Chapter 6. Lines and Angles

Important Note :

1. If two lines intersect each other, then the vertically opposite angles are equal.
2. If a transversal intersects two parallel lines, then
(i) each pair of corresponding angles is equal,
(ii) each pair of alternate interior angles is equal,
(iii) each pair of interior angles on the same side of the transversal is supplementary.
3. If a transversal intersects two lines such that, either
(i) any one pair of corresponding angles is equal, or
(ii) any one pair of alternate interior angles is equal, or
(iii) any one pair of interior angles on the same side of the transversal is supplementary, then the lines are parallel.
4. Lines which are parallel to a given line are parallel to each other.
5. The sum of the three angles of a triangle is 180°.
6. If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

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EXERCISE 6.1

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

    

Solution:  Given,  and  .

       

Since ,  is a straight line .

 

 

 

and  is a straight line .

 

 

Reflex  

2. In Fig. 6.14 , lines XY and MN intersect at O . If  and  , find .

  

Solution: Given,  and  .

 Let  and  .

Since,   is a straight line .

  and

and  is a straight line .

 

Therefore, the value of  is 126° .

3. In Fig. 6.15, , then prove that  .

 

Solution:  Given ,  , then we prove that  .

Proof : Since, ray   stands on line  .

 ………. (i)

Ray   stands on line  .

 …………. (ii)  

From (i) and (ii) , we get  

But,

 

   Proved .

4. In Fig. 6.16 , If  , then prove that AOB is a line .

  

Solution : Given,  , then we that prove that is a line .

Ray   stands on line  .  

 …….(i)  [Linear pair of angles]

Ray  stands on line  .

 ……(ii) [Linear pair of angles]

Adding (i) and (ii) , we get 

 

 

   [Linear pair of angles]

 Therefore,  is a line .   Proved .

5. In Fig. 6.17 , POQ is a line . Ray OR is perpendicular to line PQ . OS is another ray lying between rays OP and OR . Prove that  .

 

Solution:  Given , POQ is a line . Ray OR is perpendicular to line PQ . OS is another ray lying between rays OP and OR .

To Prove :

Proof  : Since, ray OR is perpendicular to line PQ .

i.e.,  

 

 ……….. (i)

And 

  [Add both sides  ]

 

……….. (ii)

 From (i) and (ii) , we get

 

 

   Proved .

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

Solution : Given,  

   

Ray YQ bisects  , then  .

Let

Since,   is a straight line .

 

 

          

And

EXERCISE 6.2

1. In Fig. , find the values of  and  and then show that .

    

Solution:  Since,  [vertically opposite angle]

Again, [Linear pair of angles]

   

   

But    [Alternative interior angle]

   Proved.

2. In Fig. ,if  ,  and find  .

    

Solution:  Given,  

Let,   and

Since,   and  is transversal .

          

Again,   and  is transversal .

         

   

   

   

   

 

And    

Therefore, the value  is 126° .

3. In Fig .6.30, if   and  , find  and  .

   

Solution:  Given,  

Since,  and  is a transversal .

 [alternative interior angle]

Again,  

    

But 

  

  

  

And [Linear pair of angles]

 

 

 

4. In Fig. 6.31, if   and find

       

Solution: Given, and  

We draw a line parallel to ST through point R .

i.e.,   .

Since,   and  is a transversal

 

 

 

Again,  and also  , then

  [Alternative interior angle ]

5.  In Fig . 6.32,if and find  and.

    

Solution: Given, and  

Since,  and PQ is transversal .

 [Alternative interior angle]

And  and PR is transversal .

 [Alternative interior angle]

 

 

6.  In Fig .6.33,  and  are two mirrors placed parallel to each other .An incident ray strikes the mirror  at  , the reflected ray moves along the path  and strikes the mirror  at  and again reflects back along  . prove that  .

      

Solution: Given,  and  are two mirrors placed parallel to each other .An incident ray strikes the mirror  at  , the reflected ray moves along the path  and strikes the mirror  at  and again reflects back along  .

 To prove that :   .

Construction : We draw  and  .

Proof:  We know that , the incidence angle and the reflection angle are equal .

So,   and

     

Since,  and  and also given   .

So,  

Again,  and BC is a transversal .

So,   [Alternative interior angle]

 Again,

 [Alternative interior angle]

So,  and is a transversal .

 proved.

EXERCISE 6.3

1. In Fig. 6.39, sides  and  of  are produced to points  and  respectively . If  and find

   

Solution: Given,  and

Ray PQ stand on the line TR .

 

 

We know that, an exterior angle of a triangle is equal to the sum of the two interior opposite angles.

   

2.In Fig .6.40 , If  and  are the bisectors of  and   respectively of  find  and  .

  

Solution:  Given,  and .

In  , we have

 

Since,  and  are the bisectors of  and   respectively.

In  , we have

 

Therefore,  and .

3. In Fig .6.41 ,if  and , find  .

        

Solution:  Given,  and  

     and  is a transversal .

So,

And  

In  , we have

4. In Fig .6.42 , if lines  and  intersect at point ,such that  and find   

      

Solution:  Given,  ,  and

In  , we have

 

Again,  [Vertically opposite angle]

In , we have

 

5. In Fig. 6.43 , if  ,and then find the values of  and .

    

Solution:  Given,  and  .

Since,    

So,

We know that, an exterior angle of a triangle is equal to the sum of the two interior opposite angles.

  

Again,  and  is a transversal .

[alternative interior angle]

In  , we have

   

Therefore,  and  

6. In Fig . 6.44, the sides of  is produced to a point  .If the bisectors of  and meet at point , then prove that

   

Solution:  Given, the sides   of  is produced to a point  .If the bisectors of  and meet at point .

 To prove :   

Proof: Since,  and  are the bisectors of  and respectively.

So,

and

We know that, an exterior angle of a triangle is equal to the sum of the two interior opposite angles.

In  , we have

In  , we have

From (i) and (ii) , we get

   

    Proved.