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#### Page No 188:

#### 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:

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**(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:

#### 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:

#### 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:

#### Answer:

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

(b) $CEFG$

(c) No point is concurrent.

#### Page No 189:

#### 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:

#### 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:

#### 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:

#### Answer:

(b) squares and circles

#### Page No 190:

#### 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:

#### Answer:

(c) nine

#### Page No 190:

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(b) 4 : 2 : 1

#### Page No 190:

#### Answer:

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

Hence, the correct answer is option (a).

#### Page No 191:

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(b) Greece

#### Page No 191:

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(c) Greece

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(ii) Thales

#### Page No 191:

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(d) theorem

#### Page No 191:

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(a) a definition

#### Page No 191:

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(b) an axiom

#### Page No 191:

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(d) any polygon

#### Page No 191:

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(a) triangles

**â€‹**

#### Page No 191:

#### 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:

#### 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:

#### 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:

#### Answer:

(c) surfaces

#### Page No 191:

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(b) curves

#### Page No 191:

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(d) 1

#### Page No 191:

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(d) universal truths in all branches of mathematics

#### Page No 192:

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

#### Page No 192:

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

#### Page No 192:

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

#### Page No 192:

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

#### Page No 192:

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

#### Page No 192:

#### 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:

#### 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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