Circle properties and their proofs

Prove the circle properties — the angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc; an angle in a semicircle is a right angle; angles at the circumference of a circle subtended by the same arc are equal; the alternate segment theorem; the opposite angles of a cyclic quadrilateral are supplementary and its converse; a tangent drawn to a circle is perpendicular to the radius at the point of contact and its converse.