This lesson will begin with a Do Now that reviews two important topics for this lesson, triangles and angles in a circle. Teachers may want to review triangle types like equilateral and isosceles in preparation for our exploration. Further, before beginning the Angles in Circles Exploration, teachers should review the total number of degrees in a circle and semicircle and the relationship between a central angle and intercepted arc.
Paper Folding Activity
After reviewing the Do Now with students, teachers can introduce the exploration for today, which is the second page of the document, Angles in Circles Exploration. Students will follow the directions on the hand-out to fold circles into shapes. Within this activity, students will hopefully discover that inscribed angles and half the size of intercepted arcs. This is a great opportunity for teachers to let students explain and describe their findings before introducing formal class notes!
Proving Inscribed Angles:
Once the activity is completed, teachers can lead students through the first page of notes which provides definitions, diagrams and examples of inscribed angles in circles. I find that asking students to break down the work in-scribed, as in = inside and scribe = write or draw, helps them to remember that these angles are inside the circle. There are diagrams provided in notes for teachers to discuss the proofs with students for three major cases of inscribed angles. I would suggest that teachers review the Khan Academy video as a resource to proving these: https://www.khanacademy.org/math/geometry/circles/v/inscribed-and-central-angles?v=MyzGVbCHh5M
I’ve included a basic diagram to get teachers and students started on these proofs. This is a great place to differentiate for students with different levels. Some teachers may want to provide radii drawn in for students to get them started (on the first case), while other groups of students may not need this extra support. The third proof relies heavily on the work done in the second proof, which is a great opportunity for students to work and brainstorm on their own, or in small groups.
Students can work on practice questions, examples 1-15 in class notes for the rest of class. The last question, #16, is a proof which is a great extension for students, and requires a strong knowledge of triangle congruence ideas. Before class is over, students can write and show their work of at least one or two examples on the board. I would suggest that teacher specifically review #1, 4 and 5 before asking students to start their exit ticket. #1 is a great question because it specifically reviews both central angles and inscribed angles. While question #2 reviews the idea that a right angle is formed when we have a triangle inscribed in a circle with the diameter as a side. Lastly, #5 just asks students to review inscribed angles, and so would be a great question to determine if students understood the key concepts in this lesson.
The exit ticket asks students to find the measure of an inscribed angle and an arc, which will help teachers to determine if students understand how to use the main idea for today.