Exercise

In a parallelogram $ABCD, AB=10 \mathrm{cm}$ and $\mathrm{AD}=6\mathrm{cm}$. The bisector of $\angle A$ meets $DC$ in $E$. If $AE$ and $BC$ produced meet at $F$, the length of $CF$ is $k \mathrm{cm}$, then find the value of $k.$

$E$ and $F$ are respectively the mid-points of the non-parallel sides $AD$ and $BC$ of a trapezium $ABCD$. Prove that $EF$$\Vert \mathrm{AB}$ and $\mathrm{EF}=\frac{1}{2}(\mathrm{AB}+\mathrm{CD})$ [Hint: Join $BE$ and produce it to meet $CD$ produced at $G$]

Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.

$P$ and $Q$ are points on opposite sides $AD$ and $BC$ of a parallelogram $ABCD$ such that $PQ$ passes through the point of intersection $O$ of its diagonals $\mathrm{AC}$ and $\mathrm{BD}$. Show that $\mathrm{PQ}$ is bisected at $\mathrm{O}$.

$ABCD$ is a rectangle in which diagonal $BD$ bisects $\angle \mathrm{B}$. Show that $ABCD$ is a square.

$D, E$ and $F$ are respectively the mid-points of the sides $AB, BC$ and $CA$ of a triangle $ABC$. Prove that by joining these mid-points $D, E$ and $F$, the triangle $ABC$ is divided into four congruent triangles.

Prove that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.

$P$ is the midpoint of the side $CD$ of a parallelogram $ABCD$. A line through $C$ parallel to $PA$ intersects $AB$ at $Q$ and $DA$ produced at $R$. Prove that $\mathrm{DA}=\mathrm{AR}$ and $\mathrm{CQ}=\mathrm{QR}$.