Show that the line segment which joins the midpoints of the oblique sides of a trapezium is parallel to the parallel sides.

In the adjoining figure, $\mathrm{ABCD}$ is a trapezium in which $CD\parallel \mathrm{AB}$ and its diagonals intersect at $\mathrm{O}$. If $\mathrm{AO}=(2x+1) \mathrm{cm}, \mathrm{OC}=(5x-7) \mathrm{cm}, DO=(7\mathrm{x}-5) \mathrm{cm}$ and $OB=(7x+1) \mathrm{cm}$. Find the value of $\mathrm{x}$.

$∆ABC$ and $\Delta DBC$ lie on the same side of $BC,$ as shown in the figure. From a point $P$ on $BC, PQ \left|\right| AB$ and $PR \left|\right| BD$ are drawn, meeting $AC$ at $Q$ and $CD$ at $R,$ respectively. Prove that $QR \left|\right| AD.$

In the given figure, side $BC$ of $∆ABC$ is bisected at $D$ and $O$ is any point on $AD. BO$ and $\mathrm{CO}$ produced meet $A\mathrm{C}$ and $AB$ at $E$ and $F$ respectively, and $AD$ is produced to $X$ so that $D$ is the midpoint of $OX.$ Prove that $AO:AX=AF:AB$ and show that $EF\parallel BC.$

$ABCD$ is a parallelogram in which $P$ is the midpoint of $DC$ and $Q$ is a point on $AC$ such that $CQ=\frac{1}{4}AC$. If $PQ$ produced meets $BC$ at $R$, prove that $R$ is the midpoint of $BC$.

In the adjoining figure, $\mathrm{ABC}$ is a triangle in which $\mathrm{AB}=\mathrm{AC}$. If $\mathrm{D}$ and $\mathrm{E}$ are points on $\mathrm{AB}$ and $\mathrm{AC}$ respectively such that $\mathrm{AD}=\mathrm{AE}$, show that the points $\mathrm{B}, \mathrm{C}, \mathrm{E}$ and $\mathrm{D}$ are concyclic.

In $\mathrm{\Delta ABC}$, the bisector of $\angle \mathrm{B}$ meets $\mathrm{AC}$ at $\mathrm{D}$. A line $\mathrm{PQ}\parallel \mathrm{AC}$ meets $\mathrm{AB}, \mathrm{BC}$ and $\mathrm{BD}$ at $P$, $\mathrm{Q}$ and $\mathrm{R}$ respectively. Show that $\mathrm{BP}\times \mathrm{QR}=\mathrm{BQ}\times \mathrm{PR}$.

In each of the given pairs of triangles, find which pair of triangles are similar. State the similarity criterion and write the similarity relation in symbolic form.