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. 

 

Answer 1

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