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9th CBSE Mathematics Unit 5 Introduction to Euclid's Geometry Important Question Paper

Published on: 31/07/2019

Introduction to Euclid's Geometry

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Questions:

Generic 3

4 x 3 = 12

(i) Why is Axiom 5, in the list of Euclid's axioms, considered a 'universal truth'? (Note that the question is not about the fifth postulate).
(ii) How would you rewrite Eulid's fifth postulate so that it would be easier to understand?

Point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.

Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?
(i) parallel lines
(ii) perpendicular lines
(iii) line segment
(iv) radius of a circle
(v) square.

Which of the following statements are true and which are false? Give reasons for your answers:
(i) Only one line can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are Equal.

Generic 4

3 x 4 = 12

In the given figure, it is given that \(\angle \)1 = \(\angle \)4 and \(\angle \)3 = \(\angle \)2. By which Euclid's axiom, it can be shown that if \(\angle \)2 = \(\angle \)4, then \(\angle \)1 = \(\angle \)3.

In the given figure, if AB = CD, then prove that AC = BD. Also, write the Euclid's Axiom used for proving it.

Solve the equation x -15 = 25 and state Euclid's Axiom used here.

Generic 2

5 x 2 = 10

Ram and Ravi have the same weight. If they each gain weight by 2kg, how will their new weights be compared?

In the given figure, we have AB=AD and AC=AD. Prove that AB=AC. State the Euclid's axiom to support this.

10.

State any two Euclid's axioms.

11.

State any two Eulis's axioms.

12.

Consider the following statement: There exists a pair of straight lines that are everywhere equidistant from one another. Is this statement a

Multiple Choice Question

5 x 1 = 5

13.

Two planes intersect each other to form a:

(a)

plane

(b)

Point

(c)

straight line

(d)

angle

14.

The number of line segments determined by three collinear points is:

(a)

two

(b)

Three

(c)

Only one

(d)

Four

15.

The number of lines that can pass through a given point is:

(a)

two

(b)

none

(c)

only one

(d)

infinite many

16.

In ancient India, alters with combination of shapes like rectangles, triangles and trapeziums were used for

(a)

public workship

(b)

household rituals

(c)

both (a) and (b)

(d)

none of the above

17.

In Indus Valley Civilisation (about 300 b.C), The brick used for construction work were having dimension in the ration

(a)

1 : 3 : 4

(b)

4 : 2 : 1

(c)

4 : 4 : 1

(d)

4 : 3 : 2

1 Marks

5 x 1 = 5

18.

How can we identify parallel lines?

19.

Write the number of dimension(s) of a surface.

20.

What is a surface?

21.

Give any one example of a geometrical line from your surroundings.

22.

What is a straight line?

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Answers Key:

Generic 3

4 x 3 = 12

(i) Since this is true for anything in any part of the world, this is a universal truth.
(ii) If the sum of the cointerior angles made by a transversal intersect two straight lines at distinct points is less than 180^o, then the lines cannot be parallel.

Let a line AB have two mid-points, say, C and D. Then,
AC = \(1\over2\) AB ...(1)
AD = \(1\over2\)AB ...(2)
From (1) and (2),
AC = AD.
Things which are equal to the same thing are equal to one another

(i) Parallel lines. Lines which do not intersect anywhere are called parallel lines.
(ii) Perpendicular lines. Two lines which are at a right angle to each other are called perpendicular lines.
(iii) Line segment. It is a terminated line.
(iv) Radius. The length of the line segment joining the centre of a circle to any point on its circumference is called its radius.
(v) Square. A quadrilateral with all the four sides equal and all the four angles of measure 90° each is called a square.

(i) False. This can be seen visually.
(ii) False. This contradicts the Axiom.
[Given two distinct points, there is a unique line that passes through them.]
(iii) True by Euclid's Postulate
[A terminated line can be produced indefinitely.]
(iv) True. If we superimpose the region bounded by one circle on the other, then they coincide. So, their centres and boundaries coincide, therefore, their radii will coincide.
(v) True by the first Axiom of Euclid.
[Things which are equal to the same thing are equal to one another.]

Generic 4

3 x 4 = 12

\(\angle \)2 = \(\angle \)4 ,,,,,,,(1) | Given
\(\angle \)1= \(\angle \)4 ........(2) | Given
\(\angle \)3 =\(\angle \)2 .......(3) | Given
From (1), \(\angle \)4 = \(\angle \)2 ......(4)
From (3) and (4),
\(\angle \)3 = \(\angle \)4 .....(5)
I Things which are equal to the same thing are equal to one another
From (2) and (4),
\(\angle \)1 = \(\angle \)3
I Things which are equal to the same thing are equal to one another

We have, AX=CY
AB=CD
\(\Rightarrow \) AB + BC = CD + BC
| If equals are added to equals, the wholes are equal (Euclid's Axiom (ii))
AC= BD
I AB + BC coincides with AC;
CD + BC coincides with BD
[Things which coincide with one another are equal to one another (Euclid's Axiom (iv))]

We have,
x - 15 = 25
\(\Rightarrow \) x - I5 + 15 = 25 + 15 | If equals are added to equals, the wholesare equal (Euclid's Axiom (ii))
\(\Rightarrow \) x = 40

Generic 2

5 x 2 = 10

Let x be the weight of Ram and Ravi each. On gaining 2 kg, weight of Ram and Ravi will be (x+2) kg each. According to Euclid's second axiom, when equals are added to equals, the wholes are equal. So, weights of Ram and Ravi are again equal.

Things which are equal to the same thing are equal to one another.

10.

Euclid's 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.

11.

Euclid's 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.

12.

Take any line l and a point P not on l. Then, by Playfair’s axiom, which is equivalent to the fifth postulate, we know that there is a unique line m through P which is parallel to l.

Multiple Choice Question

5 x 1 = 5

13.

14.