CBSE Class 9 Mathematics Euclids Geometry Worksheet Set B

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1.A terminated line can be produced

(A) definitely

(B) indefinitely

(C) both A & B

(D) None

2.Two lines are said to be coincident if they have only one common Point. (T/F)

3.Which of the following has two dimension ?

(A) solid

(B) surface

(C) line

(D) point

4.The Euclid's geometry was developed by

(A) Thales

(B) Pythagoras

(C) Egypt

(D) Euclid

5.A solid has

(A) one dimension

(B) two dimension

(C) three dimension

(D) none

6.In Euclid's axioms, If equals are added to equals, the wholes are

(A) unequal

(B) equal

(C) many or many not be equal

(D) none of these

7.The edges of a surface are

(A) points

(B) length

(C) lines

(D) none

8.If a straight line falling on two straight lines makes the interior angles on the same side of it taken

together their sum is less than the sum of two right angles, then the two straight lines, if produced

indefinitely, meet on that side on which the sum of angles is :

(A) < 180º

(B) > 180º

(C) 180º

(D) 180º

9.Two distinct intersecting lines cannot be parallel to the

(A) same line

(B) distinct line

(C) both a & b

(D) none

10.For all positive integer n, 2n + 1 is

(A) even

(B) odd

(C) prime

(D) none

11.If a point C lies between two points A and B such that AC = BC, then AC =

(A) AB

(B) BC

1. Euclid’s Definitions, Axioms and Postulates
2. Equivalent Versions of Euclid’s Fifth Postulate
The Greeks developed geometry is a systematic manner Euclid (300 B.C.) a greek mathematician, father of geometry introduced the method of proving mathematical results by using deductive logical reasoning and the previously proved result. The Geometry of plane figure is known as "Euclidean Geometry". 
Axioms: The basic facts which are taken for granted without proof are called axioms some Euclid's axioms are:
(i) Things which are equal to the same thing are equal to one another. i.e. a = b, b = c⇒a = c
(ii) If equals are added to equals, the wholes are equal i.e. a = b ⇒ a + c = b + c
(iii) If equals are subtracted from equals, the remainders are equal i.e. a = b⇒a - c = b - c
(iv) Things which coincide with one another are equal to one another.
(v) The whole is greater than the part.
Postulates: Axioms are the general statements, postulates are the axioms relating to a particular field.
Educlid's five postulates are.
(i) A straight line may be drawn from any one point to any other point.
(ii) A terminated line can be produced indefinitely.
(iii) A circle can be drawn with any centre and any radius.
(iv) All right angles are equal to one another.
(v) If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely meet on that side on which the angles are less than two right angles.
Statements: A sentence which is either true or false but not both, is called a statement.
eg. (i) 4+9=6 If is a false sentence, so it is a statement.
(ii) Sajnay is tall. This is not a statement because he may be tall for certain persons and may not be taller for others.
Theorems: A statement that requires a proof is called a theorem.
eg. (i) The sum of the angles of triangle is 180o .
(ii) The angles opposite to equal sides of a triangles are equal.
Corollary - Result deduced from a theorem is called its corollary.

Question. 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: A

Question. The number of dimension, a point has:
(a) 0
(b) 1
(c) 2
(d) 3

Answer: A

Question. The total number of propositions in the Elements are:
(a) 465
(b) 460
(c) 13
(d) 55

Answer: A

Question. The number of dimensions, a solid has:
(a) 1
(b) 2
(c) 3
(d) 0

Answer: C

Question. Boundaries of solids are:
(a) surfaces
(b) curves
(c) lines
(d) points

Answer: A

Question. Boundaries of surfaces are:
(a) surfaces
(b) curves
(c) lines
(d) points

Answer: B

Question. 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: A

Question. 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: D

Question. The side faces of a pyramid are:
(a) Triangles
(b) Squares
(c) Polygons
(d) Trapeziums

Answer: A

Question. 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: B

Question. 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: B

Question. Euclid divided his famous treatise “The Elements” into:
(a) 13 chapters
(b) 12 chapters
(c) 11 chapters
(d) 9 chapters

Answer: A

Question. The number of interwoven isosceles triangles in Sriyantra (in the Atharva Veda) is:
(a) Seven
(b) Eight
(c) Nine
(d) Eleven

Answer: C

Question. 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: A

Question. 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: C

Question. Euclid belongs to the country:
(a) Babylonia
(b) Egypt
(c) Greece
(d) India

Answer: C

Question. Greek’s emphasised on:
(a) Inductive reasoning
(b) Deductive reasoning
(c) Both A and B
(d) Practical use of geometry

Answer: B

Question. Thales belongs to the country:
(a) Babylonia
(b) Egypt
(c) Greece
(d) Rome

Answer: C

Question. The number of dimensions, a surface has:
(a) 1
(b) 2
(c) 3
(d) 0

Answer: B

Question. Which of the following needs a proof?
(a) Theorem
(b) Axiom
(c) Definition
(d) Postulate

Answer: A

Question. ‘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: B

Question. Pythagoras was a student of:
(a) Thales
(b) Euclid
(c) Both A and B
(d) Archimedes

Answer: A

Question. The boundaries of the solids are curves.
Answer. The given statement is false because boundaries of solids are surfaces.

Question. The statements that are proved are called axioms.
Answer. The given statement is false because the statement that are proved are called theorems.

Question. The things which are double of the same thing are equal to one another.
Answer. True Since, it is one of the Euclid’s axioms. Some of Euclid’s axiom:
(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) If equals are subtracted from equals, the remainder are equal.
(4) Things which coincide with one another are equal to one another.
(5) The whole is greater than the part.
(6) Things which are double of the same things are equal to one another.
(7) Thing which are halves of the same things are equal to one another.

Question. Euclidean geometry is valid only for curved surfaces.
Answer. The given statement is false because Euclidean geometry is valid only for the figures in the plane.

Question. Attempts to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries.
Answer. The given statement is true because these geometries are different from Euclidean geometry called non-Euclidean geometry.

Question. 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. The given statement is true because it is one of Euclid’s axiom.

Question. “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 Play fair’s axiom.
Answer. The given statement is true, because it is an equivalent version of Euclid’s fifth postulate.

Question. The edges of a surface are curves.
Answer. The given statement is false because the edges of surfaces are line.

Question. Two distinct intersecting lines cannot be parallel to the same line.
Answer. The given statement is true, because it is an equivalent version of Euclid’s fifth postulate.

Question. 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 the sales of two salesmen in the month of August be x and y. As, they make equal sale during the month of August, x = y. In September, each salesman double his sale of the month of August, So 2x = 2y.
Now, by Euclid’s axiom, thing which are double of the same things are equal to one another.
Hence, we can say that in the month of September also, two salesmen make equal sales.

Question. Look at the figure, and show that
Length AH > sum of lengths of AB + BC + CD.

Answer. We see the AB, BC and CD are parts of line.
Now, AB + BC + CD = AD                                                         …(1)
By Euclid’s axiom 5, the whole is greater than the part, so AH > AD
i.e., Length AH> sum of length of AB + BC + CD                [Using (1)]

Question. It is known that x + y = 10 and that x = z. Show that z + y= 10?
Answer. It is known that x + y = 10 and that x = z.
∴ x + y = z + y                [∵By Euclid’s axiom 2, if equals of are added to equals, the wholes are equal]
⇒ 10 = y + z                   [Using (1), x + y = 10]
Hence, z + y = 10.

Question. 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. [Hint: Identify the two
intersecting lines l and m and the line n in the above statement.]
Answer. Two intersecting lines cannot be both perpendicular to the same line because if two lines l and m are perpendicular to the same line n, then l and m must be parallel. The given statement is not an equivalent version of Euclid’s fifth postulate. 

Question. 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 180o, then for two lines which intersect each other, the vertically opposite angles becomes equal.

Question. 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 term need to be defined are:
Polygon: A simple closed figure made up of three or more line segments.
Line segment: Part of a line with two end points.
Line: Undefined term.
Point: Undefined term.
Angle: A figure formed by two rays with a common initial point.
Acute angle: Angle whose measure is between 0o and 90o.
Undefined terms used are: Line, part Two line segments are equal to third line segment Therefore, all three sides of an equilateral triangle are equal All its angle are 60o each. Therefore, all angles are equal (by Euclid’s first axiom, things which are equal to same things are equal to one another.)
Hence, we can say that all sides and all angles are equal in an equilateral triangle.

Question. 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. The given system of axioms is consistent because (i), (ii) and (iii) are Euclid’s axiom.

Question. 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 intersects two parallel lines, then alternate interior angles are equal. Is this system of axioms consistent? Justify your answer.
Answer. No, this system of axioms is not consistent because if a transversal intersects two parallel lines and if corresponding angles are not equal, then alternate interior angles cannot be equal.