If $\triangle ABC \sim \triangle DEF$,$AB = 4 \, cm$,$DE = 6 \, cm$,$EF = 9 \, cm$,and $FD = 12 \, cm$,find the perimeter of $\triangle ABC$ (in $cm$).

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QD}$ is an altitude to the hypotenuse $PR$. If $PD = 25 DR$,prove that $PQ = 5 QR$.

Is the following statement true? Why?
"Two quadrilaterals are similar,if their corresponding angles are equal".

If $\Delta ABC \sim \Delta PQR$ under the correspondence $ABC \leftrightarrow RPQ$,and $m\angle A = 35^{\circ}$ and $m\angle B = 50^{\circ}$,find $m\angle Q$. (in $^{\circ}$)

Which of the following correctly matches the information in Part $I$ and Part $II$?

Part $I$	Part $II$
$1.$ In $\Delta ABC$,$\angle B$ is a right angle and $\overline{BM}$ is a median.	$a. AB^2 + BC^2 = 2(BD^2 + CD^2)$
$2.$ In $\Delta ABC$,$\angle A$ is a right angle and $\overline{AD}$ is an altitude.	$b. BC = \frac{1}{2} AB$
$3.$ In $\Delta ABC$,$m\angle C = 90^\circ$ and $m\angle A = 30^\circ$.	$c. AC^2 = CD \cdot BC$
$4.$ In $\Delta ABC$,$\overline{BD}$ is a median.	$d. BM = \frac{1}{2} AC$

The length of a diagonal of a square is $6$. Then,its area is............