Question

"Question 6 In the given figure, AC = AE, AB = AD and ∠BAD = ∠EAC. Show that BC = DE.

Class 9 - Math - Triangles Page 120"

Answer 1

Congruence of triangles:

Two ∆’s are congruent if sides and angles of a triangle are equal to the corresponding sides and angles of the other ∆.

 

In Congruent Triangles corresponding parts are always equal and we write it in short CPCT i e, corresponding parts of Congruent Triangles.

 

It is necessary to write a correspondence of vertices correctly for writing the congruence of triangles in symbolic form.

 

Criteria for congruence of triangles:

There are 4 criteria for congruence of triangles.

SAS( side angle side):

Two Triangles are congruent if two sides and the included angle of a triangle are equal to the two sides and included angle of the the other triangle.

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Solution:

First show that ΔABC ≅ ΔADE by using SAS rule and then use  CPCT  to show given result.

 

Given,
AC = AE, AB = AD and ∠BAD = ∠EAC

 

To prove:
BC = DE

Proof: We have
∠BAD = ∠EAC

(Adding ∠DAC to both sides)
∠BAD + ∠DAC = ∠EAC + ∠DAC

⇒ ∠BAC = ∠EAD

In ΔABC and ΔADE,
AC = AE (Given)
∠BAC = ∠EAD                    (proved above)
AB = AD                                   (Given)

Hence, ΔABC ≅ ΔADE             (by SAS congruence rule)

Then,
BC = DE                                      ( by CPCT.)

 

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Answer 2

Proof:

triangle BAC, triangle ADE
AB=AD - S
angle A=angle A
AC=AE -S

therefore triangle BAC is congruent to triangle ADE (SAS congruent)
BC=DE (cpct)

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