UPSC CDS Math Triangles Questions

ΔABC is an equilateral triangle
∴ AB = BC = CA
Also AZ = BY = CY
∴ AB – AZ = BC – BX = CA – CY
⇒ BZ = CX = AY
In ΔBXZ and ΔCXY
BX = CY (Given)
∠XBZ =∠YCX (60° each)
BZ = CX(Proved)
∴ ΔPXY ≅ ΔPRQ (SAS congruence axiom)
⇒ XZ = XY(CPCT)
Similarly, XY = YZ
∴ XZ = XY = YZ
⇒ ΔXYZ is an equilateral triangle.
Hence, statement 1 is correct.
Remember:
Two equilateral triangles are always similar as their corresponding angles are equal as each angle is 60°.
∴ ΔXYZ ∼ΔABC (AAA similarity)
Hence, statement 2 is correct.