In the given figure, if x = y and AB = CB then prove that AE = CD.

1.	In the given figure, if x = y and AB = CB then prove that AE = CD.
Answer» It is given that x = y and AB = CB By considering the △ ABE We know that Exterior ∠ AEB = ∠ EBA + ∠ BAE By substituting ∠ AEB as y we get y = ∠ EBA + ∠ BAE By considering the △ BCD We know that x = ∠ CBA + ∠ BCD It is given that x = y So we can write it as ∠ CBA + ∠ BCD = ∠ EBA + ∠ BAE On further calculation we can write it as ∠ BCD = ∠ BAE Based on both △ BCD and △ BAE We know that B is the common point It is given that AB = BC It is proved that ∠ BCD = ∠ BAE Therefore, by ASA congruence criterion we get △ BCD ≅ △ BAE We know that the corresponding sides of congruent triangles are equal Therefore, it is proved that AE = CD.

Answer»

It is given that x = y and AB = CB

By considering the △ ABE

We know that

Exterior ∠ AEB = ∠ EBA + ∠ BAE

By substituting ∠ AEB as y we get

y = ∠ EBA + ∠ BAE

By considering the △ BCD

We know that

x = ∠ CBA + ∠ BCD

It is given that x = y

So we can write it as

∠ CBA + ∠ BCD = ∠ EBA + ∠ BAE

On further calculation we can write it as

∠ BCD = ∠ BAE

Based on both △ BCD and △ BAE

We know that B is the common point

It is given that AB = BC

It is proved that ∠ BCD = ∠ BAE

Therefore, by ASA congruence criterion we get

△ BCD ≅ △ BAE

We know that the corresponding sides of congruent triangles are equal

Therefore, it is proved that AE = CD.

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