Question

In the adjoining figure, ABCD is a parallelogram. Perpendicular DN and BP are drawn on diagonal AC. Prove that:

AND THE DIAGRAM AND THE QUESTION IS IN THE ATTACHMENT,
Please answer​

Answer 1

|| ☆》Question -:

• In the adjoining figure, ABCD is a parallelogram. Perpendicular DN and BP are drawn on diagonal AC. Prove that:

Prove that ;-

(i) ∆DCN ≈ ∆BAP

(ii) AN = CP

||☆》Given -:

• ABCD is a parallelogram.
• Perpendicular DN and BP are drawn on diagonal AC.

||☆》To Prove -:

(i) ∆DCN ≈ ∆BAP

(ii) AN = CP

||♡》Proof -:

Here,

ABCD is a parallelogram , then AB || CD & AD || DC.

(i) ∆DCN ≈ ∆BAP

Since ,

AB || DC & AC is a transversal.

∠ PAB = ∠ NCD

( Alternate Interior Angles )

In ∆DCN & ∆BAP ;-

$\Rightarrow$∠ DCN = ∠ BAP ( 90° each )

$\Rightarrow$∠ NCD = ∠ PAB ( Proved Above!)

$\Rightarrow$DC = AB ( Opposite Sides of ||gm are equal )

$\longrightarrow \boxed {∆DCN ≈ ∆BAP} \quad ( AAS ) \\ \therefore \; \bold { DN = BP} \qquad ( CPCT )$

(ii) AN = CP

In ∆DNA & ∆BPC ;-

$\Rightarrow$AD = BC ( Opposite Sides of a ||gm are equal )

$\Rightarrow$DN = BP ( Proved Above! )

$\longrightarrow∆DNA ≈ ∆BPC \quad ( RHS ) \\ \therefore \; \boxed { AN = CP} \qquad( CPCT )$

Hence , Proved !

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