RD Sharma Solutions: Congruent Triangles- 3

 Page 1


 
      
Question:17
In two right triangles one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the
triangles are congruent.
Solution:
It is given that 
We are asked to show that
Let us assume
, and are right angled triangle.
Thus in  and , we have
And  given
Hence by AAs congruence criterion we have Proved.
Question:18
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
Solution:
We have to prove that  is isosceles.
Let ? be such that the bisector of  is parallel to
The base , we have 
Correspondingangles
Page 2


 
      
Question:17
In two right triangles one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the
triangles are congruent.
Solution:
It is given that 
We are asked to show that
Let us assume
, and are right angled triangle.
Thus in  and , we have
And  given
Hence by AAs congruence criterion we have Proved.
Question:18
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
Solution:
We have to prove that  is isosceles.
Let ? be such that the bisector of  is parallel to
The base , we have 
Correspondingangles
Alternateangle
(Since )
Hence is isosceles.
Question:19
In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.
Solution:
In the triangle ABC it is given that the vertex angle is twice of base angle.
We have to calculate the angles of triangle.
Now, let  be an isosceles triangle such that
Then 
Given
( ) 
Now
propertyoftriangle
Hence 
Question:20
Prove that each angle of an equilateral triangle is 60°
Solution:
We have to prove each angle of an equilateral triangle is .
Page 3


 
      
Question:17
In two right triangles one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the
triangles are congruent.
Solution:
It is given that 
We are asked to show that
Let us assume
, and are right angled triangle.
Thus in  and , we have
And  given
Hence by AAs congruence criterion we have Proved.
Question:18
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
Solution:
We have to prove that  is isosceles.
Let ? be such that the bisector of  is parallel to
The base , we have 
Correspondingangles
Alternateangle
(Since )
Hence is isosceles.
Question:19
In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.
Solution:
In the triangle ABC it is given that the vertex angle is twice of base angle.
We have to calculate the angles of triangle.
Now, let  be an isosceles triangle such that
Then 
Given
( ) 
Now
propertyoftriangle
Hence 
Question:20
Prove that each angle of an equilateral triangle is 60°
Solution:
We have to prove each angle of an equilateral triangle is .
Here 
Sideofequilateraltriangle
       ...........
1
And 
Sideofequilateraltriangle
          ..........2
From equation
1 and
2 we have 
Hence 
Now 
That is  (since )
Hence Proved.
Question:21
Angles A, B, C of a triangle ABC are equal to each other. Prove that ?ABC is equilateral.
Solution:
It is given that 
 
We have to prove that triangle ?ABC is equilateral.
Since 
Given
So,      ..........1
And 
given
So        ........2
From equation
1 and
2 we have 
Page 4


 
      
Question:17
In two right triangles one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the
triangles are congruent.
Solution:
It is given that 
We are asked to show that
Let us assume
, and are right angled triangle.
Thus in  and , we have
And  given
Hence by AAs congruence criterion we have Proved.
Question:18
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
Solution:
We have to prove that  is isosceles.
Let ? be such that the bisector of  is parallel to
The base , we have 
Correspondingangles
Alternateangle
(Since )
Hence is isosceles.
Question:19
In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.
Solution:
In the triangle ABC it is given that the vertex angle is twice of base angle.
We have to calculate the angles of triangle.
Now, let  be an isosceles triangle such that
Then 
Given
( ) 
Now
propertyoftriangle
Hence 
Question:20
Prove that each angle of an equilateral triangle is 60°
Solution:
We have to prove each angle of an equilateral triangle is .
Here 
Sideofequilateraltriangle
       ...........
1
And 
Sideofequilateraltriangle
          ..........2
From equation
1 and
2 we have 
Hence 
Now 
That is  (since )
Hence Proved.
Question:21
Angles A, B, C of a triangle ABC are equal to each other. Prove that ?ABC is equilateral.
Solution:
It is given that 
 
We have to prove that triangle ?ABC is equilateral.
Since 
Given
So,      ..........1
And 
given
So        ........2
From equation
1 and
2 we have 
Now from above equation if  we have
Given condition satisfy the criteria of equilateral triangle.
Hence the given triangle is equilateral.
Question:22
ABC is a right angled triangle in which ?A = 90° and AB = AC. Find ?B and ?C.
Solution:
It is given that 
We have to find  and .
Since  so, 
Now
propertyoftriangle
(Since  )
Here 
Then 
Hence 
Question:23
PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting
PR at T. Prove that PS = PT.
Solution:
It is given that
We have to prove 
In  we have 
Given
Page 5


 
      
Question:17
In two right triangles one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the
triangles are congruent.
Solution:
It is given that 
We are asked to show that
Let us assume
, and are right angled triangle.
Thus in  and , we have
And  given
Hence by AAs congruence criterion we have Proved.
Question:18
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
Solution:
We have to prove that  is isosceles.
Let ? be such that the bisector of  is parallel to
The base , we have 
Correspondingangles
Alternateangle
(Since )
Hence is isosceles.
Question:19
In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.
Solution:
In the triangle ABC it is given that the vertex angle is twice of base angle.
We have to calculate the angles of triangle.
Now, let  be an isosceles triangle such that
Then 
Given
( ) 
Now
propertyoftriangle
Hence 
Question:20
Prove that each angle of an equilateral triangle is 60°
Solution:
We have to prove each angle of an equilateral triangle is .
Here 
Sideofequilateraltriangle
       ...........
1
And 
Sideofequilateraltriangle
          ..........2
From equation
1 and
2 we have 
Hence 
Now 
That is  (since )
Hence Proved.
Question:21
Angles A, B, C of a triangle ABC are equal to each other. Prove that ?ABC is equilateral.
Solution:
It is given that 
 
We have to prove that triangle ?ABC is equilateral.
Since 
Given
So,      ..........1
And 
given
So        ........2
From equation
1 and
2 we have 
Now from above equation if  we have
Given condition satisfy the criteria of equilateral triangle.
Hence the given triangle is equilateral.
Question:22
ABC is a right angled triangle in which ?A = 90° and AB = AC. Find ?B and ?C.
Solution:
It is given that 
We have to find  and .
Since  so, 
Now
propertyoftriangle
(Since  )
Here 
Then 
Hence 
Question:23
PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting
PR at T. Prove that PS = PT.
Solution:
It is given that
We have to prove 
In  we have 
Given
So, 
Now 
Given
Since corresponding angle are equal, so
That is, 
Hence proved.
Question:24
In a ?ABC, it is given that AB = AC and the bisectors of ?B and ?C intersect at O. If M is a point on BO produced  prove that
?MOC = ?ABC.
Solution:
It is given that
In , 
We have to prove that
Now 
Given
Thus
    ........1
In , we have
So,  {from equation
1}
Hence Proved.
Question:25
P is a point on the bisector of an angle ?ABC. If the line through P parallel to AB meets BC at Q, prove that triangle BPQ is
isosceles.
Solution:
In the following figure it is given that sides AB and PQ are parallel and BP is bisector of