7. Triangles

NCERT

 

7. TRIANGLES

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Important Note :

1. If ∆ PQR is congruent to ∆ ABC, we write ∆ PQR ≅ ∆ ABC.

2. CPCT : corresponding parts of congruent triangles.

3. SAS Congruence Rule :  If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent).
4. ASA Congruence Rule : If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent
5. AAS Congruence Rule : If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent .
6. Angles opposite to equal sides of a triangle are equal.
7. Sides opposite to equal angles of a triangle are equal.
8. Each angle of an equilateral triangle is of 60°.
9. SSS Congruence Rule : If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent
10. RHS (Right angle-Hypotenuse-Side Congruence rule) : If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle, then the two triangles are congruent .
11. In a triangle, angle opposite to the longer side is larger (greater).
12. In a triangle, side opposite to the larger (greater) angle is longer.
13. Sum of any two sides of a triangle is greater than the third side.

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EXERCISE 7.1

1. In quadrilateral ABCD,  and AB bisects (see Fig. 7.16) .Show that  . What can you say about BC and BD ?

Solution : Given, ABCD is a quadrilateral,  and AB bisects  . Then show we that  

Proof: Since ,  AB bisects  .

So,

In  and  , we have 

    [Given ]

  [Given]

 [Common side]

  [SAS  rule]

  [CPCT]

2. ABCD is a quadrilateral in which and  (see Fig. 7.17) .

 Prove that (i)   (ii)     (iii) 

Solution : Given, ABCD is a quadrilateral , and  .

To prove  (i)     (ii)   (iii) 

Proof : (i) In  and  , we have

  [ Common side]

   [Given]

  [Given]

  [SAS rule]   Proved

(ii)  In  and  , we have

  [ Common side]

   [Given]

  [Given]

  [SAS rule]

    [CPCT]   Proved

(iii)  In  and  , we have

    [ Common side]

   [Given]

  [Given]

  [SAS rule]

   [CPCT] Proved

3. AD and BC are equal perpendicular to a line segment AB (see Fig. 7.18) . Show that CD bisects AB .

Proof : In  and  , we have

 [Given]

  [ given]

  [Vertically opposite angle]

 [ASA rule]

   [CPCT]

So,  CD bisects AB .    Proved

4.  and  are two parallel lines intersected by another pair of parallel lines  and  (see Fig. 7.18) . Show that  .

Solution : Given,  and are two parallel lines intersected by another pair of parallel lines and.Then we show that   .

Proof : In  and  , we have

  [Alternative interior angle]

   [Common side]

 [Alternative interior angle]

  [ASA rule]

5. Line is the bisector of an angle  and B is any point on  . BP and BQ are perpendiculars from B to the arms of  (see Fig. 7.20) . Show that :  (i)    (ii)  or B is equidistant from the arms of  .

Solution:  Given , the line is the bisector of an angle  and B is any point on  . BP and BQ are perpendiculars from B to the arms of  .

 Then we show that :  (i)    (ii)  or B is equidistant from the arms of  .

Proof : (i) Since,  is the bisector of an angle  . 

So,  

In  and  we have

  [Given]

   [Common side]

 [Given] 

   [ASA rule]

(ii) Since,  is the bisector of an angle . 

So,  

In  and  we have

  [Given]

  [Common side]

 [Given] 

   [ASA rule]

     [CPCT]

Or B is equidistant from the arms of  .  

                                                    Proved .

6. In Fig. 7.21, , and . Show that  .

Solution: Given, ,and .Then we show that  .

Proof : We have, 

  [Add both side  ]

 

In  and  , we have

 

 [SAS rule]

  [CPCT]   Proved .

7. AB is a line segment and P is its mid-point D and E are points on the same side of AB such that and  (see Fig. 7.22) . Show that (i)   (ii)  .

Solution : Given,  AB is a line segment and P is its mid-point D and E are points on the same side of AB such that and  .

Then we show that (i)   (ii)  .

Proof : Since, P is the mid-point of AB .

   

Again,  

  [ Add both side  ]

 

And

In  and  , we have

   [ Given]

  [Given]

  [Given]

  [ASA rule]

(ii)  Proof : Since, P is the mid-point of AB .

     

Again, 

  [ Add both side  ]

 

And 

In  and  , we have

   [ Given]

  [Given]

  [Given]

 [ASA rule]

  [CPCT]   Proved.

                OR

(ii) Since,   [ASA rule]

  [CPCT]   Proved .

8. In right triangle ABC, right-angled at C , M is the mid-point of hypotenuse AB . C is joined to M and produced to a point D such that  . Point D is joined to point B (see Fig. 7.23) . Show that :

  (i)     (ii)  is a right angle .   (iii)    (iv)

Solution :  Given , ABC is a right triangle angled at C , M is the mid-point of hypotenuse AB . C is joined to M and produced to a point D such that  . Point D is joined to point B .

Then we show that : (i)     (ii)  is a right angle    (iii)    (iv)

Proof :  (i) Since, M is the mid-point of hypotenuse AB .

      

In  and  , we have

   [Given]

   [Vertically opposite angle]

   [Given]

 [SAS rule]       

(ii) Proof :  Since ,  [SAS rule]       

    [CPCT]

 

So,   and BC is a transversal .

 

 [ ]

 

 is a right angle .

(iii)  Proof : Since, we have

           [CPCT]

In  and  , we have

      [Given ]

 [Given]

     [Common side]

  [SAS rule]

(iv)  Since,   [SAS rule]

        [ CPCT]  

Now,

  [ ]

  Proved .                                 

 EXERCISE 7.2

1.In an isosceles triangle ABC , with , the bisectors of  and  intersect each other at O . join A to O . Show that :

(i)        (ii) bisects  

2. In ABC , AD is the perpendicular bisectors of  BC (see Fig. 7.30) . Show that  ABC is an isosceles triangle in which .

3.  ABC is an isosceles triangle in which altitudes BE and CF are draw to equal sides AC and AB respectively (see Fig. 7.31) . Show that thee altitudes are equal .

4. ABC is a triangle in which altitudes  BE and  CF to sides AC and AB are equal (see Fig. 7.32) . Show that

(i) 

(ii)   i.e,  is an isosceles triangle .

5. ABC and BDC are two isosceles triangle on the same base BC  (see Fig. 7.33) .Show that  .

6. is an isosceles triangle in which  .(see Fig. 7.34 ).Show that  is a right triangle .

7. ABC is right angled triangle in which and Find  and .

8. Show that the  triangles of an equilateral triangles are  each .

EXERCISE 7.3

1. ABC and DBC  are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig. 7.39).If AD is extended to intersect BC at P, show that

(i)    

(ii)

(iii)   bisects  as well as .

(iv)  is the perpendicular bisector of .

2. AD is an alititudes of an isosceles triangle ABC in which . Show that    (i)  is bisects           (ii)  bisects .

3. Two sides AB and BC and median AM of the triangle ABC are respectively  equal to sides PQ and QR and median PN of  (see Fig. 7.40). Show that :

(i)

(ii)

4. BE and CF are two equal alititudes of a triangle ABC . Using RHS congruence rule , prove that the triangle ABC is isosceles .

5. ABC is an isosceles triangle with .Draw  to show that

EXERCISE 7.4

1. Show that in aright angled triangle, the hypotenuse is the longest side .

2. In Fig. 7.48, sides AB and AC of  ABC  are extended to points P and Q respectively. Also , .Show that .

3. In Fig .7.49 ,   and . Show that

4.  AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD (see Fig .7.50). Show that and .

5. In  Fig .7.51,  and PS bisects .Prove that

6.Show that of all line segments drawn from a given point not on it ,the perpendicular line segment is the shortest.

EXERCISE 7.5

1. ABC is a triangle . Locate a point in the interior of ABC which is quidistance from all the vertices of ABC.

2. In a triangle locate appoint in its interior which is quidistance from all the sides of the tringle .

       

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Posted 3 years ago