The diagonals of the parallelogram formed by the lines $a_{1}x+b_{1}y+c_1=0$, $a_{1}x+b_{1}y+{c_1}^1=0$, $a_{2}x+b_{2}y+c_1=0$ and $a_{2}x+b_{2}y+{c_1}^1=0$ will be right angles if

In any quadrilateral, if the diagonals are perpendicular, then that quadrilateral is either Rhombus or Square

It implies that the distance between the opposite sides must be equal

hence, 

$\frac{{c_1}^1-c_1}{\sqrt{{a_1}^2+{b_1}^2}}=\frac{{c_1}^1-c_1}{\sqrt{{a_2}^2+{b_2}^2}}$

It implies that 

$\sqrt{{a_1}^2+{b_1}^2}=\sqrt{{a_2}^2+{b_2}^2}$

Squaring on both sides 

${a_1}^2+{b_1}^2={a_2}^2+{b_2}^2$