$AB$ is a chord of a circle with centre $P$. Point $C$ is a point other than $A$ and $B$ on the major arc $AB$. If $\angle APB = 130^{\circ}$,then find $\angle ACB$. (in $^{\circ}$)

$XY$ is a diameter of the circumcircle of $\Delta XYZ$. What is the value of $\angle XZY$ (in $^{\circ}$)?

The longest chord of a circle is a .......... of the circle.

If $BM$ and $CN$ are the altitudes drawn on the sides $AC$ and $AB$ of the triangle $ABC$,prove that the points $B, C, M$ and $N$ are concyclic.

In cyclic quadrilateral $ABCD$,$\angle B = \angle D - 30^{\circ}$,then find $\angle B$. (in $^{\circ}$)

$A$ circle has a radius of $\sqrt{2} \text{ cm}$. It is divided into two segments by a chord of length $2 \text{ cm}$. Prove that the angle subtended by the chord at a point in the major segment is $45^{\circ}$.