# Lesson 6

Construction Techniques 4: Parallel and Perpendicular Lines

### Lesson Narrative

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 warm-up, students identify rigid motions as a review of their study of congruence in grade 8. In the cool-down, 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

### Learning Goals

Teacher Facing

• 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.

### Student Facing

• Let’s use tools to draw parallel and perpendicular lines precisely.

### 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.

### Student Facing

• I can construct a parallel line through a given point.
• I can construct a perpendicular line through a given point.

Building On

Building Towards

### Glossary Entries

• angle bisector

A line through the vertex of an angle that divides it into two equal angles.

• circle

A circle of radius $$r$$ with center $$O$$ is the set of all points that are a distance $$r$$ units from $$O$$

To draw a circle of radius 3 and center $$O$$, use a compass to draw all the points at a distance 3 from $$O$$.

• conjecture

A reasonable guess that you are trying to either prove or disprove.

• inscribed

We say a polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. We say a circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.

• line segment

A set of points on a line with two endpoints.

• parallel

Two lines that don't intersect are called parallel. We can also call segments parallel if they extend into parallel lines.