Lesson 6: Construction Techniques 4: Parallel and Perpendicular Lines
About this lesson
In this lesson, students use previous constructions to create new constructions. This provides an example of a more general theme of learning—that discoveries build on one another and lay the foundation for new knowledge. The subsequent sections and units will guide students through a process of establishing a theoretical foundation and building new knowledge on itself within the framework of transformational geometry. Students make use of the structure that two circles of the same radius that go through each other’s center can be used to construct perpendicular lines to think strategically about how to construct a line perpendicular to a given line that goes through a given point not on the line (MP7). As students continue to apply the method for constructing a perpendicular line to construct a parallel line, they are engaging in repeated reasoning (MP8).
In the warmup, students identify rigid motions as a review of their study of congruence in grade 8. In the cooldown, students use a construction to reflect a point across a line—although they may not realize that was what they did—as a preview of subsequent lessons.
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.
Lesson overview
 6.1 Warmup: Math Talk: Transformations (10 minutes)

6.2 Activity: Standing on the Shoulders of Giants (10 minutes)
 Digital applet in this activity
 Includes "Are you Ready for More?" extension problem

6.3 Activity: Parallel Constructions Challenge (15 minutes)
 Digital applet in this activity
 Lesson Synthesis
 6.4 Cooldown: Find the Missing Endpoint (5 minutes)
Learning goals:
 Construct a line parallel to a given line that goes through a point not on the given line.
 Construct a line perpendicular to a given line that goes through a point not on the given line.
Learning goals (student facing):
 Let’s use tools to draw parallel and perpendicular lines precisely.
Learning targets (student facing):
 I can construct a parallel line through a given point.
 I can construct a perpendicular line through a given point.
Required materials:
 Geometry toolkits
Required preparation:
 For the Math Talk: Transformations, you may wish to use GeoGebra to perform and display students’ transformations. Alternatively, you might prepare several copies of the images and notate each student’s strategy on a different copy.
 Create a display that inventories figures students know how to construct for Standing on the Shoulders of Giants.
Standards:
 This lesson builds on the standards: CCSS.8.G.A.2MS.8.G.2CCSS.HSGCO.A.1MS.GCO.1MO.8.GM.A.2aMO.G.CO.A.1
 This lesson builds towards the standards: CCSS.HSGCO.A.2MS.GCO.2CCSS.HSGCO.A.4MS.GCO.4MO.G.CO.A.2MO.G.CO.A.4
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