In $\Delta ABC$,$AB = 8 \text{ cm}$ and $BC = 10 \text{ cm}$. Then the perimeter of $\Delta ABC$ is less than how many centimeters?

$ABCD$ is a quadrilateral such that $AB = AD$ and $CB = CD$. Prove that $AC$ is the perpendicular bisector of $BD$.

In the figure,$AD$ is the bisector of $\angle BAC$. Prove that $AB > BD$.

In $\Delta ABC$,$AB = AC$ and $\angle C = 75^{\circ}$,then find $\angle B$. (in $^{\circ}$)

Show that in a quadrilateral $ABCD$,$AB + BC + CD + DA < 2(BD + AC)$.

The line segment joining the mid-points $M$ and $N$ of the parallel sides $AB$ and $DC$ respectively of a trapezium $ABCD$ is perpendicular to both the sides $AB$ and $DC$. Prove that $AD = BC$.