Lesson 6

A Special Point

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.

Prove (using words and other representations) that the angle bisectors of a triangle are concurrent.

Let’s see what we can learn about a triangle by watching how salt piles up on it.

If desired, prepare a plate, bottle, container of salt, and a triangle made out of cardboard for the salt demonstration in the warm-up. 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.

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.

Building On

HSG-CO.C.9

Addressing

Building Towards

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.