Class 9 Mathematics · 48 questions · 35 with answers
Euclid’s second axiom (as per order given in the Textbook for Class IX) is INTRODUCTION TO EUCLID’S GEOMETRY 45
Solution : Answer (B)
Euclid’s fifth postulate is
Solution : Answer (D)
The things which are double of the same thing are
Solution : Answer (A)
Axioms are assumed
Solution : Answer (A)
John is of the same age as Mohan. Ram is also of the same age as Mohan. State the Euclid’s axiom that illustrates the relative ages of John and Ram
Solution : Answer (A)
If a straight line falling on two straight lines makes the interior angles on the same side of it, whose sum is 120°, then the two straight lines, if produced indefinitely, meet on the side on which the sum of angles is
Solution : Answer (C)
Write whether the following statements are True or False? Justify your answer.
Solution : (i) False. The side faces of a pyramid are triangles not necessarily equilateral triangles. (ii) True. The geometry of Vedic period originated with the construction of vedis and fireplaces for performing vedic rites. The location of the sacred fires had to be in accordance to the clearly laid down instructions about their shapes and area. (iii) True. To define a point, a line and a plane in geometry we need to define many other things that give a long chain of definitions without an end. For such reasons, mathematicians agree to leave these geometric terms undefined. (iv) True. Things equal to the same thing are equal. (v) True. It is the justification of the principle of superposition. (vi) True. It fails on the curved surfaces. For example on curved surfaces, the sum of angles of a triangle may be more than 180°.
Ram and Ravi have the same weight. If they each gain weight by 2 kg, how will their new weights be compared ?
Solution : Let x kg be the weight each of Ram and Ravi. On gaining 2 kg, weight of Ram and Ravi will be (x + 2) each. According to Euclid’s second axiom, when equals are added to equals, the wholes are equal. So, weight of Ram and Ravi are again equal.
Solve the equation a – 15 = 25 and state which axiom do you use here.
Solution : a – 15 = 25. On adding 15 to both sides, we have a – 15 + 15 = 25 + 15 = 40 (using Euclid’s second axiom). or a = 40
In the Fig. 5.1, if ∠1 = ∠3, ∠ 2 = ∠ 4 and ∠ 3 = ∠ 4, write the relation between ∠1 and ∠2, using an Euclid’s axiom.
Solution : Here, ∠3 = ∠4, ∠1 = ∠3 and ∠2 = ∠4. Euclid’s first axiom says, the things which are equal to equal thing are equal to one aother. Fig. 5.1 So, ∠1 = ∠2.
In Fig. 5.2, we have : AC = XD, C is the mid-point of AB and D is the mid-point of XY. Using an Euclid’s axiom, show that AB = XY.
Solution : AB = 2AC (C is the mid-point of AB) XY = 2XD (D is the mid-point of XY) Also, AC = XD (Given) Fig. 5.2 Therefore, AB = XY, because things which are double of the same things are equal to one another.
Read the following statement: “A square is a polygon made up of four line segments, out of which, length of three line segments are equal to the length of fourth one and all its angles are right angles”. Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all angles and sides of a square are equal?
Solution : The terms 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. Ray : Part of a line with one end point. Right angle : Angle whose measure is 90°. Undefined terms used are : line, point. Euclid’s fourth postulate says that “all right angles are equal to one another.” In a square, all angles are right angles, therefore, all angles are equal (From Euclid’s fourth postulate). Three line segments are equal to fourth line segment (Given). Therefore, all the four sides of a square are equal. (by Euclid’s first axiom “things which are equal to the same thing are equal to one another.”)
The three steps from solids to points are :
(A) Solids - surfaces - lines - points
The number of dimensions, a solid has :
(C) 3
The number of dimensions, a surface has :
(B) 2
The number of dimension, a point has :
(A) 0
Euclid divided his famous treatise “The Elements” into :
(A) 13 chapters
The total number of propositions in the Elements are :
(A) 465
Boundaries of solids are :
(A) surfaces
Boundaries of surfaces are :
(B) curves
In Indus Valley Civilisation (about 3000 B.C.), the bricks used for construction work were having dimensions in the ratio
(B) 4:2:1
A pyramid is a solid figure, the base of which is
(D) any polygon
The side faces of a pyramid are :
(A) Triangles
It is known that if x + y = 10 then x + y + z = 10 + z. The Euclid’s axiom that illustrates this statement is :
(B) Second Axiom
In ancient India, the shapes of altars used for house hold rituals were :
(A) Squares and circles
The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is:
(C) Nine
Greek’s emphasised on :
(B) Deductive reasoning
In Ancient India, Altars with combination of shapes like rectangles, triangles and trapeziums were used for :
(A) Public worship
Euclid belongs to the country :
(C) Greece
Thales belongs to the country :
(C) Greece
Pythagoras was a student of :
(A) Thales
Which of the following needs a proof ?
(A) Theorem
Euclid stated that all right angles are equal to each other in the form of
(C) a postulate
‘Lines are parallel if they do not intersect’ is stated in the form of
(B) a definition
Euclidean geometry is valid only for curved surfaces.
False, it is valid only for the figures in the plane.
The boundaries of the solids are curves.
False, boundaries of the solids are surfaces.
The edges of a surface are curves.
False, the edges of surfaces are line.
The things which are double of the same thing are equal to one another.
True, one of the Euclid’s axioms.
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.
True, because of one of Euclid’s axioms.
The statements that are proved are called axioms. INTRODUCTION TO EUCLID’S GEOMETRY 49
False, statements that are proved are theorms.
“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 Playfair’s axiom.
True, it is an equivalent version of Euclid’s fifth postulate.
Two distinct intersecting lines cannot be parallel to the same line.
True, it is an equivalent version of Euclid’s fifth postulate.
Attempts to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries.
True, these geometries are different from Euclidean geometry.
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.
It is known that x + y = 10 and that x = z. Show that z + y = 10?
Look at the Fig. 5.3. Show that length AH > sum of lengths of AB + BC + CD. Fig. 5.3
This question refers to a figure in the original PDF.
In the Fig.5.4, we have AB = BC, BX = BY. Show that AX = CY.
This question refers to a figure in the original PDF.
In the Fig.5.5, we have X and Y are the mid-points of AC and BC and AX = CY. Show that AC = BC. Fig. 5.4 Fig. 5.5
This question refers to a figure in the original PDF.
In the Fig.5.6, we have BX = AB BY = BC and AB = BC. Show that Fig. 5.6 BX = BY. INTRODUCTION TO EUCLID’S GEOMETRY 51
This question refers to a figure in the original PDF.
In the Fig.5.7, we have ∠1 = ∠ 2, ∠2 = ∠3. Show that ∠1 = ∠ 3.
This question refers to a figure in the original PDF.
In the Fig. 5.8, we have ∠1 = ∠3 and ∠ 2 = ∠ 4. Show that ∠A = ∠ C. Fig. 5.7 Fig. 5.8
This question refers to a figure in the original PDF.
In the Fig. 5.9, we have ∠ABC = ∠ ACB, ∠3 = ∠ 4. Show that ∠ 1 = ∠2. D
This question refers to a figure in the original PDF.
In the Fig. 5.10, we have AC = DC, CB = CE. Show that AB = DE. Fig. 5.9 Fig. 5.10
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1 11. In the Fig. 5.11, if OX = XY, PX = XZ
This question refers to a figure in the original PDF.
2 and OX = PX, show that XY = XZ. Fig. 5.11 12. In the Fig.5.12 :
This question refers to a figure in the original PDF.
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. INTRODUCTION TO EUCLID’S GEOMETRY 53 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 this question on the same manner as given in the solution of Sample Question 1 in (E).
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.]
Read the following statements which are taken as axioms :
No
Read the following two statements which are taken as axioms :
No
Read the following axioms:
Consistent ANSWERS 159