In $\triangle PQR,$ if $\angle R > \angle Q,$ then

$\angle ACD$ is an exterior angle of $\Delta ABC$. If $\angle A = 50^{\circ}$ and $\angle B = 65^{\circ}$,then $\angle ACD = \dots$ (in $^{\circ}$)

If $\Delta XYZ$ has $XY = 7 \, cm$ and $YZ = 10 \, cm$,the perimeter of $\Delta XYZ$ is greater than how many centimeters?

Bisectors of the angles $B$ and $C$ of an isosceles triangle with $AB = AC$ intersect each other at $O$. $BO$ is produced to a point $M$. Prove that $\angle MOC = \angle ABC$.

State whether each of the following statements is true or false:
$(1)$ In $\Delta XYZ$,if $XY > XZ$,then $\angle Z > \angle Y$.
$(2)$ In $\Delta ABC$ and $\Delta PQR$,if $\frac{AB}{PR} = \frac{BC}{QP} = \frac{CA}{RQ} = 1$,then $\Delta ABC \cong \Delta RPQ$.

$ABC$ and $DBC$ are two triangles on the same base $BC$ such that $A$ and $D$ lie on the opposite sides of $BC$,$AB = AC$ and $DB = DC$. Show that $AD$ is the perpendicular bisector of $BC$.