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NCERT Exemplar · Class 9 Mathematics Introduction to Euclid's Geometry

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EXERCISE 5.2 1–9 (part 3 of 5)

  1. Write whether the following statements are True or False? Justify your answer :

    Exercise 1

    Euclidean geometry is valid only for curved surfaces.

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    NCERT’s answer
    False, it is valid only for the figures in the plane.
    False. Euclid's postulates describe flat surfaces: figures and lines behave as stated only in the plane. On a curved surface, such as a sphere, they fail — so the claim has the surfaces reversed.

  2. Exercise 2

    The boundaries of the solids are curves.

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    NCERT’s answer
    False, boundaries of the solids are surfaces.
    False. A solid's boundary is a surface.

  3. Exercise 3

    The edges of a surface are curves.

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    NCERT’s answer
    False, the edges of surfaces are line.
    False. By Euclid's Definition $\displaystyle 6$, the edges of a surface are lines, not curves.

  4. Exercise 4

    The things which are double of the same thing are equal to one another.

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    NCERT’s answer
    True, one of the Euclid's axioms.
    True — this is Euclid's Axiom $\displaystyle 6$ word for word: quantities that are each double a common quantity must equal each other.

  5. Exercise 5

    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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    NCERT’s answer
    True, because of one of Euclid's axioms.
    True. "Part" means a remainder is left over: \[A = B + C \] with \(\displaystyle C\) the quantity by which \(\displaystyle B\) falls short of \(\displaystyle A\) — Euclid's whole–part relation.

  6. Exercise 6

    The statements that are proved are called axioms.

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    NCERT’s answer
    False, statements that are proved are theorms.
    False. Axioms are the self-evident truths accepted without proof; a statement that is actually proved from them is called a theorem, not an axiom.

  7. Exercise 7

    "For every line l\displaystyle l and for every point P not lying on a given line l\displaystyle l, there exists a unique line m\displaystyle m passing through P and parallel to l\displaystyle l " is known as Playfair's axiom.

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    NCERT’s answer
    True, it is an equivalent version of Euclid's fifth postulate.
    True — this is the standard modern restatement of Euclid's fifth postulate, and it is exactly the form named after Playfair.

  8. Exercise 8

    Two distinct intersecting lines cannot be parallel to the same line.

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    NCERT’s answer
    True, it is an equivalent version of Euclid's fifth postulate.
    True. By Playfair's axiom a point admits only one parallel to a given line \(\displaystyle m\); two distinct lines through it can't both be parallel to \(\displaystyle m\) without coinciding, so lines that meet cannot share a parallel.

  9. Exercise 9

    Attempts to prove Euclid's fifth postulate using the other postulates and axioms led to the discovery of several other geometries. (D) Short Answer Questions Sample Question 1\displaystyle 1 : Ram and Ravi have the same weight. If they each gain weight by 2\displaystyle 2 kg, how will their new weights be compared? Solution : Let x kg\displaystyle x \mathrm{~kg} be the weight each of Ram and Ravi. On gaining 2\displaystyle 2 kg, weight of Ram and Ravi will be ( x+2\displaystyle 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. Sample Question 2\displaystyle 2 : Solve the equation a−15=25\displaystyle a-15=25 and state which axiom do you use here. Solution : a−15=25\displaystyle a-15=25. On adding 15\displaystyle 15 to both sides, we have a−15+15=25+15=40\displaystyle a-15+15=25+15=40 (using Euclid's second axiom). or a=40\displaystyle a=40 Sample Question 3\displaystyle 3 : In the Fig. 5.1\displaystyle 5.1, if ∠1=∠3,∠2=∠4\displaystyle \boldsymbol{\angle} 1=\boldsymbol{\angle} 3, \boldsymbol{\angle} 2=\boldsymbol{\angle} 4 and ∠3=∠4\displaystyle \boldsymbol{\angle} 3=\boldsymbol{\angle} 4, write the relation between ∠1\displaystyle \angle 1 and ∠2\displaystyle \angle 2, using an Euclid's axiom. Solution : Here, ∠3=∠4,∠1=∠3\displaystyle \angle 3=\angle 4, \angle 1=\angle 3 and ∠2=∠4\displaystyle \angle 2=\angle 4. Euclid's first axiom says, the things which are equal to equal thing are equal to one aother. So, ∠1=∠2\displaystyle \angle 1=\angle 2. Sample Question 4\displaystyle 4 : In Fig. 5.2\displaystyle 5.2, we have : AC=XD,C\displaystyle \mathrm{AC}=\mathrm{XD}, \mathrm{C} is the mid-point of AB and D is the mid-point of XY. Using an Euclid's axiom, show that AB=XY\displaystyle \mathrm{AB}=\mathrm{XY}. NCERT_Question_Class9_Maths_Exemplar_Ch5_Ex5-2_Q9 NCERT_Question_Class9_Maths_Exemplar_Ch5_Ex5-2_Q9

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    NCERT’s answer
    True, these geometries are different from Euclidean geometry.
    True. Euclid's fifth postulate could not be derived from the other four; every attempt to prove it instead produced a new, consistent set of axioms, giving rise to the non-Euclidean geometries.