# Construct Parallel Lines Through a Point Not on the Line

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## Objective

SWBAT to construct a parallel line through a point not on the line using at least three different methods.

#### Big Idea

By making connections to the last unit--angle relationships--students will make sense of how they can perform this construction.

## Warm-Up: Triangle Triumph

25 minutes
In Triangle Triumph, students review and apply the definitions of angle bisector, medians, perpendicular bisectors, and altitudes when sketching these for each triangle.  Since students' understanding of each of these terms is still new, the direction, "All of the segments will intersect at a common point" helps them to ensure they are sketching correctly.

A question I hope emerges is, "Are there times when an angle bisector might also be a median (or perpendicular bisector or altitude) in a triangle?"  I have found that taking time to let students wonder about what types of triangles have angle bisectors, medians, perpendicular bisectors, and/or altitudes concurrent engages them in proof-like thinking, where they have to consider different cases of triangles to test.

Triangle Triumph previews a lesson later in the unit where students will discover points of concurrency (incenter, centroid, circumcenter, orthocenter) and how their location depends on the kind of triangle they are working with.

## Proof Practice 3

10 minutes

In Proof Practice 3, students again focus on the given information from which they can draw logical conclusions.

I ask a student to present their ideas for the proof.  Like Proof Practice 2, I project the diagram on the whiteboard so that students can write, use color to clarify their ideas, and invite other students in the audience to comment on their work.  Since being able to clearly explain one's ideas for the proof is difficult, I make sure to use encourage students to "take risks" so they can get feedback and push their understanding further.

## Construction: Construct Parallel Lines

25 minutes

To begin this activity I pass out tracing paper to students and ask them to try to construct parallel lines by folding.  As students experiment in their groups, I circulate the room, looking for a student who will share the idea of folding to construct a perpendicular, and then folding again to construct a perpendicular to the perpendicular.

After I find a student to presents the construction, and he/she presents, I will formalize the idea by getting students to complete the conjecture: "If two lines are perpendicular to the same line, then they are ________."

Next, I ask students to show this idea with their compass and straightedge.  I then ask them to think back to the last unit, where we explored angle relationships on parallel lines, and to consider the kinds of angles that must be congruent in order for lines to be parallel.  I give students time in their groups to experiment with these ideas with the goal of constructing parallel lines.

Ideally, I want students to say that two lines are guaranteed to be parallel if formed by congruent corresponding angles or congruent alternate interior angles.  I circulate the room as students experiment, finding a student volunteer for each of these constructions.  I then re-cap all three compass and straightedge constructions for the students.

## Group Constructions Quiz

45 minutes

This is a great time in the unit to have a Constructions Group Quiz.  Students have such varying levels of understanding; some are better at recalling certain "steps" to take while others are better at focusing on the ideas needed to perform certain constructions.

By creating group accountability--and requiring that each person does his/her own constructions at all times--I have found that students finish the quiz with a much stronger understanding of how to perform foundational geometry constructions.

While groups work on the Group Constructions Quiz, I alert students that I will circulate the room, tallying the groupwork behaviors I see at each group.  This is important because every group needs to make sure everyone understands at a high level.  I tell students that after the quiz, they will reflect on their behavior and how it impacted their understanding.

## Groupwork Reflection

5 minutes

Since I asked students to work on this group quiz, I hold them accountable for their behaviors by asking them to reflect on their group work in writing.

Here is an explanation of my Group and Individual Reflection strategy.