Congruent Triangles- 1 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

 Page 1


Question:1
In the given figure, the sides BA  and CA have been produced such that BA = AD and CA = AE. Prove that segment DE || BC.
Solution:
It is given that
We have to prove that 
Now considering the two triangles we have
In 
Given
Given
We need to show to prove .
Now
Verticallyoppositeangle
So by  congruence criterion we have
So  and 
Then
, and
Hence from above conditions .
Question:2
In a ?PQR, if PQ =  QR and L, M and N are the mid-points of the sides PQ, OR, and RP respectively. Prove that LN = MN.
 
Solution:
It is given that
Page 2


Question:1
In the given figure, the sides BA  and CA have been produced such that BA = AD and CA = AE. Prove that segment DE || BC.
Solution:
It is given that
We have to prove that 
Now considering the two triangles we have
In 
Given
Given
We need to show to prove .
Now
Verticallyoppositeangle
So by  congruence criterion we have
So  and 
Then
, and
Hence from above conditions .
Question:2
In a ?PQR, if PQ =  QR and L, M and N are the mid-points of the sides PQ, OR, and RP respectively. Prove that LN = MN.
 
Solution:
It is given that
And is the mid point of 
So 
And is the mid point of 
So 
And is the mid point of 
So 
We have to prove that 
In we have 
Equilateraltriangle
Then 
, and 
, and 
Similarly comparing and we have
, and 
And  (Since N is the mid point of )
So by  congruence criterion, we have 
Hence .
Question:3
Prove that the medians of an equilateral triangle are equal.
Solution:
We have to prove that the median of an equilateral triangle are equal.
Let  be an equilateral triangle with as its medians.
Let 
In  we have
Page 3


Question:1
In the given figure, the sides BA  and CA have been produced such that BA = AD and CA = AE. Prove that segment DE || BC.
Solution:
It is given that
We have to prove that 
Now considering the two triangles we have
In 
Given
Given
We need to show to prove .
Now
Verticallyoppositeangle
So by  congruence criterion we have
So  and 
Then
, and
Hence from above conditions .
Question:2
In a ?PQR, if PQ =  QR and L, M and N are the mid-points of the sides PQ, OR, and RP respectively. Prove that LN = MN.
 
Solution:
It is given that
And is the mid point of 
So 
And is the mid point of 
So 
And is the mid point of 
So 
We have to prove that 
In we have 
Equilateraltriangle
Then 
, and 
, and 
Similarly comparing and we have
, and 
And  (Since N is the mid point of )
So by  congruence criterion, we have 
Hence .
Question:3
Prove that the medians of an equilateral triangle are equal.
Solution:
We have to prove that the median of an equilateral triangle are equal.
Let  be an equilateral triangle with as its medians.
Let 
In  we have
(Since  similarly )
(In equilateral triangle, each angle )
And 
commonside
So by  congruence criterion we have
This implies that, 
Similarly we have
Hence .
Question:4
In a ? ABC, if ?A = 120° and AB = AC. Find ?B and ?C.
Solution:
In , it is given that
, and 
We have to find , and
Since  and 
Then 
asAB = AC
Now 
Bypropertyoftriangle
Thus, 
, as 
given
So, 
Since, , so
Hence .
Question:5
In a ?ABC, if AB = AC and ?B = 70°, find ?A.
Solution:
In  it is given that
, and 
We have to find .
Page 4


Question:1
In the given figure, the sides BA  and CA have been produced such that BA = AD and CA = AE. Prove that segment DE || BC.
Solution:
It is given that
We have to prove that 
Now considering the two triangles we have
In 
Given
Given
We need to show to prove .
Now
Verticallyoppositeangle
So by  congruence criterion we have
So  and 
Then
, and
Hence from above conditions .
Question:2
In a ?PQR, if PQ =  QR and L, M and N are the mid-points of the sides PQ, OR, and RP respectively. Prove that LN = MN.
 
Solution:
It is given that
And is the mid point of 
So 
And is the mid point of 
So 
And is the mid point of 
So 
We have to prove that 
In we have 
Equilateraltriangle
Then 
, and 
, and 
Similarly comparing and we have
, and 
And  (Since N is the mid point of )
So by  congruence criterion, we have 
Hence .
Question:3
Prove that the medians of an equilateral triangle are equal.
Solution:
We have to prove that the median of an equilateral triangle are equal.
Let  be an equilateral triangle with as its medians.
Let 
In  we have
(Since  similarly )
(In equilateral triangle, each angle )
And 
commonside
So by  congruence criterion we have
This implies that, 
Similarly we have
Hence .
Question:4
In a ? ABC, if ?A = 120° and AB = AC. Find ?B and ?C.
Solution:
In , it is given that
, and 
We have to find , and
Since  and 
Then 
asAB = AC
Now 
Bypropertyoftriangle
Thus, 
, as 
given
So, 
Since, , so
Hence .
Question:5
In a ?ABC, if AB = AC and ?B = 70°, find ?A.
Solution:
In  it is given that
, and 
We have to find .
Since
Then 
isoscelestriangles
Now 
(As  given)
Thus 
Propertyoftriangle
Hence .
Question:6
The vertical angle of an isosceles triangle is 100°. Find its base angles.
Solution:
Suppose in the isosceles triangle ?ABC it is given that
We have to find the base angle.
Now vertical angle 
given
And 
Since  then 
Now
Bypropertyoftriangle
So 
Hence the base angle is .
Question:7
In the given figure, AB = AC and ?ACD = 105°, find ?BAC.
 
Page 5


Question:1
In the given figure, the sides BA  and CA have been produced such that BA = AD and CA = AE. Prove that segment DE || BC.
Solution:
It is given that
We have to prove that 
Now considering the two triangles we have
In 
Given
Given
We need to show to prove .
Now
Verticallyoppositeangle
So by  congruence criterion we have
So  and 
Then
, and
Hence from above conditions .
Question:2
In a ?PQR, if PQ =  QR and L, M and N are the mid-points of the sides PQ, OR, and RP respectively. Prove that LN = MN.
 
Solution:
It is given that
And is the mid point of 
So 
And is the mid point of 
So 
And is the mid point of 
So 
We have to prove that 
In we have 
Equilateraltriangle
Then 
, and 
, and 
Similarly comparing and we have
, and 
And  (Since N is the mid point of )
So by  congruence criterion, we have 
Hence .
Question:3
Prove that the medians of an equilateral triangle are equal.
Solution:
We have to prove that the median of an equilateral triangle are equal.
Let  be an equilateral triangle with as its medians.
Let 
In  we have
(Since  similarly )
(In equilateral triangle, each angle )
And 
commonside
So by  congruence criterion we have
This implies that, 
Similarly we have
Hence .
Question:4
In a ? ABC, if ?A = 120° and AB = AC. Find ?B and ?C.
Solution:
In , it is given that
, and 
We have to find , and
Since  and 
Then 
asAB = AC
Now 
Bypropertyoftriangle
Thus, 
, as 
given
So, 
Since, , so
Hence .
Question:5
In a ?ABC, if AB = AC and ?B = 70°, find ?A.
Solution:
In  it is given that
, and 
We have to find .
Since
Then 
isoscelestriangles
Now 
(As  given)
Thus 
Propertyoftriangle
Hence .
Question:6
The vertical angle of an isosceles triangle is 100°. Find its base angles.
Solution:
Suppose in the isosceles triangle ?ABC it is given that
We have to find the base angle.
Now vertical angle 
given
And 
Since  then 
Now
Bypropertyoftriangle
So 
Hence the base angle is .
Question:7
In the given figure, AB = AC and ?ACD = 105°, find ?BAC.
 
Solution:
It is given that
We have to find .
Isoscelestriangle
Now 
Since exterior angle of isosceles triangle is the sum of two internal base angles
Now 
So, 
Bypropertyoftriangle
Hence .
Question:8
Find the measure of each exterior angle of an equilateral triangle.
Solution:
We have to find the measure of each exterior angle of an equilateral triangle.
It is given that the triangle is equilateral