In the diagram ,N lies on a side of the square ABCD,AM and LC are perpendicular to DN,prove that ADN=LCD.

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