In the figure,$\angle ABC = 69^{\circ}$ and $\angle ACB = 31^{\circ}$. Find $\angle BDC$. (in $^{\circ}$)

$AC$ and $BD$ are chords of a circle that bisect each other. Prove that $ABCD$ is a rectangle.

$ABCD$ is a parallelogram. The circle passing through $A, B$ and $C$ intersects $CD$ (produced if necessary) at $E$. Prove that $AE = AD$.

The lengths of two parallel chords of a circle are $6\, cm$ and $8\, cm$. If the smaller chord is at a distance of $4\, cm$ from the centre,what is the distance of the other chord from the centre (in $, cm$)?

In the figure,$A, B$ and $C$ are three points on a circle with centre $O$ such that $\angle BOC = 30^{\circ}$ and $\angle AOB = 60^{\circ}$. If $D$ is a point on the circle other than the arc $ABC$,find $\angle ADC$. (in $^{\circ}$)

Two circles of radii $5\, cm$ and $3\, cm$ intersect at two points and the distance between their centres is $4\, cm.$ Find the length of the common chord. (in $, cm$)