Question

Suppose in a quadeilateral ABCD, it is given that $AB=AD$ and $\angle ABC=\angle ADC$. Prove that ABCD is a kite.

Answer 1

$\Rightarrow AB=AD$ and $\angle ABC\cong \angle ADC$

Consider $△ABC$ and $△ADC$

They both are congruent by SAS,

Since $△ABC=\angle ADC$ and $AB=AD.$

They are congruent ( by hypothesis )

The sum of right angles are equal because of the congruent angles

$\Rightarrow \angle ADC=\angle ADB+\angle BDC$

$\angle ABC=\angle ABD+\angle DBC$

$\Rightarrow △CBD$ is isosceles, $\angle CBD=\angle CDB$

$△ABD$ is isosceles, $\angle ABD=\angle ADB$

$\therefore ABCD$ is a kite.

Hence, the answer is kite.