Question

In the diagram N lies on a side of a square abcd, am and close are perpendicular to deprive that

Answer 1

Given :

N lies on a side of the square ABCD,AM and LC are perpendicular to DN,

To Find :  prove that ∠ADN= ∠LCD.

Solution

Let say

∠ADN= α

∠LCD = β

∠ADN + ∠CDN = 90°   ( ∠D = angle of square )

=> α  + ∠CDN = 90°

∠CDN = ∠CDL

=>  α  + ∠CDL = 90°

in ΔCDL

∠CDL + ∠LCD + ∠DLC  = 180°   ( sum of angles of triangle )

∠DLC  = 90° as LC is perpendicular to DN

=> ∠CDL + β + 90°   = 180°

=>  ∠CDL + β = 90°

 α  + ∠CDL = 90°

 ∠CDL + β = 90°

Equating both

 α  + ∠CDL  =   ∠CDL + β

=> α   = β

=> ∠ADN=  ∠LCD

QED

Hence Proved

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Answer 2

Let, ∠ADN = a

∠LDC = 90° – a

∠LDC = 180° – (90° – a + 90° )

= 180° – ( 180° – a )

= 180° – 180° + a

= a

∴ ∠ADN = ∠LCD (proved)