NCERT Exemplar Class 9 Maths Solutions Chapter 5 Introduction to Euclids Geometry

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

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Question:2 The number of dimensions, a solid has:
(A) 1
(B) 2
(C) 3
(D) 0

Answer:
We know that a solid is a three-dimensional object having a definite shape, size, and volume.
Three-dimensional means that to represent the shape, we need three axes (x, y, z).
The three dimensions of solids are generally known as length, width, and height.
For example, a cuboid is a solid with three dimensions as shown:

In the case of a cube, we know that length = width = height.
Hence all are called side lengths.
$\because$ Solid has three dimensions.

Hence, option (C) is correct.

Question:3 The number of dimensions, a surface has:
(A) 1
(B) 2
(C) 3
(D) 0

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Question:4 The number of dimensions, a point has:
(A) 0
(B) 1
(C) 2
(D) 3

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Question:5 Euclid divided his famous treatise “The Elements” into:
(A) 13 chapters
(B) 12 chapters
(C) 11 chapters
(D) 9 chapters

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Question:6 The total number of propositions in the Elements are:
(A) 465
(B) 460
(C) 13
(D) 55

Answer:
Euclid divided his famous treatise “The Elements” into 13 chapters. It is a collection of definitions, postulates, theorems, and mathematical proofs of these theorems.
By propositions, we mean theorems and constructions. Euclid’s total propositions are 465.
Hence, Euclid deduced 465 propositions in the Elements using his axioms, postulates, definitions, and theorems.

Hence, option (A) is correct.

Question:7 Boundaries of solids are:
(A) surfaces
(B) curves
(C) lines
(D) points

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Question:8 Boundaries of surfaces are:
(A) Surfaces
(B) curves
(C) lines
(D) points

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Question:9: In the Indus Valley Civilization (about 3000 B.C.), the bricks used for construction work had dimensions in the ratio
(A) 1 : 3: 4
(B) 4: 2: 1
(C) 4: 4 1
(D) 4: 3:2

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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

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Question:11 The side faces of a pyramid are:
(A) Triangles
(B) Squares
(C) Polygons
(D) Trapeziums

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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

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Question:13 In ancient India, the shapes of altars used for household rituals were:
(A) Squares and circles
(B) Triangles and rectangles
(C) Trapeziums and pyramids
(D) Rectangles and squares

Answer:

In India, the shapes of altars used for household rituals were squares and circles. Other shapes like rectangles, triangles, trapeziums, pyramids, and rectangles were used in public worship. In ancient times, a Hindu home was required to have three fires burning at three different altars. The three altars were to be of the same area but different shapes. Hence, in ancient India, square and circular shapes of altars were used for household rituals.

Hence, option (A) is correct.

Question:14 The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is:
(A) Seven
(B) Eight
(C) Nine
(D) Eleven

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Question:15 Greeks emphasized on:
(A) Inductive reasoning
(B) Deductive reasoning
(C) Both A and B
(D) Practical use of geometry

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Question:16 In Ancient India, Altars with a 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

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Question:17 Euclid belongs to the country:
(A) Babylonia
(B) Egypt
(C) Greece
(D) India

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Question:18 Thales belongs to the country:
(A) Babylonia
(B) Egypt
(C) Greece
(D) Rome

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Question:19 Pythagoras was a student of:
(A) Thales
(B) Euclid
(C) Both A and B
(D) Archimedes

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Question:20 Which of the following needs a proof?
(A) Theorem
(B) Axiom
(C) Definition
(D) Postulate

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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

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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

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Question:1 Write whether the following statements are True or False. Justify your answer: Euclidean geometry is valid only for curved surfaces.

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Question:2 Write whether the following statements are True or False. Justify your answer: The boundaries of the solids are curves.

Answer:

Solid: A solid is an object that has length, width, and height, it is not a flat shape. Hence if we notice a solid then we can see that the boundaries of solids are not curves but surfaces.

Therefore, the given statement is false.

Question:3 Write whether the following statements are True or False.? Justify your answer: The edges of a surface are curves.

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Question:4 Write whether the following statements are True or False. Justify your answer: The things which are double of the same thing are equal to one another.

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Question:5 Write whether the following statements 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.

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Question:6 Write whether the following statements are True or False. Justify your answer: The statements that are proved are called axioms.

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Question:7 Write whether the following statements are True or False. Justify your answer:
“For every line l and for every point P not lying on a given line, there exists a unique line m passing through P and parallel to l is known as Playfair’s axiom.”

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Question:8 Write whether the following statements are True or False. Justify your answer: Two distinct intersecting lines cannot be parallel to the same line.

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Question:9 Write whether the following statements are True or False. Justify your answer:
Attempts to prove Euclid’s fifth postulate using other postulates and axioms led to the discovery of several other geometries.

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Question:1 Solve each of the following questions using appropriate Euclid’s axiom: Two salesmen make equal sales during the month of August. In September, each salesman doubles his sales for the month of August. Compare their sales in September.

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Question:2 Solve each of the following questions using appropriate to Euclid’s axiom: It is known that x + y = 10 and that x = z. Show that z + y = 10.

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Question:3 Solve each of the following questions using appropriate Euclid’s axiom: Look at Fig. 5.3. Show that length AH > sum of lengths of AB + BC + CD.

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Question:4 Solve each of the following questions using appropriate Euclid’s axiom: In Fig.5.4, we have AB = BC, BX = BY. Show that AX = CY.

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Question:5 Solve each of the following questions using appropriate Euclid’s axiom: In Fig.5.5, we have X and Y are the mid-points of AC and BC, and AX = CY. Show that AC = BC.

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Question:6 Solve each of the following questions using appropriate Euclid’s axiom: In the Fig.5.6, we have
$BX=\frac{1}{2}AB,BY=\frac{1}{2}BC$
and AB = BC. Show that BX = BY.

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Question:7 Solve each of the following questions using appropriate Euclid’s axioms: In Fig.5.7, we have $\mathrm{\angle }$1 = $\mathrm{\angle }$2, $\mathrm{\angle }$2 = $\mathrm{\angle }$3. Show that $\mathrm{\angle }$1 = $\mathrm{\angle }$3.

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Question:8 Solve each of the following questions using appropriate Euclid’s axiom: In Fig. 5.8, we have $\mathrm{\angle }$1 = $\mathrm{\angle }$3 and $\mathrm{\angle }$2 = $\mathrm{\angle }$4. Show that $\mathrm{\angle }$A = $\mathrm{\angle }$C.

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Question:9 Solve each of the following questions using appropriate Euclid’s axioms: In Fig. 5.9, we have $\mathrm{\angle }$ABC = $\mathrm{\angle }$ACB, $\mathrm{\angle }$3 = $\mathrm{\angle }$4. Show that $\mathrm{\angle }$1 = $\mathrm{\angle }$2.

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Question:10 Solve each of the following questions using appropriate Euclid’s axiom: In Fig. 5.10, we have AC = DC, CB = CE. Show that AB = DE.

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Question:11 Solve each of the following questions using appropriate Euclid’s axiom: In Fig. 5.11, if OX=0.5XY, PX=0.5XZ and OX = PX, show that XY = XZ.

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Question:12 i) Solve each of the following questions using appropriate Euclid’s axiom: In the Fig. 5.12:

AB = BC, M is the mid-point of AB and N is the mid-point of BC. Show that AM = NC.
ii) Solve each of the following questions using appropriate to Euclid’s axiom: In Fig. 5.12:

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

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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 that you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in an equilateral triangle?

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Question:2 Study the following statement: “Two intersecting lines cannot be perpendicular to the same line”. Check whether it is an equivalent version to Euclid’s fifth postulate.

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Question:3 Read the following statements which are taken as axioms:

1. If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.
2. If a transversal intersects two parallel lines, then alternate interior angles are equal.

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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.

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Question:5 Read the following axioms:

1. Things which are equal to the same thing are equal to one another.
2. If equals are added to equals, the wholes are equal.
3. Things which are double of the same thing are equal to one another.

Answer:

1. Things that are equal to the same thing are also equal to one another
2. If equals are added to equals, then the wholes are equal
3. If equals are subtracted from equals, then the differences are equal
4. Things that coincide with one another are equal to one another
5. The whole is greater than the part.