Lesson 5

In this lesson, students learn two constructions:

• a line perpendicular to a given line through a point on the line
• an angle bisector

For the perpendicular line construction, students rely on their experience with the perpendicular bisector construction. The angle bisector construction is then connected to the perpendicular line construction with the observation that constructing a perpendicular line is the same as bisecting a straight angle. Students make use of structure when they decide how to apply what they already know about constructions to construct perpendicular lines and angle bisectors (MP7). Students are likely to struggle to do so; this is an opportunity to encourage them to persevere in solving problems (MP1).

There is a significant connection between the angle bisector and the perpendicular bisector in triangles that is made in this lesson and built on in the next unit. For isosceles triangles, in particular, the angle bisector of the vertex between the congruent sides is the same as the perpendicular bisector of the side opposite that vertex. This connection is essential for proving that the perpendicular bisector and the set of points equidistant to 2 given points are the same set.

If students have ready access to digital materials in class, they can choose to perform all construction activities with the GeoGebra Construction tool accessible in the Math Tools or available at https://www.geogebra.org/m/VQ57WNyR.