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).
Lesson overview
 2.1 Warmup: Notice and Wonder: A New Angle (5 minutes)

2.2 Activity: A Central Relationship (15 minutes)
 There is a digital applet in this activity
 Includes "Are you Ready for More?" extension problem
 2.3 Activity: Similarity Returns (15 minutes)
 Lesson Synthesis
 2.4 Cooldown: Inscribed Angle Measures (5 minutes)
Learning goals:
 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.
Learning goals (student facing):
 Let’s analyze angles made from chords.
Learning targets (student facing):
 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.
Required materials:
 Geometry toolkits
Required preparation:
 If students will do the digital activity for the activity A Central Relationship, prepare class access to internetenabled devices, ideally 1 for every 12 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.
Glossary:
 inscribed angle  An angle formed by two chords in a circle that share an endpoint.
 Access the complete Geometry Course glossary.
Standards:
 This lesson builds towards the standard: CCSS.HSGC.A.2MS.GC.2MO.G.C.A.2
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