In square $ABCD$,$AC = 16 \text{ cm}$,then $\operatorname{ar}(ABCD) = \dots \text{ cm}^2$.

State whether each of the following statements is true or false:
$(1)$ $ABCD$ is a parallelogram. If $AB = 12 \text{ cm}$ and $BC = 5 \text{ cm}$,then $AC = 13 \text{ cm}$.
$(2)$ $ABCD$ is a trapezium. If $AB \parallel CD$ and $AB = 10 \text{ cm}$,then $CD = 10 \text{ cm}$.

In trapezium $ABCD$,$AB \parallel CD$. If $\angle A = 70^{\circ}$ and $\angle B = 85^{\circ}$,then $\angle D = \ldots$ (in $^{\circ}$)

$ABCD$ is a rhombus in which the altitude from $D$ to side $AB$ bisects $AB$. Find the angles of the rhombus.

$A$ rectangular plot is given for constructing a house, having a measurement of $40 \, m$ long and $15 \, m$ in the front. According to the laws, a minimum of $3 \, m$ wide space should be left in the front and back each, and $2 \, m$ wide space on each of the other sides. Find the largest area where the house can be constructed (in $m^2$).

For any convex quadrilateral $ABCD$,prove that $AB + BC + CD + DA > AC + BD$.