............... is not a condition for the similarity of two triangles.

$\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow EFD$. $AB : BC : CA = 4 : 3 : 5$. If the perimeter of $\Delta DEF$ is $36$,find the measures of all the sides of $\Delta DEF$.

It is given that $\triangle DEF \sim \triangle RPQ.$ Is it true to say that $\angle D = \angle R$ and $\angle F = \angle P?$ Why?

$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. If $AB = 12, XY = 8$ and the area of $\Delta ABC$ is $81$,find the area of $\Delta XYZ$.

In an isosceles right-angled triangle, the length of the hypotenuse is $24$. Find the area of the triangle.

$\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow XZY$. If $\frac{PQ}{XZ} = \frac{7}{3}$ and the area of $\Delta XYZ$ is $72$,find the area of $\Delta PQR$.