If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides, prove that the quadrilateral so formed is cyclic.

In Figure, $\angle ADC=130°$ and chord $BC=$chord $BE$. If $\angle CBE=k°$. Then find the value of k.

In Figure, $\angle \text{ACB}=40°$ and $\angle \text{OAB=k°}$. Then find the value of k.

Two circles with centres $O$ and $O\text{'}$intersect at two points $A$ and $B$. A line $PQ$ is drawn parallel to $O{O}^{\text{'}}$ through $A$ (or $B$ ) intersecting the circles at $P$ and $Q$. Prove that $PQ=2 OO\text{'}.$

In Figure, $AOB$ is a diameter of the circle and $C,D, E$ are any three points on the semi-circle. If the value of $\angle ACD+\angle BED=k°$. Then find the value of $k$.