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Page No 188:

Question 1:

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.
ii) The sum of all the angles of triangle is 180.        

Page No 188:

Question 2:

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.
ii) The sum of all the angles of triangle is 180.        

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:

(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.

Answer:

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

(ii) EF¯, GH¯, FH¯ , EG¯, MN¯

(iii) EP, GR, HS, FQ

(iv) AB, CD, PQ, RS

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



Page No 189:

Question 4:

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

(ii) EF¯, GH¯, FH¯ , EG¯, MN¯

(iii) EP, GR, HS, FQ

(iv) AB, CD, PQ, RS

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

Answer:

(i) Two pairs of intersecting lines and their point of intersection are
EF, GH, point R, AB, CD, point P

(ii) Three concurrent lines are
AB, EF, GH, point R

(iii) Three rays are
RB, RH, RF

(iv) Two line segments are
RQ¯ and RP¯

Page No 189:

Question 5:

(i) Two pairs of intersecting lines and their point of intersection are
EF, GH, point R, AB, CD, point P

(ii) Three concurrent lines are
AB, EF, GH, point R

(iii) Three rays are
RB, RH, RF

(iv) Two line segments are
RQ¯ and RP¯

Answer:


(a) Line PQLineRS and LineAB
(b) CEFG
(c) No point is concurrent.

Page No 189:

Question 6:


(a) Line PQLineRS and LineAB
(b) CEFG
(c) No point is concurrent.

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
AB, BC¯ and AC.

Page No 189:

Question 7:

(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
AB, BC¯ and AC.

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. BA and 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:

(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. BA and 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

Answer:


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

∴ AL = BL = 12AB       .....(1)

Also, M is the mid-point of BC.

∴ BM = MC = 12BC     .....(2)

AB = BC       (Given)

⇒ 12AB = 12BC        (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 = 12AB

⇒ 2AL = 2BL = AB        .....(3)

Also, M is the mid-point of BC.

∴ BM = MC = 12BC    

⇒ 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:


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

∴ AL = BL = 12AB       .....(1)

Also, M is the mid-point of BC.

∴ BM = MC = 12BC     .....(2)

AB = BC       (Given)

⇒ 12AB = 12BC        (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 = 12AB

⇒ 2AL = 2BL = AB        .....(3)

Also, M is the mid-point of BC.

∴ BM = MC = 12BC    

⇒ 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)]

Answer:

(b) squares and circles

Page No 190:

Question 2:

(b) squares and circles

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:

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).

Answer:

(c) nine

Page No 190:

Question 4:

(c) nine

Answer:

(b) 4 : 2 : 1

Page No 190:

Question 5:

(b) 4 : 2 : 1

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:


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

Hence, the correct answer is option (a).

Answer:

(b) Greece

Page No 191:

Question 7:

(b) Greece

Answer:

(c) Greece

Page No 191:

Question 8:

(c) Greece

Answer:

(ii) Thales

Page No 191:

Question 9:

(ii) Thales

Answer:

(d) theorem

Page No 191:

Question 10:

(d) theorem

Answer:

(a) a definition

Page No 191:

Question 11:

(a) a definition

Answer:

(b) an axiom

Page No 191:

Question 12:

(b) an axiom

Answer:

(d) any polygon

Page No 191:

Question 13:

(d) any polygon

Answer:

(a) triangles
​

Page No 191:

Question 14:

(a) triangles
​

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:


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

Hence, the correct answer is option (c).

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:


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).

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:


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).

Answer:

(c) surfaces

Page No 191:

Question 18:

(c) surfaces

Answer:

(b) curves

Page No 191:

Question 19:

(b) curves

Answer:

(d) 1

Page No 191:

Question 20:

(d) 1

Answer:

(d) universal truths in all branches of mathematics



Page No 192:

Question 21:

(d) universal truths in all branches of mathematics

Answer:

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

Page No 192:

Question 22:

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

Answer:

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

Page No 192:

Question 23:

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

Answer:

(c) Ray AB = Ray BA 

Page No 192:

Question 24:

(c) Ray AB = Ray BA 

Answer:

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

Page No 192:

Question 25:

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

Answer:

(c) points A, C and B are collinear

Page No 192:

Question 26:

(c) points A, C and B are collinear

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:


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).

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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