Question

show that the diagonals of a rhombus are perpendicular to each other

Answer 1

In the above pic it's proved that diagonals of a room us are perpendicular bisectors

Answer 2

$\Large{\textbf{\underline{\underline{According\:to\:the\:Question}}}}$

Assumption

QWER be a rhombus

Then,

QW = WE = ER = RQ

Now,

In ∆QKR and ∆EKR

KQ = KE (Diagonal of ||gm and rhombus bisect each other)

KR = KR (Common)

QR = ER (Given)

Hence,

∆QKR ≅ ∆EKR (SSS rule)

∠QKR = ∠EKR (CPCT)

But,

∠QKR + ∠EKR = 180° (Linear Pair)

2∠QKR = 180°

∠QKR = 90

Hence,

Diagonal of a rhombus are perpendicular to each other