7. An inscribed angle in a circle is always half as big as the corresponding central angle.

There are three cases to consider. The first is when one of the arms of the angle, call it AB goes through the center of the circle.

Since A is a point on the circle, angle BAC is an inscribed angle. O is the center of the circle, so OA = OC since they are both radii of the same circle. That makes triangle OAC an isosceles triangle, so the angle at A is the same size as the angle at C, since they are base angles in an isosceles triangle. Now angle COB, the central angle which corresponds to the inscribed angle at A, is an exterior angle to triangle COA, so it is the same size as the sum of the other two interior angles, namely A and C. Since they are both equal in size, we conclude that the interior angle is twice the size as the inscribed angle.

The next case is the one where the center of the circle is inside the angle.

In the picture, angle CAD is the inscribed angle, and the central angle corresponding to it will be angle COD. If we draw in the diameter AB, we break each angles into two angles which satisfy the conditions of the last case. Angle COB is twice as big as angle CAB, and angle DOB is twice as big as angle DAB. When we add them we get that angle COD is twice as big as angle CAD.

The third case is when the center of the circle lies outside of the angle.

In the picture, angle CAD is the inscribed angle, and the central angle corresponding to it will be angle COD. If we draw in the diameter AB, by the first case, angle COB is twice angle CAB and angle DOB is twice angle DAB. When we subtract them we get that angle COD is twice as big as angle CAD.