In this lesson, students encounter inscribed angles in circles, or angles formed by 2 chords which share an endpoint. Through experiment, students explore the relationship between inscribed angles and their associated central angles. They develop the conjecture that the measure of an inscribed angle is half the measure of the central angle that defines the same arc. Then, taking this conjecture as an assertion, they show that 2 intersecting chords and the segments joining adjacent endpoints of the chords create similar triangles.
The relationship between central angles, arcs, and inscribed angles will be used in a subsequent lesson to prove properties of quadrilaterals inscribed in circles.
One of the activities in this lesson works best when students have access to devices that can run the GeoGebra applet because students will benefit from seeing the relationship in a dynamic way.
When students use experiments to form conjectures about angle relationships, they are making sense of a problem (MP1).
- Calculate measures of central and inscribed angles.
- Use the relationship between central and inscribed angles to prove (using words and other representations) a theorem about intersecting chords.
- Let’s analyze angles made from chords.
If students will do the digital activity for the activity A Central Relationship, 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 protractors.
Students will need compasses and index cards from their geometry toolkits for the lesson synthesis.
- I can use the relationship between central and inscribed angles to calculate angle measures and prove geometric theorems.
- I know that an inscribed angle is half the measure of the central angle that defines the same arc.
The part of a circle lying between two points on the circle.
An angle formed by two rays whose endpoints are the center of a circle.
A chord of a circle is a line segment both of whose endpoints are on the circle.
An angle formed by two chords in a circle that share an endpoint.