In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM : MB = 4 : 13$ and $AC = 20.4$,find $AN$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = 3x - 1$,$MB = 2x + 1$,$AN = 3x + 1$ and $NC = 5x + 1$,find the value of $x$.

In triangles $PQR$ and $MST$,$\angle P = 55^{\circ}$,$\angle Q = 25^{\circ}$,$\angle M = 100^{\circ}$ and $\angle S = 25^{\circ}$. Is $\triangle QPR \sim \triangle TSM$? Why?

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 2x$,$BM = 3x + 5$,and $CM = 8x$,find the value of $x$.

In the given figure,if $\angle D = \angle C$,then is it true that $\triangle ADE \sim \triangle ACB$? Why?

$\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow XYZ$. If $PQ : QR : PR = 3 : 5 : 7$ and the perimeter of $\Delta XYZ$ is $22.5$,then $YZ = \ldots$