Inequalities in a Triangle

Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

In shown figure, $\mathrm{P}\mathrm{R}>\mathrm{P}\mathrm{Q}$ and $\mathrm{P}\mathrm{S}$ bisects $\mathrm{\angle }\mathrm{Q}\mathrm{P}\mathrm{R}.$ Prove that $\mathrm{\angle }\mathrm{P}\mathrm{S}\mathrm{R}>\mathrm{\angle }\mathrm{P}\mathrm{S}\mathrm{Q}.$

$\mathrm{A}\mathrm{B}$ and $\mathrm{C}\mathrm{D}$ are respectively the smallest and longest sides of a quadrilateral $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ as shown in figure. Show that $\mathrm{\angle }\mathrm{A}>\mathrm{\angle }\mathrm{C}$ and $\mathrm{\angle }\mathrm{B}>\mathrm{\angle }\mathrm{D}.$

In figure shown below, $\mathrm{\angle }\mathrm{B}<\mathrm{\angle }\mathrm{A}$ and $\mathrm{\angle }\mathrm{C}<\mathrm{\angle }\mathrm{D}$. Show that $\mathrm{A}\mathrm{D}<\mathrm{B}\mathrm{C}.$

In figure shown below, sides $\mathrm{A}\mathrm{B}$ and $\mathrm{A}\mathrm{C}$ of $∆\mathrm{A}\mathrm{B}\mathrm{C}$ are extended to points $\mathrm{P}$ and $\mathrm{Q}$ respectively. Also, $\mathrm{\angle }\mathrm{P}\mathrm{B}\mathrm{C}<\mathrm{\angle }\mathrm{Q}\mathrm{C}\mathrm{B}.$ Show that $\mathrm{A}\mathrm{C}>\mathrm{A}\mathrm{B}.$

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