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NCERT Exemplar · Class 9 Mathematics Lines and Angles

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EXERCISE 6.2 1–10 (part 2 of 4)

  1. Exercise 1

    For what value of x+y\displaystyle x+y in Fig. 6.4\displaystyle 6.4 will ABC be a line? Justify your answer. NCERT_Question_Class9_Maths_Exemplar_Ch6_Ex6-2_Q1

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    NCERT’s answer
    $\displaystyle x+y$ must be equal to $\displaystyle 180$°. For ABC to be a line, the sum of the two adjacent angles must be $\displaystyle 180$°.
    NCERT_Solution_Class9_Maths_Exemplar_Ch6_Ex6-2_Q1 \[\angle ABC = \angle ABD + \angle DBC = y + x \quad \text{(ray } BD \text{ lies between rays } BA \text{ and } BC\text{)} \] \[ABC \text{ is a line} \iff \angle ABC = 180^\circ \] \[\Rightarrow\ x + y = 180^\circ \] Answer: \(\displaystyle x + y = 180^\circ\).

  2. Exercise 2

    Can a triangle have all angles less than 60\displaystyle 60°? Give reason for your answer.

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    NCERT’s answer
    No, angle sum will be less than $\displaystyle 180$°.
    \[\angle 1 + \angle 2 + \angle 3 = 180^\circ \quad \text{(angle sum property of a triangle)} \] \[\angle 1, \angle 2, \angle 3 < 60^\circ \implies \angle 1 + \angle 2 + \angle 3 < 180^\circ \] contradicts the angle sum property.Answer: No — a triangle cannot have all three angles less than \(\displaystyle 60^\circ\).

  3. Exercise 3

    Can a triangle have two obtuse angles? Give reason for your answer.

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    NCERT’s answer
    No, angle sum cannot be more than $\displaystyle 180$°.
    \[\angle 1 > 90^\circ, \ \angle 2 > 90^\circ \implies \angle 1 + \angle 2 > 180^\circ \] \[\angle 1 + \angle 2 + \angle 3 = 180^\circ \quad \text{(angle sum property)} \] forces \(\displaystyle \angle 3 < 0^\circ\), impossible.Answer: No — a triangle cannot have two obtuse angles.

  4. Exercise 4

    How many triangles can be drawn having its angles as 45\displaystyle 45°, 64\displaystyle 64° and 72\displaystyle 72°? Give reason for your answer.

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    NCERT’s answer
    None, angle sum cannot be $\displaystyle 181$°.
    \[45^\circ + 64^\circ + 72^\circ = 181^\circ \neq 180^\circ \] violates the angle sum property of a triangle.Answer: None — no such triangle exists, since the angles do not add to \(\displaystyle 180^\circ\).

  5. Exercise 5

    How many triangles can be drawn having its angles as 53\displaystyle 53°, 64\displaystyle 64° and 63\displaystyle 63°? Give reason for your answer.

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    NCERT’s answer
    Infinitely many triangles. sum of the angles of every triangle is $\displaystyle 180$°.
    \[53^\circ + 64^\circ + 63^\circ = 180^\circ \] satisfies the angle sum property, for a triangle of any size.Answer: Infinitely many — any size, all similar to one another.

  6. Exercise 6

    In Fig. 6.5\displaystyle 6.5, find the value of x\displaystyle x for which the lines l\displaystyle l and m\displaystyle m are parallel. NCERT_Question_Class9_Maths_Exemplar_Ch6_Ex6-2_Q6

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    $\displaystyle 136$°.
    NCERT_Solution_Class9_Maths_Exemplar_Ch6_Ex6-2_Q6 \[x + 44° = 180° \quad \text{(co-interior angles, }l\parallel m\text{)} \] \[x = 180° - 44° \] \[x = 136° \] Answer: \(\displaystyle x = 136°\).

  7. Exercise 7

    Two adjacent angles are equal. Is it necessary that each of these angles will be a right angle? Justify your answer.

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    NCERT’s answer
    No, each of these will be a right angle only when they form a linear pair.
    NCERT_Solution_Class9_Maths_Exemplar_Ch6_Ex6-2_Q7 adjacent angles need only share a vertex and a ray, not add to \(\displaystyle 180^\circ\). \[\angle AOB = \angle BOC = 40^\circ, \quad \angle AOC = 80^\circ \neq 180^\circ \] neither angle is \(\displaystyle 90^\circ\).Answer: No — equal adjacent angles are right angles only when they also form a linear pair.

  8. Exercise 8

    If one of the angles formed by two intersecting lines is a right angle, what can you say about the other three angles? Give reason for your answer.

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    NCERT’s answer
    Each will be a right angle. Linear pair axiom .
    NCERT_Solution_Class9_Maths_Exemplar_Ch6_Ex6-2_Q8 \[\angle AOD = 90^\circ \quad \text{(given)} \] \[\angle AOD + \angle DOB = 180^\circ \quad \text{(linear pair, } A,O,B \text{ collinear)} \implies \angle DOB = 90^\circ \] \[\angle BOC = \angle AOD = 90^\circ \quad \text{(vertically opposite angles)} \] \[\angle COA = \angle DOB = 90^\circ \quad \text{(vertically opposite angles)} \] Answer: the other three angles, \(\displaystyle \angle DOB\), \(\displaystyle \angle BOC\) and \(\displaystyle \angle COA\), are also \(\displaystyle 90^\circ\) each.

  9. Exercise 9

    In Fig.6.6, which of the two lines are parallel and why? NCERT_Question_Class9_Maths_Exemplar_Ch6_Ex6-2_Q9

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    NCERT’s answer
    $\displaystyle l \| m$ because $\displaystyle 132^{\circ}+48^{\circ}=180^{\circ}, p$ is not parallel to $\displaystyle q$, because $\displaystyle 73^{\circ}+106^{\circ} \neq 180^{\circ}$.
    NCERT_Solution_Class9_Maths_Exemplar_Ch6_Ex6-2_Q9 \[\angle P + \angle Q = 132^\circ + 48^\circ = 180^\circ \quad \text{(co-interior angles, transversal } n\text{)} \] \[\Rightarrow l \parallel m \quad \text{(converse of the co-interior angles axiom)} \] \[\angle S + \angle T = 73^\circ + 106^\circ = 179^\circ \neq 180^\circ \quad \text{(co-interior angles, transversal } r\text{)} \] \[\Rightarrow p \nparallel q \] Answer: \(\displaystyle l \parallel m\); \(\displaystyle p\) is not parallel to \(\displaystyle q\).

  10. Exercise 10

    Two lines l\displaystyle l and m\displaystyle m are perpendicular to the same line n\displaystyle n. Are l\displaystyle l and m\displaystyle m perpendicular to each other? Give reason for your answer.

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    NCERT’s answer
    No, they are parallel
    NCERT_Solution_Class9_Maths_Exemplar_Ch6_Ex6-2_Q10 \[\angle P = \angle Q = 90^\circ \quad \text{(} l \perp n, \ m \perp n \text{)} \] \[\angle P = \angle Q \quad \text{(corresponding angles, transversal } n\text{)} \] \[\Rightarrow l \parallel m \] Parallel lines never meet, so \(\displaystyle l\) is not perpendicular to \(\displaystyle m\). Answer: No; \(\displaystyle l \parallel m\), so they are not perpendicular to each other.