In the figure x is a point in the interior of square ABCD, AXYZ is also a square. Prove that BX = DZ.

Since ABCD and AXYZ both are squares
∠AZY = ∠AXY
       = ∠AXB ... (each 90°) and AX = AZ and AB = AD
Now in ΔABX and ΔADZ,
∠AZD = ∠AXB ...... (each 90°)
  AZ = AX ...... (sides of a square)
  AB = AD ...... (sides of a square)
ΔABX ≅ ΔADZ
  BX = DZ ....... (c.p.c.t.)