Question

In the given figure AC = AE, AB = AD and angle BAD = angle EAC. Prove that BC = DE
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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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