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Lesson 11: Circles Outside of Quadrilaterals
• Prove properties of angles for a quadrilateral inscribed in a circle. Lesson Narrative
In this lesson, students take a closer look at cyclic quadrilaterals and their properties. A cyclic quadrilateral is a quadrilateral that has a circumscribed circle. Students rst generate the conjecture that opposite angles of a cyclic quadrilateral are supplementary, and then prove their conjecture.
Students make use of structure when they use the geometry of the circle to analyze the opposite angles of cyclic quadrilaterals and prove that they are supplementary (MP7).
Required Materials Geometry toolkits
Student Learning Goals
• Let’s investigate quadrilaterals on a circle. 11.1 Circled
Warm Up: 5 minutes
In this warm-up, students apply what they learned about circumcenters and circumscribed circles in the previous lesson to nd the circumcenter of a quadrilateral. The quadrilateral is designed to be cyclic and to contain a right angle. From their experiences with right, acute, and obtuse triangles in the previous lesson, students should look to decide whether angle traces out more than, less than, or exactly half a circle. If it does, then the circumcenter should lie on segment . Thinking about angle measures by examining how much arc they trace out will be helpful for one particular strategy for proving opposite angles of cyclic quadrilaterals are supplementary in an upcoming activity.
Student Task Statement
Use a straightedge and a compass or paper folding to construct a circle that circumscribes this quadrilateral.
Without using a protractor, nd the measure of angle . Explain your reasoning.
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Teacher Guide