Question

the sides BA and CA have been produced such that BA is equal to AD and CA is equal to AE Prove that DE is parallel to BC

Answer 1

Given that, the sides BA and CA have been produced such that BA= AD and CA=AE and given to prove  DE|| BC

Consider triangle BAC and DAE,

We have

BA = ADand CA = AE  [∴ given in the data]

And also ∠BAC =∠DAE  [∴ vertically opposite angles]

So, by SAS congruence criterion, we have ΔBAC ≅ ΔDAE

⇒ BC = DE and ∠DEA=∠BCA,∠EDA=∠CBA

[Corresponding parts of congruent triangles are equal]

Now, DE and BC are two lines intersected by a transversal DB such that ∠DEA=∠BCA,

i.e., alternate angles are equal

Therefore, DE || BC


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