In figure AN and CP are perpendiculars to the diagonal BD of a parallelogram ABCD.
Prove that:
(1) $△ A D N ≅ △ C B P$ (2) AN = CP

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Solution

Since ABCD is a paraallelogram.
$∴ A D | | B C N o w, A D | | B C a n d t r a n s v e r s a l B D i n t e r s e c t s t h e m a t D a n d B ∴ ∠ 1 = ∠ 2 [A l t e r n a t e i n t e r i o r a n g l e s a r e e q u a l]$
Now, in $△ A D N a n d △ C B P, w e h a v e ∠ 1 = ∠ 2 ∠ A N D = ∠ C P B (r i g h t a n g l e)$
AD = BC [Opposite sides of a || gm are equal]
So, by AAS criterion of congruence
$△ A D N ≅ △ C B P$
AN = CP
[Corresponding parts of congruent triangles are equal]
Hence proved.

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