Prove that a quadrilateral with both pairs of opposite sides equal, is a parallelogram. [4 MARKS]

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Solution

Concept : 1 Mark
Application : 1 Mark
Proof : 2 Marks

Let $A B C D$ be the given quadrilateral with $A B = D C$ and $A D = B C .$

Consider $Δ A B C$ and $Δ C D A .$

$A B = C D$ [Given]

$A D = B C$ [Given]

$A C = C A$ [Common]

$∴ Δ A B C ≅ Δ C D A .$ [SSS congruency]

$\Rightarrow ∠ D C A = ∠ B A C$ [CPCTC]

So, $A B ∥ C D$ .......(i) [Since the alternate angles are equal]

$∠ D A C = ∠ B C A$ [CPCTC]

So, $A B ∥ D C$ .......(ii) [Since the alternate angles are equal]

From (i) and (ii) both pairs of opposite sides are parallel.

$\Rightarrow A B C D$ is a parallelogram.

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