Given that ABCD is a square, X and Y are points on sides AD and BC respectively such that AY =BX

We have to prove BY = AX and ∠BAY = ∠ABX.

Join B and X, A and Y.

Since, ABCD is a square

∠DAB = ∠CBA = 90°

∠XAB =∠YBA = 90° .....(1)

Now, consider triangle XAB and YBA

We have

So, by RHS congruence criterion,

we have ΔXAB≅ΔYBA

Now, we know that corresponding parts of congruent triangles are equal.

∴ BY = AX and ∠BAY = ∠ABX.

∴ Hence proved