Chapter 6. Triangles 

 Class 10 Maths Chapter 6. Triangles Multiple Choice Questions and Solutions :

Question :  In figure , ,then EC is equal to :

           

(a)  2 cm          (b) 3 cm       (c) 5 cm      (d) 6 cm

Solution: (a)  2 cm     

 [  In  and  we have ,

  

   ]

Question:  D and E are respectively the points on the sides AB and AC of a triangle ABC such that AD = 2 cm , BD = 3 cm , BC = 7.5 cm and  . Then, length of DE (in cm) is :

(a) 2.5          (b) 3            (c) 5             (d) 6

Solution :  (b) 3

[ Here, AD = 2 cm , BD = 3 cm , AB = 2 + 3 = 5 cm , BC = 7.5 cm

In  and  , we have

  [Common angles]

  [corresponding angles]

  [A.A.]

  ]

Question : In given figure and , then the value of is :

      

(a) 2.3 cm      (b) 2.5 cm     (c) 2.4 cm    (d) 2.8 cm

Solution :   (c) 2.4 cm

[  Here,

 In  and  , we have

  ]

Question :  If , and , then RQ is :   

 (a) 6 cm       (b)  12 cm      (c) 10 cm     (d) 3 cm

Solution:   (b)  12 cm 

[  In figure ,

  Since  , we have

    So, 

 

 

   ]

Question :  In given figure , and , then the length of BN is :

  

 (a) 5 cm         (b)  4 cm          (c)  2 cm       (d) 8 cm

Solution:   (a) 5 cm

[ In and , we have 

  

 

   ]                                                                  

Question:  In . If and  ,then the value of DB is : 

 

    (a) 12 cm         (b) 24 cm       (c) 8 cm        (d) 4 cm

Solution:  (c) 8 cm

 [      In , we have

 

   ]

Question: DE is drawn parallel to the base BC of a   , meeting AB at D and AC at E . If  and CE = 2 cm , then  AE is :

 (a)  5 cm        (b)  4 cm      (c) 6 cm        (d) 7 cm 

Solution:   (c) 6 cm 

 [   In and , We have                           

 

We have ,  

Again, 

  ] 

  Class 10 Triangles  Fill in the blank :

Question :  All circles are .  [ congruent / similar]

Answer :  Similar .

Question :  All squares are   .  [ similar / congruent ]

Answer :  Similar .

Question :  All    triangles are similar .   [ isosceles / equilateral / acute triangle  ]

Answer :  Equilateral .

Question: Two polygons of the same number of sides are similar , if (a) their corresponding angles are  and (b) their corresponding sides are  . [congruent / equal / proportional /Similar ]

Answer :  Equal  , Proportional .

 Class 10 Maths Chapter 6 .Triangles 2 Marks Questions and Answers   

Question :  In given figure ,  and  , prove that 

 

Solution:  Given,  and .

To prove that  

Proof : In  and  , we have

In  and, we have

From  and  , we get

    Proved .  

Question :  Prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side .

Solution: Given, PQR is a triangle whose   and S is a mid-point of the side PQ .

To prove  : T is a mid-point of PR .

Proof : In figure, 

 Since S is a mid-point of PQ , then

      

In  and  , we have

   

 and  , we get

     

Thus, T is a mid-point of PR . Proved .

Question :  In Fig. 6.18, if   and, prove that

Solution:  In given figure ,

  In  and   we have ,

Again,  and  we have ,

 

 and we have ,  

  Proved .

Question : In Fig. 6.36,    and . Show that  .

Solution: In given figure,

  Since,

So, 

PQR is an isosceles triangle .

Again,   

    [from (i) ]

In  and  , we have

     [Common angle]

   [ given]

 [SAS]          

Question :  In Fig. 6.37 , if  , show that .

Solution: Given,  . Then we show that  .

Proof :  Since, , We have

   

  

     

and 

[SAS]      Proved                            

 Class 10 Maths Chapter 6.Triangles 3 Marks Questions and Solutions :      

Question : E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F . Show that   .

Solution: Given, E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F .

To prove :   .

Proof: In given figure,

In   and  , we have

  [ Opposite angle of the parallelogram ]

  [ Alternative interior angle]

        [A.A rule]        proved.

Question :  Diagonals AC and BD of a trapezium ABCD with   intersect each other at the point O . Using a similarity criterion for two triangles , show that   .

Solution:  Given, Diagonals AC and BD of a trapezium ABCD with   intersect each other at the point O . Then we show that  .

Proof : Given figure ,

In and  , we have

  [ Vertically opposite angle]

  [ Alternative interior angle]

  [ Alternative interior angle]

  [ AAA  similarity criterion]

 

      Proved .                                             

Question : The diagonals of a quadrilateral ABCD intersect each other at the point O such that  . Show that ABCD is a trapezium .

Solution: Given, the diagonals of a quadrilateral ABCD intersect each other at the point O such that .Then we show that ABCD is a trapezium .

Construction : We join OP such that  .

Proof: In given figure,

 

In  and  .

 

But  

 and  we get, 

So,  then

  ABCD is a trapezium .   Proved

Question : In Fig. 6.40 , E is a point on side CB produced of an isosceles triangle ABC with  . If  and , prove that  .

Solution:  Given, E is a point on side CB produced of an isosceles triangle ABC with ,  and .

To prove that  .

Proof : In given figure,

In , we have

         AB = AC

  

i. e.  

 In  and  , we have   

  

     [ Given]

  [Third angle]

   [ AAA rule ]

 Class 10 Maths Chapter 6 . Triangles 4 Marks Questions and Solutions :                                             

Question :  ABCD is a trapezium with . E and F are points on non-parallel sides AD and BC respectively such that EF is parallel to AB . Show that .

                             

Solution:  Given, ABCD is a trapezium with . E and F are points on non-parallel sides AD  and BC respectively such that .

To Prove :     

Construction :  Join AC to intersect EF at G .

Proof : In given figure,

and and also

 In and         (   )

So, 

 In and   (   )

So, 

From (i) and (ii) , we get

      Proved. 

Question :  If AD and PM are medians of triangle ABC and PQR , respectively where  , prove that 

Solution: Given,  AD and PM are medians of triangle ABC and PQR respectively and  .

To prove :     

Proof :  In figure,

Since , D and M are mid-point of the sides BC and QR .

So,   and

Given,

Then,  

     

   

In  and , we have

  [  ]

         

    [S.A.S.]

       

    Proved.

 Class 10 Triangles 5 Marks Questions and Answers                                                 

SECTION = E

Question :  Prove that  a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio .

Solution:  Given, a triangle ABC in which a line parallel to side BC intersects other two sides AB and AC at D and E respectively .

To prove :     .

Construction : Join BE and CD  and also , draw and .

Proof : In figure,

 

We know that , Area of triangle

    

 

 

and    

 

   and  

   Since,  and  are on the same base DE and between the same parallels BC and DE.

So, 

From  ,   and  , we  have          

 

       Proved.