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

$A$ circular park of radius $20 \, m$ is situated in a colony. Three boys Ankur,Syed and David are sitting at equal distances on its boundary,each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.

Let the vertex of an angle $\angle ABC$ be located outside a circle and let the sides of the angle intersect equal chords $AD$ and $CE$ with the circle. Prove that $\angle ABC$ is equal to half the difference of the angles subtended by the chords $DE$ and $AC$ at the centre.

If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection,prove that the chords are equal.

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 $, cm$)?

If two equal chords of a circle intersect within the circle,prove that the segments of one chord are equal to corresponding segments of the other chord.