Lines and Angles - Solutions

 CBSE Class 9 Mathematics

NCERT Solutions
CHAPTER 6
Lines and Angles(Ex. 6.1)

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1. In Fig. 6.13, lines AB and CD intersect at O. If  and, find and reflex .

Ans. We are given that and .

We need to find .

From the given figure, we can conclude that ∠AOC,∠COE and ∠BOE$\mathrm{\angle }AOC,\mathrm{\angle }COEand\mathrm{\angle }BOE$ form a linear pair.

We know that sum of the angles of a linear pair is .

∠AOC+∠COE+∠BOE=180∘$\mathrm{\angle }AOC+\mathrm{\angle }COE+\mathrm{\angle }BOE={180}^{\circ }$

Reflex 


(Vertically opposite angles), or

But, we are given that

Therefore, we can conclude that and  .

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2. In Fig. 6.14, lines XY and MN intersect at O. If and a:b = 2 : 3, find c.

Ans. We are given that and .

We need find the value of c in the given figure.

Let a be equal to 2x and b be equal to 3x.

    

Therefore

Now [Linear pair]

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3. In the given figure,, then prove that .

Ans. We need to prove that .
We are given that .
From the given figure, we can conclude thatform a linear pair.
We know that sum of the angles of a linear pair is .
and (i)
 (ii)
From equations (i) and (ii), we can conclude that



Therefore, the desired result is proved.

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4. In Fig. 6.16, if x + y = w + z, then prove that AOB is a line.

Ans. We need to prove that AOB is a line.

We are given that .

We know that the sum of all the angles around a fixed point is .

Thus, we can conclude that

But, (Given).

From the given figure, we can conclude that y and x form a linear pair.

We know that if a ray stands on a straight line, then the sum of the angles of linear pair formed by the ray with respect to the line is.
.
Therefore, we can conclude that AOB is a line.

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5. In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that

Ans. We need to prove that.

We are given that OR is perpendicular to PQ, or

From the given figure, we can conclude thatform a linear pair.

We know that sum of the angles of a linear pair is.
 , or
.
From the figure, we can conclude that.

 , or

.(i)

From the given figure, we can conclude thatform a linear pair.

We know that sum of the angles of a linear pair is.
, or
.(ii)
Substitute (ii) in (i), to get


Therefore, the desired result is proved.

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6. It is given thatand XY is produced to point P. Draw a figure from the given information. If ray YQ bisects, find 

Ans. We are given that, XY is produced to P and YQ bisects.

We can conclude the given below figure for the given situation:

We need to find .

From the given figure, we can conclude thatform a linear pair.

We know that sum of the angles of a linear pair is .
.
But.

Ray YQ  bisects , or
.


Reflex 

Therefore, we can conclude that and Reflex