In $\Delta PQR$,$PQ = 7.5 \, cm$,$QR = 6.2 \, cm$ and $a < PR < b$,then find the values of $a$ and $b$ in $cm$.

"If two sides and an angle of one triangle are equal to two sides and an angle of another triangle,then the two triangles must be congruent." Is the statement true? Why?

In $\Delta PQR$,$X$ is the midpoint of $QR$. $XY$ and $XZ$ are altitudes from $X$ on $PQ$ and $PR$ respectively. If $XY = XZ$,then prove that $\Delta PQR$ is an isosceles triangle.

In the figure,$D$ and $E$ are points on side $BC$ of a $\triangle ABC$ such that $BD = CE$ and $AD = AE$. Show that $\triangle ABD \cong \triangle ACE$.

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 $AD = BC$.

$ABC$ is a right triangle such that $AB = AC$ and the bisector of angle $C$ intersects the side $AB$ at $D$. Prove that $AC + AD = BC$.