A good way of understanding why opposite angles in an inscribed shape are supplementary is by splitting up the shape itself. For example, if you had a quadrilateral inscribed within a circle, you could "cut" up the shape by the diagonal into two triangles. If you put the opposite angles next to each other (by lining up the triangles), you would notice a straight line. This line would prove that the opposite angles are equal to 180 degrees.
