State the necessary conditions,as per definition,for two triangles to be similar.

In rectangle $ABCD$,$AC = 29$ and $AB + BC = 41$. Find $AB$ (given $AB > BC$).

In $\Delta ABC$,$\overline{AD}$ is a median. If $AD = 7$ and $AB^2 + AC^2 = 148$,find $BC$.

If $\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow EFD$.
If $m \angle X : m \angle Y : m \angle Z = 2 : 3 : 5$,then in $\Delta DEF$,$\ldots \ldots$ is a right angle.

In $\Delta ABC$ and $\Delta XYZ$,$\frac{AB}{XY} = \frac{BC}{YZ}$ and $\angle B \cong \angle Y$. If $AB = 6$,$AC = 15$,and $XY = 15$,find $XZ$.

In rectangle $ABCD$,$AB = 9$ and $BC = 12$. Then,$BD = \ldots$