In this lesson, students consider an outcome of the Inscribed Angle Theorem in an analysis of cyclic quadrilaterals. A cyclic quadrilateral is a quadrilateral for which a circumscribed circle, or a circle that passes through each vertex of the quadrilateral, can be drawn.

First, students try to draw circumscribed circles for several quadrilaterals. They observe that it is possible to circumscribe some but not all quadrilaterals. Then students use inscribed angles to prove that those quadrilaterals that are cyclic have supplementary pairs of opposite angles. 

Students construct the circumscribed circle for a cyclic quadrilateral with a 90-degree angle, and they explore the idea that the center of the circumscribed circle is equidistant from each vertex of the figure.

As students draw a conclusion from repeated calculations of the measures of angles in cyclic quadrilaterals, they are expressing regularity in repeated reasoning (MP8).