The key point for discussion is that all points equidistant to the two rays are on the angle bisector, and that all points on the angle bisector are equidistant to the two rays. Here are some questions for discussion:

• “How does this relate to the salt pile activity?” (If the angle were one of the angles in the triangle in the salt pile, the angle bisector would represent the ridge. The salt that forms the ridge is the same distance from either side, so it doesn’t fall in one direction or the other.)
• “What is the difference between what you showed in the first question and what you showed in the second question?” (In the first question, we showed that if a point is equidistant from the rays that form an angle, then it’s on the angle bisector. In the second, we proved the converse: If a point is on the angle bisector, it’s equidistant from the rays that form the angle.)
• “Have we proven these conjectures for all angles or just this one?” (This works for all angles, because we didn’t rely on any specific measurements or placements. If we drew a new angle, the same arguments would all apply.)

Tell students that what they’re learning will be useful when they construct another special circle in an upcoming lesson.

In a previous lesson, students reasoned that the perpendicular bisectors of a triangle’s sides meet at a single point. Here, students use a similar line of argument, combined with the results from the previous activity about points on angle bisectors, to determine that the three angle bisectors of a triangle also meet at a single point. This point is called the triangle’s “incenter.” The fact that this point is the same distance from all three sides of a triangle will lead to the construction of an inscribed circle in the next lesson.

If students are having trouble sketching segments that show the distance from point to the sides of the triangle, suggest that they use an index card to estimate a right angle.

The goal is to further explore how to visualize the distance between a point and the side of a triangle, in order to strengthen students’ understanding that the incenter of a triangle is the same distance from all three of the triangle’s sides. 

Tell students that this point at which the angle bisectors meet is called the triangle’s incenter. We will add a theorem about a triangle’s incenter to the reference chart in the next lesson, after we look at a special circle related to incenters.

Ask students, “How does the incenter relate to the salt pile?” (This point is the same distance from all the sides of a triangle, so grains of salt that land on this point balance there and don’t fall toward any of the sides.)

Then display this image for all to see, and explain that the dashed lines are the angle bisectors for triangle .

• “Which side or sides are closest to the point?” (Point is closest to side . None of the distances from this point to the other sides are equal. Point is closest to side . It is equidistant from sides and .)
• “Which sides are equidistant from the point, if any?” (Point is equidistant from sides and . It is closer to these two sides than it is to side . Point is equidistant from all three sides.)

Students may need to be reminded that an angle bisector divides an angle into two congruent halves.