Lesson 6
A Special Point
Lesson Narrative
In this lesson, students explore properties of angle bisectors. To build intuition, students first observe that pouring salt on a triangle forms ridges that meet at a peak, and the ridges appear to be angle bisectors. Students go on to prove that a point is on an angle bisector if and only if it is equidistant from the rays that form the angle. Then, they show that all 3 angle bisectors of a triangle meet at a single point, the incenter of the triangle. This will lead to constructing a triangle’s inscribed circle in a subsequent lesson.
Students create viable arguments (MP3) when they use what they know about triangle congruence to prove facts about angle bisectors.
Learning Goals
Teacher Facing
 Prove (using words and other representations) that the angle bisectors of a triangle are concurrent.
Student Facing
 Let’s see what we can learn about a triangle by watching how salt piles up on it.
Required Preparation
If desired, prepare a plate, bottle, container of salt, and a triangle made out of cardboard for the salt demonstration in the warmup. Alternatively, prepare a method to show the embedded video for this activity.
The activity Point and Angle includes a digital and print version of the launch. For the digital version, be prepared to display an applet for all to see.
Learning Targets
Student Facing
 I can explain why the angle bisectors of a triangle meet at a single point.
 I know any point on an angle bisector is equidistant from the rays that form the angle.
CCSS Standards
Building On
Addressing
Building Towards
Glossary Entries

circumcenter
The circumcenter of a triangle is the intersection of all three perpendicular bisectors of the triangle’s sides. It is the center of the triangle’s circumscribed circle.

circumscribed
We say a polygon is circumscribed by a circle if it fits inside the circle and every vertex of the polygon is on the circle.

cyclic quadrilateral
A quadrilateral whose vertices all lie on the same circle.

incenter
The incenter of a triangle is the intersection of all three of the triangle’s angle bisectors. It is the center of the triangle’s inscribed circle.