Prove that if the diagonals of a quadrilateral bisect each other, then it is a parallelogram. [4 MARKS]

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Solution

Diagram : 1 Mark
Concept : 1 Mark
Proof : 2 Marks

Consider $Δ A O D$ and $Δ C O B .$

$A O = C O$ [Given]

$B O = D O$ [Given]

$∠ A O D = ∠ C O B$ [vertically opposite]

$∴ Δ A O D ≅ Δ C O B .$ [SAS conguency]

$∠ A D O = ∠ C B O$ [CPCTC]

Thus, $A D | | B C$ (Alternate angles are equal)--------(i)

Similarly, We can prove

$Δ D O C ≅ Δ B O A .$

$\Rightarrow A B | | D C$ -----(ii)

Combining (i) and (ii), we can conclude that ABCD is a parallelogram.

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