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K, L, M and N are points on the sides AB, BC, CD and DA respectively of a square ABCD such that AK = BL = CM = DN. Prove that KLMN is a square.

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It is given that AK = BL = CM = DN

ABCD is a square

So we get

BK = CL = DM = AN …… (1)

Consider △ AKN and △ BLK

It is given AK = BL

From the figure we know that ∠ A = ∠ B = 90o

Using equation (1)

AN = BK

By SAS congruence criterion

△ AKN ≅ △ BLK

We get

∠ AKN = ∠ BLK and ∠ ANK = ∠ BKL (c. p. c. t)

We know that

∠ AKN + ∠ ANK = 90o

∠ BLK + ∠ BKL = 90o

By adding both the equations

∠ AKN + ∠ ANK + ∠ BLK + ∠ BKL = 90o + 90o

On further calculation

2 ∠ ANK + 2 ∠ BLK = 180o

Dividing the equation by 2

∠ ANK + ∠ BLK = 90o

So we get

∠ NKL = 90o

In the same way

∠ KLM = ∠ LMN = ∠ MNK = 90o

Therefore, it is proved that KLMN is a square.

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