There can be $\ldots \ldots \ldots$ one-one correspondences between two triangles.

Prove that the sum of any two sides of a triangle is greater than twice the median with respect to the third side.

Two lines $l$ and $m$ intersect at the point $O$ and $P$ is a point on a line $n$ passing through the point $O$ such that $P$ is equidistant from $l$ and $m$. Prove that $n$ is the bisector of the angle formed by $l$ and $m$.

In $\Delta PQR$,$PQ = PR$ and $\angle R = 40^{\circ}$,then $\angle P = \dots$ (in $^{\circ}$)

In $\Delta ABC$,$\angle A = 90^{\circ}$ and $AB = AC$. The bisector of $\angle A$ meets $BC$ at $D$. Prove that $BC = 2 AD$.

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 $AD = BC$.