Inequalities in a Triangle

Try to construct a triangle with $5 \mathrm{cm}, 8 \mathrm{cm}$ and $1 \mathrm{cm}$. Is it possible or not? Why? Give your justification? 

If two sides of a triangle measure $4 \mathrm{cm}$ and $6 \mathrm{cm}$, find all possible measurements (positive Integers) of the third side. How many distinct triangles can be obtained? 

In adjacent figure, $PR>PQ$ and $PS$ bisects $\angle QPR$. Prove that $\angle PSR>\angle PSQ$.

$AB$ and $CD$ are respectively the smallest and longest sides of a quadrilateral $ABCD$ (see adjacent figure). Show that $\angle A>\angle C$ and $\angle B>\angle D$.

In adjacent figure, $\angle B<\angle A$ and $\angle C<\angle D$. Show that $AD< BC$. 

In adjacent figure, sides $AB$ and $AC$ of $∆ABC$ are extended to points $P$ and $Q$ respectively. Also, $\angle PBC<\angle QCB$. Show that $AC>AB$. 

Show that in a right-angled triangle, the hypotenuse is the longest side.