Find the value of $x$ for which $DE \parallel AB$ in the given figure.

$\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow DEF$. If $m \angle X = 50^{\circ}$ and $m \angle Y = 75^{\circ}$,then $m \angle F = \dots$ (in $^{\circ}$)

The areas of two similar triangles are $144$ and $81$. If the length of the longest side of the first triangle is $36$,the length of the longest side of the second triangle is........

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $AC$. If $AD = 9$ and $CD = 4$,find $BD$.

$\square ABCD$ is a trapezium in which $\overline{ AD } \| \overline{ BC }, \overline{ AC } \cap \overline{ BD }=\{ P \} .$ If $PD =9, PA =5$ and $PB =7.2,$ then $AC =\ldots \ldots \ldots \ldots$

$\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow QPR.$ If $m \angle P : m \angle Q : m \angle R = 2 : 3 : 4,$ then $m \angle A : m \angle B : m \angle C = \ldots$