In this lesson, students explore an outcome of the Inscribed Angle Theorem in an analysis of cyclic quadrilaterals. These are quadrilaterals that have a circumscribed circle, or a circle that passes through each vertex of the quadrilateral.
First, students try to draw circumscribed circles for several quadrilaterals. They observe that it is possible to circumscribe some but not all quadrilaterals. Then, they use inscribed angles to prove that those quadrilaterals that are cyclic have supplementary pairs of opposite angles. They 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 looking for regularity in repeated reasoning (MP8).
- Prove (using words and other representations) properties of angles for a quadrilateral inscribed in a circle.
- Let’s investigate quadrilaterals that fit in a circle.
If students will do the digital activity for the activity Construction Ahead, prepare class access to internet-enabled devices, ideally 1 for every 1-2 students. If students will use the activity from the printed materials, prepare access to geometry toolkits.
The scientific calculators are for the extension to the activity Inscribed Angles and Circumscribed Circles.
- I can prove a theorem about opposite angles in quadrilaterals inscribed in circles.
We say a polygon is circumscribed by a circle if it fits inside the circle and every vertex of the polygon is on the circle.
A quadrilateral whose vertices all lie on the same circle.