In a circle with centre $P$,the length of a radius is $\sqrt{2} \ cm$. It is divided into two segments by chord $AB$ of length $2 \ cm$. If $C$ is a point on the major arc $AB$,then find the value of $\angle ACB$. (in $^{\circ}$)

If non-parallel sides of a trapezium are equal,prove that it is cyclic.

State whether each of the following statements is true or false:
$(1)$ $A$ piece of a circle between two points on a circle is called an arc of the circle.
$(2)$ The length of the complete circle is called its circumference.

The circumcentre of the triangle $ABC$ is $O$. Prove that $\angle OBC + \angle BAC = 90^{\circ}$.

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.

State whether each of the following statements is true or false:
$(1)$ The collection of all the points in a plane,which are at a fixed distance from a fixed point in the plane,is called a circle.
$(2)$ The line segment joining the centre and any point on the circle is called a radius of the circle.