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

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EXERCISE 5.4 1–5 (part 5 of 5)

  1. Exercise 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\displaystyle 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.

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    NCERT’s answer
    Answer this question on the same manner as given in the solution of Sample Question $\displaystyle 1$ in (E).
    Terms needing definition: Polygon: a closed figure made up of three or more line segments. Line segment: part of a line with two end points. Angle: two rays with a common initial point. Point and line stay undefined, exactly as Euclid left them.NCERT_Solution_Class9_Maths_Exemplar_Ch5_Ex5-4_Q1 \[AB = CA, \quad BC = CA \quad \text{(the definition's own claim: two sides equal the third)} \] \[\Rightarrow\ AB = BC \quad \text{(Axiom 1: things equal to the same thing are equal to one another)} \] \[\therefore\ AB = BC = CA \] Each angle is \(\displaystyle 60°\) directly by the definition, so all three are equal (Axiom $\displaystyle 1$) too.Answer: Point and line are the undefined terms; Axiom $\displaystyle 1$ forces the three sides equal, and Axiom $\displaystyle 1$ again makes the three \(\displaystyle 60°\) angles equal.

  2. Exercise 2

    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\displaystyle l and m\displaystyle m and the line n\displaystyle n in the above statement.]

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    Let \(\displaystyle n\) meet \(\displaystyle l\) at \(\displaystyle A\) and \(\displaystyle m\) at \(\displaystyle B\), with \(\displaystyle l\perp n\) and \(\displaystyle m\perp n\); suppose \(\displaystyle l\) and \(\displaystyle m\) meet at \(\displaystyle P\). NCERT_Solution_Class9_Maths_Exemplar_Ch5_Ex5-4_Q2 \[\angle PAB=90^\circ,\quad \angle PBA=90^\circ \] \[\angle PAB+\angle PBA+\angle APB=180^\circ \quad \text{(angle sum property of a triangle)} \] \[90^\circ+90^\circ+\angle APB=180^\circ \;\Rightarrow\; \angle APB=0^\circ \] which is impossible, so \(\displaystyle l\) and \(\displaystyle m\) cannot meet. This angle-sum property is itself equivalent to Euclid's fifth postulate -- it fails once the postulate is dropped -- so the given statement is equivalent to it too. Answer: Yes, it is equivalent to Euclid's fifth postulate.

  3. Exercise 3

    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 intersect two parallel lines, then alternate interior angles are equal. Is this system of axioms consistent? Justify your answer.

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    NCERT’s answer
    No
    NCERT_Solution_Class9_Maths_Exemplar_Ch5_Ex5-4_Q3 \[\angle 1 = \angle 2 \quad \text{(Axiom (ii): alternate interior angles, }l\parallel m\text{)} \] \[\angle 2 = \angle 3 \quad \text{(vertically opposite angles)} \] \[\Rightarrow\ \angle 1 = \angle 3 \] \(\displaystyle \angle 1\) and \(\displaystyle \angle 3\) are a corresponding pair, so Axiom (ii) forces them equal -- contradicting Axiom (i), which says corresponding angles need not be equal.Answer: Inconsistent -- Axiom (ii) with the vertical-angle fact forces corresponding angles equal, contradicting Axiom (i).

  4. Exercise 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\displaystyle 180°. Is this system of axioms consistent? Justify your answer.

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    NCERT’s answer
    No
    Let lines \(\displaystyle AB\) and \(\displaystyle CD\) meet at \(\displaystyle O\). Ray \(\displaystyle OA\) stands on line \(\displaystyle CD\), and ray \(\displaystyle OD\) stands on line \(\displaystyle AB\): \[\angle AOC + \angle AOD = 180° \quad \text{(Axiom (ii): linear pair)} \] \[\angle AOD + \angle BOD = 180° \quad \text{(Axiom (ii): linear pair)} \] \[\Rightarrow\ \angle AOC = \angle BOD \]NCERT_Solution_Class9_Maths_Exemplar_Ch5_Ex5-4_Q4\(\displaystyle \angle AOC\) and \(\displaystyle \angle BOD\) are vertically opposite, and Axiom (ii) alone has just forced them equal -- contradicting Axiom (i)'s claim that vertically opposite angles are not equal.Answer: Inconsistent -- Axiom (ii) proves vertically opposite angles equal, which Axiom (i) denies.

  5. Exercise 5

    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.

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    NCERT’s answer
    Consistent
    Let \(\displaystyle x\) and \(\displaystyle y\) each be double of the same thing \(\displaystyle z\): \[x = 2z, \quad y = 2z \] \[\Rightarrow\ x = y \quad \text{(Axiom (i): equal to the same thing)} \] That is exactly Axiom (iii)'s claim — it is not an independent statement but a case of Axiom (i), so it cannot conflict with it. Axiom (ii) governs addition, a separate operation neither of the other two touches.Answer: Consistent — Axiom (iii) is simply Axiom (i) applied to doubles of the same thing; none of the three contradicts another.