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 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. 

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.