euclidean geometry.

#### Page No 50:

#### Question 1:

Solve the given 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.

#### Answer:

Let each salesman makes Rs. $x$ in August

In September, the sale of each is Rs. $2x$.

Now,

According to Euclid sixth axiom:

'Things which are double of the same thing are equal to one another, the wholes are equal.'

So, the sales of each salesman is equal.

#### Page No 50:

#### Question 2:

Solve the given question using appropriate Euclid’s axiom :

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

#### Answer:

$\mathrm{Given}:\phantom{\rule{0ex}{0ex}}x+y=10.....\left(1\right)\phantom{\rule{0ex}{0ex}}x=z.....\left(2\right)$

According to Euclid axiom, 'If equals are added to equal, the wholes are equal.'

Hence, add $y$ to both sides of (2),

$\Rightarrow x+y=z+y\phantom{\rule{0ex}{0ex}}\Rightarrow z+y=10$

Hence, verified.

#### Page No 50:

#### Question 3:

Solve the given question using appropriate Euclid’s axiom :

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

#### Answer:

From the figure, we have

AD = AB + BC + CD

Here, AD is a part of AH

i.e. AH = AB + BC + CD + DE + EF + FG + GH

= AD + DE + EF + FG + GH

According to Euclid axiom,

AH > AD

$\therefore $ length of AH > Sum of length of AB + BC + CD.

#### Page No 50:

#### Question 4:

Solve the given question using appropriate Euclid’s axiom :

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

#### Answer:

Given: AB = BC .....(1)

BX = BY .....(2)

Subtracting (2) from (1), we get

AB $-$ BX = BC $-$ BY

By Euclid axiom 3rd:

'If equals are subtracted from equals, the remainder are equal.'

Thus AX = CY

Hence, verified.

#### Page No 50:

#### Question 5:

Solve the given question using appropriate Euclid’s axiom :

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

#### Answer:

Given: X is mid-point of AC

$\therefore \mathrm{AX}=\mathrm{CX}=\frac{1}{2}\mathrm{AC}\phantom{\rule{0ex}{0ex}}\Rightarrow 2\mathrm{AX}=2\mathrm{CX}=\mathrm{AC}.....\left(1\right)$

Also, Y is mid-point of BC.

$\therefore \mathrm{BY}=\mathrm{CY}=\frac{1}{2}\mathrm{BC}\phantom{\rule{0ex}{0ex}}\Rightarrow 2\mathrm{BY}=2\mathrm{CY}=\mathrm{BC}.....\left(2\right)$

Also, AX = CY (given) .....(3)

According to Euclid axiom, 'things which are double of same things are equal to one another'.

$\Rightarrow 2\mathrm{AX}=2\mathrm{CY}\left[\mathrm{From}\left(3\right)\right]\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{AC}=\mathrm{BC}\left[\mathrm{From}\left(1\right)\mathrm{and}\left(2\right)\right]$

Hence, verified.

#### Page No 50:

#### Question 6:

Solve the given question using appropriate Euclid’s axiom :

Show that BX = BY.

#### Answer:

In the figure, we have

$\mathrm{BX}=\frac{1}{2}\mathrm{AB}\phantom{\rule{0ex}{0ex}}\mathrm{BY}=\frac{1}{2}\mathrm{BC}\phantom{\rule{0ex}{0ex}}\mathrm{and}\mathrm{AB}=\mathrm{BC}.$

So, 2BX = 2BY

Hence, BX = BX (Euclid's axiom: Things which are double of things are equal to one another)

#### Page No 51:

#### Question 7:

Solve the given question using appropriate Euclid’s axiom :

In the Figure, we have ∠1 = ∠2, ∠2 = ∠3. Show that ∠1 = ∠3.

#### Answer:

$\begin{array}{rcl}\mathrm{Given}:\angle 1& =& \angle 2\\ \angle 2& =& \angle 3\end{array}$

Now, by Euclid's axiom 1, things which are equal to sane thing are equal to one another.

Hence, $\angle 1=\angle 3.$

#### Page No 51:

#### Question 8:

Solve the given question using appropriate Euclid’s axiom :

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

#### Answer:

$\begin{array}{rcl}\mathrm{Given}:\angle 1& =& \angle 3.....\left(1\right)\\ \angle 2& =& \angle 4.....\left(2\right)\end{array}$

By Euclid axiom 2, if equals are added to equals, the whole are equal.

$\therefore $ Adding (1) & (2)

$\angle 1+\angle 2=\angle 3+\angle 4$

$\angle \mathrm{A}=\angle \mathrm{C}$

Hence, verified.

#### Page No 51:

#### Question 9:

Solve the given question using appropriate Euclid’s axiom :

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

#### Answer:

$\mathrm{Given}:\angle \mathrm{ABC}=\angle \mathrm{ACB}.....\left(1\right)\phantom{\rule{0ex}{0ex}}\angle 4=\angle 3.....\left(2\right)$

Now, subtracting (2) from (1)

$\angle \mathrm{ABC}-\angle 4=\angle \mathrm{ACB}-\angle 3\left(\mathrm{By}\mathrm{Euclid}\mathrm{axiom}3\mathrm{if}\mathrm{equal}\mathrm{are}\mathrm{subtracted}\mathrm{from}\mathrm{equal},\mathrm{the}\mathrm{remainders}\mathrm{are}\mathrm{equal}.\right)\phantom{\rule{0ex}{0ex}}\Rightarrow \angle 1=\angle 2$

#### Page No 51:

#### Question 10:

Solve the given question using appropriate Euclid’s axiom :

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

#### Answer:

$\begin{array}{rcl}\mathrm{Given}:\mathrm{AC}& =& \mathrm{DC}.....\left(1\right)\\ \mathrm{CB}& =& \mathrm{CE}.....\left(2\right)\end{array}$

By Euclid axiom 2, if equal are added to equals, the wholes are equal.

Adding (1) and (2)

AC + CB = DC + CE

⇒ AB = DE

#### Page No 51:

#### Question 11:

Solve the given question using appropriate Euclid’s axiom :

In the Figure, if OX = $\frac{1}{2}$ XY, PX = $\frac{1}{2}$ XZ and OX = PX, show that XY = XZ.

#### Answer:

$\mathrm{Given}:\mathrm{OX}=\frac{1}{2}\mathrm{XY}\phantom{\rule{0ex}{0ex}}\Rightarrow 2\mathrm{OX}=\mathrm{XY}.....\left(1\right)$

Also, PX = $\frac{1}{2}$XZ

$\Rightarrow 2\mathrm{PX}=\mathrm{XZ}.....\left(2\right)$

and OX = PX .....(3)

According to Euclid axiom, things which are double of the same things are equal to one another.

$\Rightarrow 2\mathrm{OX}=2\mathrm{PX}$

$\Rightarrow \mathrm{XY}=\mathrm{XZ}$

#### Page No 52:

#### Question 12:

Solve the given question using appropriate Euclid’s axiom :

In the Figure :

(i) AB = BC, M is the mid-point of AB and N is the mid- point of BC. Show that AM = NC.

(ii) BM = BN, M is the mid-point of AB and N is the mid-point of BC. Show that AB = BC.

#### Answer:

(i) Given: AB = BC .....(1)

Also,

AM + MB = AB [$\because $ M is a point between A and B] .....(2)

Similarly

BN + NC = BC [$\because $ N is a point between B and C] .....(3)

From (2) and (3), we get

AM + MB = BN + NC

Since, M is the mid-point of AB and N is mid-point of BC

$\therefore $ 2AM = 2NC

Using Euclid's axiom 6, things which are double of same thing are equal to one another.

$\Rightarrow $AM = NC

(ii) Given: BM = BN .....(1)

Here

BM = AM .....(2) ($\because $ M is the mid-point of AB)

Similarly,

BN = NC .....(3) ($\because $ N is the mid-point of BC)

From (1), (2) and (3) and Euclid axiom 1,

AM = NC .....(4)

Adding (1) and (4), we get

AM + BM = NC + BN

Using axiom (2), if equal are added to equals the wholes are equal.

$\Rightarrow $AB = BC

#### Page No 52:

#### Question 1:

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.

#### Answer:

The terms needed to be defined are

(i) Polygon: A closed figure bounded by three or more line segments.

(ii) Line Segment: Part of line with two end points.

(iii) Angle: A figure formed by two rays with one common initial point.

(iv) Acute Angle: Angle whose measure is between 0° and 90°.

Here, undefined terms are line and point.

Now,

All angles of equilateral triangle are 60° each (given)

Two line segment are equal to third one (given)

Using Euclid axiom, things which are equal to same thing are equal to one another.

Therefore, all three sides of an equilateral triangle are equal.

#### Page No 53:

#### Question 2:

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.

#### Answer:

Two equivalent versions of Euclid's fifth postulate are

1) Fox every line $m$ and for every point $p$ not lying on Z, there exist a unique line $n$ passing $p$ parallel to Z.

2) Two distinct intersecting lines can not be parallel to same line.

From above two statement it is clear that given statement is not an equivalent version to the Euclid's fifth postulate.

#### Page No 53:

#### Question 3:

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.

#### Answer:

A system of axiom is called consistent, if there is no statement which can be deduced from these axioms such that it contradicts any axiom.

This system of axioms is not consistent.

(i) When two parallel lines are intersected by a transversal, then corresponding angles are equal.

Here $\angle 1=\angle 5,\angle 2=\angle 6,\angle 3=\angle 7\mathrm{and}\angle 4=\angle 8$

(ii) The alternate interior angle will always be equal in parallel lines.

#### Page No 53:

#### Question 4:

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.

#### Answer:

The given system of axioms is not consistent because if a ray stands on a line and the sum of two adjacent angles so formed is equal to 180°, then for two lines which intersect each other, the vertically opposite angle become equal.

#### Page No 53:

#### Question 5:

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.

#### Answer:

(i) Things which are equal to some thing are equal to one another. This statement is Euclid first axiom.

(ii) If equals are added to equals, the wholes are equal. This statement is Euclid second axiom.

(iii) Things which are double of same thing are equal to one another. This statement is true if we apply Euclid's first axiom.

Thus, given three axioms are Euclid's axioms. So here we cannot deduce any statement from these axioms which contradict any axiom.

So, given system of axiom is consistent.

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