Investigation 1: Draw a circle. Choose four
points on the circle. Connect these points with line segments
to form a quadrilateral. Measure each interior angle. Do you notice
any relationship between the angles? Particularly, what appears
to be true about opposite angles of an inscribed quadrilateral?
Drag the four vertices of the quadrilateral. Does your conjecture
hold for various locations of the vertices? Prove your conjecture.
Investigation 2: Draw a circle. Construct a diameter of the circle. Construct an inscribed angle whose sides intersect the endpoints of the diameter. What appears to be true about the angle you just inscribed? Measure this angle and drag its vertex. Does your conjecture appear to be true for various locations of the vertex? Prove your conjecture.