Question

Suppose in a quadrilateral $ABCD,AC=BD$ and $AD=BC$. Prove that $ABCD$ is a trapezium

Answer 1

Given:ABCD is a quadrilateral and $AC=BD$ and $AD=BC$

To prove: $ABCD$ is trapezium

Proof: In $\Delta ADB$ and $\Delta BCA$

(1) $AD=BC$ (Given)

(2) $AC=BD$ (Given )

(3) $AB=AB$ (Common side)

∴ $\Delta ADB\cong \Delta BCA$

∴ $\angle A=\angle B$ (Congruency property)

$AC=BD$ (Given), $AD=BC$ (Given)

Hence $ABCD$ is an isosceles trapezium