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Question
ABCD is a square, X and Yare points on sides AD and BC respectively such that AY = BX. Prove that BY = AX and ∠BAY = ∠ABX.
Solution
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
∠XAB=∠YBA=90° ...........[From (1)]
BX=AY [given]
And AB=BA [Common side]
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
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