RD Sharma 2020 solution class 9 chapter 12 Congruent Triangles Exercise 12.2

Exercise 12.2

Page-12.25

Question 1:

BD and CE are bisectors of ∠B and ∠C of an isosceles ΔABC with AB = AC. Prove that BD = CE.

Answer 1:

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 2:

In the given figure, it is given that RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ΔSAT.
 

Answer 2:

It is given that 

We have to prove that

Now

In we have

(Isosceles triangle)     .......(1)

Now we have

(Vertically opposite angles)

(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 3:

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.

Answer 3:

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.