Class 9 Exam  >  Class 9 Notes  >  RD Sharma Solutions for Class 9 Mathematics  >  RD Sharma Solutions -Ex-10.2, Congruent Triangles, Class 9, Maths

Ex-10.2, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics PDF Download

Q. (1) In fig. (10).40, it is given that RT = TS, ∠ 1 = 2 ∠ 2 and 4 = 2 ∠ (3) Prove that ΔRBT ≅ ΔSAT.

Solution:

In the figure, given that

RT = TS                                                                          ……(i)

∠ 1 = 2 ∠ 2            ……(ii)

And ∠ 4 = 2 ∠ 3    ……(iii)

To prove that ΔRBT ≅ ΔSAT.

Let the point of intersection RB and SA be denoted by O

Since RB and SA intersect at O

∠ AOR = ∠ BOS          [Vertically opposite angles]

  • ∠ 1 = ∠ 4
  • 2 ∠ 2 = 2 ∠ 3 [From (ii) and (iii)]
  • ∠ 2 = ∠ 3 ……(iv)

Now we have RT =TS in Δ TRS

Δ TRS is an isosceles triangle

∠ TRS = ∠ TSR              ……(v)

But we have

∠ TRS = ∠ TRB + ∠ 2                                                              ……(vi)

∠ TSR = ∠ TSA + ∠ 3                                                              ……(vii)

Putting (vi) and (vii) in (v) we get

∠ TRB + ∠ 2 =  ∠ TSA + ∠ B

=> ∠ TRB = ∠ TSA        [From (iv)]

Now consider Δ RBT and Δ SAT

RT = ST [From (i)]

∠ TRB = ∠ TSA            [From (iv)]  ∠ RTB = ∠ STA    [Common angle]

From ASA criterion of congruence, we have

Δ RBT = Δ SAT

 

Q. (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 that lines AB and CD Intersect at O

  Ex-10.2, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Such that BC ∥ AD and  BC = AD  …….(i)

We have to prove that AB and CD bisect at O.

To prove this first we have to prove that Δ AOD ≅ Δ BOC

 

Q. (3) BD and CE are bisectors of ∠ B and ∠ C of an isosceles Δ ABC with AB = AC. Prove that BD = CE

Solution:

Given that Δ ABC is isosceles with AB = AC and BD and CE are bisectors of ∠ B and ∠ C We have to prove BD = CE

Since AB = AC

=> Δ ABC = Δ ACB ……(i)

[Angles opposite to equal sides are equal]

Since BD and CE are bisectors of ∠ B and ∠ C

  • ∠ ABD = ∠ DBC = ∠ BCE = ECA = Ex-10.2, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Now,

Consider Δ EBC = Δ DCB

Ex-10.2, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

∠ EBC = ∠ DCB [∠ B = ∠ C] [From (i)]

BC = BC [Common side]

∠ BCE = ∠ CBD [From (ii)]

So, by ASA congruence criterion, we have Δ EBC ≅ Δ DCB

Now,

CE = BD     [Corresponding parts of congruent triangles we equal]

or, BD = CE

Hence proved

Since AD ∥ BC and transversal AB cuts at A and B respectively

∠ DAO = ∠ OBC  …….(ii) [alternate angle]

And similarly AD ∥ BC and transversal DC cuts at D and C respectively

∠ ADO = ∠ OBC                 ..…(iii) [alternate angle]

Since AB end CD intersect at O.

∠ AOD = ∠ BOC               [Vertically opposite angles]

Now consider Δ AOD and Δ BOD

∠ DAO = ∠ OBC [From (ii)]

AD = BC                                                                        [From (i)]

And ∠ ADO = ∠ OCB  [From (iii)]

So, by ASA congruence criterion, we have

ΔAOD≅ΔBOC

Now,

AO= OB and DO = OC [Corresponding parts of congruent triangles are equal)

  • Lines AB and CD bisect at O.

Hence proved

The document Ex-10.2, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics is a part of the Class 9 Course RD Sharma Solutions for Class 9 Mathematics.
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FAQs on Ex-10.2, Congruent Triangles, Class 9, Maths RD Sharma Solutions - RD Sharma Solutions for Class 9 Mathematics

1. What are congruent triangles?
Ans. Congruent triangles are triangles that have the same shape and size. This means that their corresponding sides and angles are equal.
2. How can we prove two triangles are congruent?
Ans. There are several methods to prove that two triangles are congruent. Some common methods include the Side-Side-Side (SSS) congruence criterion, Side-Angle-Side (SAS) congruence criterion, Angle-Side-Angle (ASA) congruence criterion, and the Hypotenuse-Leg (HL) congruence criterion.
3. What is the importance of congruent triangles?
Ans. Congruent triangles are important in geometry as they allow us to establish relationships between different parts of triangles. By proving that two triangles are congruent, we can conclude that their corresponding angles and sides are equal, which helps in solving various geometrical problems and proofs.
4. How do congruent triangles help in solving real-life problems?
Ans. Congruent triangles have practical applications in real-life problems. For example, in construction, congruent triangles are used to ensure that different parts of a structure are equal in size and shape. They are also used in navigation and surveying to calculate distances and angles accurately.
5. Can two triangles be congruent if they have different angles?
Ans. No, two triangles cannot be congruent if they have different angles. Congruent triangles have all their corresponding angles equal. If the angles of two triangles are different, they cannot be congruent. However, if the corresponding sides and angles of two triangles are equal, they can be proven to be congruent using appropriate congruence criteria.
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