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radius related to the side lengths? How is the radius related to the angles? How could trigonometric ratios relate the radius to the various lengths and angles in the triangle?)
• “How do we know that the circle is tangent to all three sides of the triangles?” (The circle goes through three points on the triangle, and the sides of the triangle meet the radii of the circle at right angles at those points. The sides must be tangent to the circle because the radius is only perpendicular to a line at a point on the circumference if the line is tangent to the circle at that point.)
9.3 Angles in Circles and Triangles
15 minutes
The purpose of this activity is for students to synthesis what they have learned about incircles, tangent lines, and central angles to derive the relationship between a central angle and a circumscribed angle. It is not necessary to introduce new vocabulary for circumscribed angle. Examining angles in a quadrilateral will happen again in an upcoming lesson about cyclic quadrilaterals.
Instructional Routines
• Think pair share
What: Students have quiet time to think about a problem and work on it individually, and then time to share their response or their progress with a partner. Once these partner conversations have taken place, some students are selected to share their thoughts with the class.
Why: This is a teaching routine useful in many contexts whose purpose is to give all students enough time to think about a prompt and form a response before they are expected to try to verbalize their thinking. First they have an opportunity to share their thinking in a low-stakes way with one partner, so that when they share with the class they can feel calm and con dent, as well as say something meaningful that might advance everyone’s understanding. Additionally, the teacher has an opportunity to eavesdrop on the partner conversations so that she can purposefully select students to share with the class.
Launch
Arrange students in groups of 2. Give 5 minutes of quiet work time before asking students to share their reasoning with their partner.
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Teacher Guide