Question

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

Answer 1

It is given that $x=y$ and $AB=CB$

By considering the $△ABE$

We know that

Exterior $\angle AEB=\angle EBA+\angle BAE$

By substituting $\angle AEB$ as $y$ we get

$y=\angle EBA+\angle BAE$

By considering the $△BCD$

We know that

$x=\angle CBA+\angle BCD$

It is given that $x=y$

So we can write it as

$\angle CBA+\angle BCD=\angle EBA+\angle BAE$

On further calculation, we can write it as

$\angle BCD=\angle BAE$

Based on both $△BCD$ and $△BAE$

We know that $B$ is the common point

It is given that $AB=BC$

It is proved that $\angle BCD=\angle BAE$

Therefore, by $ASA$ congruence criterion we get

$△BCD\cong △BAE$

We know that the corresponding sides of congruent triangles are equal

Therefore, it is proved that $AE=CD$.