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Lesson 9: Circle Inside a Triangle
• Construct the inscribed circle of a triangle.
• Identify and describe relationships for circumscribed angles.
Lesson Narrative
In this lesson, students complete the construction of the incircle of a triangle, which is the circle inscribed in a triangle that is tangent to all three sides. Students also apply what they have learned throughout the previous several lessons to analyze angle relationships that involve tangent lines. Towards the end of the lesson, students think about whether it is possible to inscribe circles in quadrilaterals in the same way as they did with the triangle. Thinking about how triangle constructions change when they are applied to quadrilaterals will be studied in more detail in an upcoming lesson on cyclic quadrilaterals.
Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Students use appropriate tools like tracing paper, straightedge, compass, or dynamic geometry software strategically to construct the incircle of an arbitrary triangle (MP5).
Required Materials Geometry toolkits
Required Preparation
Make sure tool kits include tracing paper and if available, dynamic software to explore and construct incenters.
Student Learning Goals
• Let’s construct circles inside of triangles and investigate the angles. 9.1 So Many Congruent Things
Warm Up: 10 minutes
The purpose of this warm-up is to review that the incenter of a triangle is the intersection point of the three angle bisectors, and that it is the same distance away from each of the three sides. This will be helpful later when students construct the incenter and incircle of a triangle.
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
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Teacher Guide