# Class 9 RD Sharma Solutions – Chapter 10 Congruent Triangles- Exercise 10.2

### Question 1: In figure, it is given that RT = TS, âˆ 1 = 2âˆ 2 and âˆ 4 = 2(âˆ 3). Prove that Î”RBT â‰… Î”SAT.

Solution:

In the figure,
Given:
RT = TS —-> (equation 1)
âˆ 1 = 2 âˆ 2 —-> (equation 2)
âˆ 4 = 2 âˆ 3 —->(equation 3)

To prove:
Î”RBT â‰… Î”SAT

Let the point of intersection RB and SA be denoted by O
âˆ AOR = âˆ BOS [Vertically opposite angles]
So, âˆ 1 = âˆ 4
2 âˆ 2 = 2 âˆ 3 [From (equation 2) and (equation 3)]
or
âˆ 2 = âˆ 3 —->(equation 4)

Now in Î” TRS,
We have RT = TS
Î” TRS is an isosceles triangle
âˆ TRS = âˆ TSR —->(equation 5)
But, âˆ TRS = âˆ TRB + âˆ 2 —->(equation 6)
âˆ TSR = âˆ TSA + âˆ 3 —->(equation 7)
Putting (equation 6) and (equation 7) in (equation 5) we get
âˆ TRB + âˆ 2 = âˆ TSA + âˆ 3
âˆ TRB = âˆ TSA [From (equation 6)]
Consider Î”RBT and Î”SAT
RT = ST [From (equation 1)]
âˆ TRB = âˆ TSA [From (equation 6)]
By ASA criterion of congruence, we have
Î”RBT â‰… Î”SAT

### Question 2: 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:

Given:
Lines AB and CD Intersect at O
Such that BC âˆ¥ AD and

To prove:
AB and CD bisect at O.

First we have to prove that Î”AOD â‰… Î”BOC
âˆ OCB =âˆ ODA {AD||BC and CD is transversal}
AD = BC {from (equation 1)}
By ASA Criterion:
Î”AOD â‰… Î”BOC
OA = OB and OD = OC (By Corresponding parts of congruent triangle )
Therefore,
AB and CD bisect each other at O.
Hence, Proved.

### Question 3: BD and CE are bisectors of âˆ B and âˆ C of an isosceles Î” ABC with AB = AC. Prove that BD = CE.

Solution:

Given:
Î”ABC is isosceles
Where,
AB = AC
BD and CE are bisectors of âˆ  B and âˆ  C

To prove:
BD = CE

Since AB = AC (given)
âˆ ABC = âˆ ACB —->(equation 1) {Angles opposite to equal sides are equal}
Since BD and CE are bisectors of âˆ  B and âˆ  C
âˆ ABD = âˆ DBC =âˆ  BCE =ECA =  âˆ B  = âˆ C   —-> (equation 2)
2        2
Now, Consider Î”EBC = Î”DCB
âˆ EBC = âˆ DCB {From (equation 1)}
BC = BC {Common side}
âˆ BCE = âˆ CBD {From (equation 2)}
By ASA congruence criterion,
Î” EBC â‰… Î” DCB
Since corresponding parts of congruent triangles are equal.(c.p.c.t.)
CE = BD
or,
BD = CE
Hence, proved.

Whether you're preparing for your first job interview or aiming to upskill in this ever-evolving tech landscape, GeeksforGeeks Courses are your key to success. We provide top-quality content at affordable prices, all geared towards accelerating your growth in a time-bound manner. Join the millions we've already empowered, and we're here to do the same for you. Don't miss out - check it out now!

Previous
Next