Congruent Triangles- 2 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

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Question:14
BD and CE are bisectors of ?B and ?C of an isosceles ?ABC with AB = AC. Prove that BD = CE.
Solution:
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Question:14
BD and CE are bisectors of ?B and ?C of an isosceles ?ABC with AB = AC. Prove that BD = CE.
Solution:
It is given that 
Is bisector of  and  is bisector of .
And  is isosceles with 
We have to prove that
If will be sufficient to prove to show that 
Now in these two triangles
Since , so
Now as BD and CE are bisector of the  respectively, so
, and
BC=BC
So by  congruence criterion we have 
Hence Proved.
Question:15
In the given figure, it is given that RT = TS, ?1 = 2 ?2 and ?4 = 2 ?3. Prove that ?RBT ??SAT.
 
Solution:
It is given that 
We have to prove that 
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Question:14
BD and CE are bisectors of ?B and ?C of an isosceles ?ABC with AB = AC. Prove that BD = CE.
Solution:
It is given that 
Is bisector of  and  is bisector of .
And  is isosceles with 
We have to prove that
If will be sufficient to prove to show that 
Now in these two triangles
Since , so
Now as BD and CE are bisector of the  respectively, so
, and
BC=BC
So by  congruence criterion we have 
Hence Proved.
Question:15
In the given figure, it is given that RT = TS, ?1 = 2 ?2 and ?4 = 2 ?3. Prove that ?RBT ??SAT.
 
Solution:
It is given that 
We have to prove that 
Now
In  we have
Isoscelestriangle     .......
1
Now we have
Verticallyoppositeangles
(Since , given)
     .......2
Subtracting equation
2 from equation
1 we have 
Now in  and  we have
Given
So all the criterion for the two triangles  and  are satisfied to be congruent
Hence by  congruence criterion we have proved.
Question:16
Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.
Solution:
It is given that
We have to prove that the lines and  bisect at .
If we prove that , then
We can prove  and  bisects at .
Now in  and 
Given
(Since  and  is transversal)
And  (since  and  is transversal)
So by congruence criterion we have,
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Question:14
BD and CE are bisectors of ?B and ?C of an isosceles ?ABC with AB = AC. Prove that BD = CE.
Solution:
It is given that 
Is bisector of  and  is bisector of .
And  is isosceles with 
We have to prove that
If will be sufficient to prove to show that 
Now in these two triangles
Since , so
Now as BD and CE are bisector of the  respectively, so
, and
BC=BC
So by  congruence criterion we have 
Hence Proved.
Question:15
In the given figure, it is given that RT = TS, ?1 = 2 ?2 and ?4 = 2 ?3. Prove that ?RBT ??SAT.
 
Solution:
It is given that 
We have to prove that 
Now
In  we have
Isoscelestriangle     .......
1
Now we have
Verticallyoppositeangles
(Since , given)
     .......2
Subtracting equation
2 from equation
1 we have 
Now in  and  we have
Given
So all the criterion for the two triangles  and  are satisfied to be congruent
Hence by  congruence criterion we have proved.
Question:16
Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.
Solution:
It is given that
We have to prove that the lines and  bisect at .
If we prove that , then
We can prove  and  bisects at .
Now in  and 
Given
(Since  and  is transversal)
And  (since  and  is transversal)
So by congruence criterion we have,
, so
Hence and  bisect each other at .