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#### Page No 188:

#### Question 1:

What is the difference between a theorem and an axiom?

#### Answer:

An axiom is a basic fact that is taken for granted without proof.

Examples:

i) Halves of equals are equal.

ii) The whole is greater than each of its parts.

Theorem: A statement that requires proof is called theorem.

Examples:

i) The sum of all the angles around a point is ${360}^{\circ}$.

ii) The sum of all the angles of triangle is ${180}^{\circ}$.

#### Page No 188:

#### Question 2:

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

#### Answer:

**(i) Line segment **:A line segment is a part of line that is bounded by two distinct end-points. A line segment has a fixed length.

**(ii) Ray:** A line with a start point but no end point and without a definite length is a ray.

**(iii) Intersecting lines: **Two lines with a common point are called intersecting lines.

** (iv) Parallel lines:** Two lines in a plane without a common point are parallel lines.

** (v) Half line:** A straight line extending from a point indefinitely in one direction only is a half line.

** (vi) Concurrent lines:** Three or more lines intersecting at the same point are said to be concurrent.

** (vii) Collinear points:** Three or more than three points are said to be collinear if there is a line, which contains all the points.

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

#### Page No 188:

#### Question 3:

In the adjoining figure, name

(i) six points

(ii) five lines segments

(iii) four rays

(iv) four lines

(v) four collinear points

#### Answer:

(i) Points are A, B, C, D, P and R.

(ii) $\overline{EF},\overline{GH},\overline{FH},\overline{EG},\overline{MN}$

(iii) $\overrightarrow{EP},\overrightarrow{GR},\overrightarrow{HS,}\overrightarrow{FQ}$

(iv) $\overleftrightarrow{AB},\overleftrightarrow{CD},\overleftrightarrow{PQ},\overleftrightarrow{RS}$

(v) Collinear points are M, E, G and B.

#### Page No 189:

#### Question 4:

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

#### Answer:

(i) Two pairs of intersecting lines and their point of intersection are

$\left\{\overleftrightarrow{EF},\overleftrightarrow{GH},pointR\right\},\left\{\overleftrightarrow{AB},\overleftrightarrow{CD},pointP\right\}$

(ii) Three concurrent lines are

$\left\{\overleftrightarrow{AB},\overleftrightarrow{EF},\overleftrightarrow{GH},pointR\right\}$

(iii) Three rays are

$\left\{\overrightarrow{RB},\overrightarrow{RH},\overrightarrow{RF}\right\}$

(iv) Two line segments are

$\left\{\overline{RQ}and\overline{RP}\right\}$

#### Page No 189:

#### Question 5:

From the given figure, name the following:

(a) Three lines

(b) One rectilinear figure

(c) Four concurrent points

#### Answer:

(a) $Line\overleftrightarrow{PQ}$, $Line\overleftrightarrow{RS}$ and $Line\overleftrightarrow{AB}$

(b) $CEFG$

(c) No point is concurrent.

#### Page No 189:

#### Question 6:

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

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

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

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

#### Answer:

(i) Infinite lines can be drawn through a given point.

(ii) Only one line can be drawn through two given points.

(iii) At most two lines can intersect at one point.

(iv) The line segments determined by three collinear points A, B and C are

$\overline{)AB},\overline{BC}\text{and}\overline{)AC.}$

#### Page No 189:

#### Question 7:

Which of the following statements are true?

(i) A line segment has no definite length.

(ii) A ray has no end-point.

(iii) A line has a definite length.

(iv) A line $\overleftrightarrow{AB}$ is same as line $\overleftrightarrow{BA}$.

(v) A ray $\underset{AB}{\to}$ is same as ray $\underset{BA}{\to}$.

(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 is the same thing as ray.

(xi) Two lines may intersect in two points.

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

#### Answer:

(i) False. A line segment has a definite length.

(ii) False. A ray has one end-point.

(iii) False. A line has no definite length.

(iv) True

(v) False. $\overleftrightarrow{BA}$ and $\overleftrightarrow{AB}$ have different end-points.

(vi) True

(vii) True

(viii) True

(ix) True

(x) True

(xi) False. Two lines intersect at only one point.

(xii) True

#### Page No 190:

#### Question 8:

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

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

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

**Hint**

(i) $AB=BC\Rightarrow \frac{1}{2}AB=\frac{1}{2}BC\Rightarrow AL=MC$.

(ii) $BL=BM\Rightarrow 2BL=2BM\Rightarrow AB=BC$.

#### Answer:

(i) It is given that L is the mid-point of AB.

∴ AL = BL = $\frac{1}{2}$AB .....(1)

Also, M is the mid-point of BC.

∴ BM = MC = $\frac{1}{2}$BC .....(2)

AB = BC (Given)

⇒ $\frac{1}{2}$AB = $\frac{1}{2}$BC (Things which are halves of the same thing are equal to one another)

⇒ AL = MC [From (1) and (2)]

(ii) It is given that L is the mid-point of AB.

∴ AL = BL = $\frac{1}{2}$AB

⇒ 2AL = 2BL = AB .....(3)

Also, M is the mid-point of BC.

∴ BM = MC = $\frac{1}{2}$BC

⇒ 2BM = 2MC = BC .....(4)

BL = BM (Given)

⇒ 2BL = 2BM (Things which are double of the same thing are equal to one another)

⇒ AB = BC [From (3) and (4)]

#### Page No 190:

#### Question 1:

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

#### Answer:

(b) squares and circles

#### Page No 190:

#### Question 2:

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)

#### Answer:

The construction of altars (or vedis) and fireplaces for performining vedic rituals resulted in the origin of the geometry of vedic period. Square and circular altars were used for household rituals whereas the altars with combination of shapes like rectangles, triangles and trapezium were used for public rituals.

Hence, the correct answer is option (b).

#### Page No 190:

#### Question 3:

*Sriyantra*is

(a) five

(b) seven

(c) nine

(d) eleven

#### Answer:

(c) nine

#### Page No 190:

#### Question 4:

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

(a) 5 : 3 : 2

(b) 4 : 2 : 1

(c) 4 : 3 : 2

(d) 6 : 4 : 2

#### Answer:

(b) 4 : 2 : 1

#### Page No 190:

#### Question 5:

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

(a) 13

(b) 12

(c) 11

(d) 9

#### Answer:

The famous treatise, '*The Elements*' by Euclid is divided into 13 chapters.

Hence, the correct answer is option (a).

#### Page No 191:

#### Question 6:

Euclid belongs to the country

(a) India

(b) Greece

(c) Japan

(d) Egypt

#### Answer:

(b) Greece

#### Page No 191:

#### Question 7:

Thales belongs to the country

(a) India

(b) Egypt

(c) Greece

(d) Babylonia

#### Answer:

(c) Greece

#### Page No 191:

#### Question 8:

Pythagoras was a student of

(i) Euclid

(ii) Thales

(iii) Archimedes

(iv) Bhaskara

#### Answer:

(ii) Thales

#### Page No 191:

#### Question 9:

Which of the following needs a proof?

(a) axiom

(b) postulate

(c) definition

(d) theorem

#### Answer:

(d) theorem

#### Page No 191:

#### Question 10:

The statement that '*the lines are parallel if they do not intersect*' is in the form of

(a) a definition

(b) an axiom

(c) a postulate

(d) a theorem

#### Answer:

(a) a definition

#### Page No 191:

#### Question 11:

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

#### Answer:

(b) an axiom

#### Page No 191:

#### Question 12:

A pyramid is a solid figure, whose base is

(a) only a triangle

(b) only a square

(c) only a rectangle

(d) any polygon

#### Answer:

(d) any polygon

#### Page No 191:

#### Question 13:

The side faces of a pyramid are

(a) triangles

(b) squares

(c) trapeziums

(d) polygons

#### Answer:

(a) triangles

**â€‹**

#### Page No 191:

#### Question 14:

The number of dimensions of a solid are

(a) 1

(b) 2

(c) 3

(d) 5

#### Answer:

A solid shape has length, breadth and height. Thus, a solid has **three** dimensions.

Hence, the correct answer is option (c).

#### Page No 191:

#### Question 15:

The number of dimensions of a surface are

(a) 1

(b) 2

(c) 3

(d) 0

#### Answer:

A plane surface has length and breadth, but it has **no** height. Thus, a plane surface has **two** dimensions.

Hence, the correct answer is option (b).

#### Page No 191:

#### Question 16:

How many dimensions does a point have

(a) 0

(b) 1

(c) 2

(d) 3

#### Answer:

A point is a fine dot which represents an exact position. It has **no** length, **no** breadth and **no** height. Thus, a point has **no** dimension or a point has **zero** dimension.

Hence, the correct answer is option (a).

#### Page No 191:

#### Question 17:

Boundaries of solids are

(a) lines

(b) curves

(c) surfaces

(d) none of these

#### Answer:

(c) surfaces

#### Page No 191:

#### Question 18:

Boundaries of surfaces are

(a) lines

(b) curves

(c) polygons

(d) none of these

#### Answer:

(b) curves

#### Page No 191:

#### Question 19:

The number of planes passing through 3 non-collinear points is

(a) 4

(b) 3

(c) 2

(d) 1

#### Answer:

(d) 1

#### Page No 191:

#### Question 20:

Axioms are assumed

(a) definitions

(b) theorems

(c) universal truths specific to geometry

(d) universal truths in all branches of mathematics

#### Answer:

(d) universal truths in all branches of mathematics

#### Page No 192:

#### Question 21:

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 parallel planes.

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

#### Answer:

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

#### Page No 192:

#### Question 22:

Which of the following is a true statement?

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

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

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

(d) A line has a definite length.

#### Answer:

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

#### Page No 192:

#### Question 23:

Which of the following is a false statement?

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

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

(c) Ray $\underset{AB}{\to}=\mathrm{ray}\underset{BA}{\to}$.

(d) A ray has one end-point.

#### Answer:

(c) **$Ray\overrightarrow{AB}=Ray\overrightarrow{BA}$ **

#### Page No 192:

#### Question 24:

A point *C* is called the mid-point of a line segment $\overline{)AB}$ if

(a) *C* is an interior point of *AB*

(b) *AC* = *CB*

(c) *C* is an interior point of *AB,* such that $\overline{)AC}=\overline{)CB}$

(d) *AC* + *CB* = *AB*

#### Answer:

(c) *C* is an interior point of *AB,* such that *AC* = *CB*

#### Page No 192:

#### Question 25:

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) None of these

#### Answer:

(c) points *A*, *C* and *B* are collinear

#### Page No 192:

#### Question 26:

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

#### Answer:

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

*x + y *= 15

Adding *z* to both sides, we get

*x + y + z *= 15 + *z*

Thus, Euclid's second axiom illustrates the statement that when *x + y *= 15, then* x + y + z *= 15 + *z*.

Hence, the correct answer is option (b).

#### Page No 192:

#### Question 27:

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

#### Answer:

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

It is given that, the age of A is equal to the age of B and the age of C is equal to the age of B.

Using Euclid's first axiom, we conclude that the age of A is equal to the age of C.

Thus, Euclid's first axiom illustrates the relative ages of A and C.

Hence, the correct answer is option (a).

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