##
* *Reflection: Developing a Conceptual Understanding
Angle Relationships Formed by Parallel Lines - Section 3: Investigation: Special Angle Relationships on Parallel Lines

Because the Common Core asks us to rethink transformations, I decided to make an Updated Parallel Lines Investigation to guide students through the investigation using a transformations lens. I had to explicitly define parallel lines in a new transformations-based way. In the past, I think my definition of parallel lines was something like, “two coplanar lines are parallel if they never intersect,” whereas this year, I have now defined parallel lines as “two coplanar lines are parallel if and only if they can map onto one another by a translation vector.” This was a big shift.

We went through the investigation, first looking at angles formed by non-parallel lines cut by a transversal. What was awesome about using a transformations lens is that when students noticed vertical angles were congruent, they could justify WHY they were congruent using transformations (each angle is a 180 degree rotation of the other around the vertex, which is the center of rotation; alternatively, each angle is a reflection of the other over the line of reflection, which passes through the vertex and bisects the other pair of vertical angles).

When students moved onto the next part of the investigation, where they look at angles formed by parallel lines cut by a transversal, they had already been primed to use transformations as a way to explain what they were seeing. It was awesome to hear students say, “if we have parallel lines, then corresponding angles are congruent because we can map them onto each other using a translation vector.” For alternate interior angles, students made one of two arguments:

- “If we have parallel lines, then alternate interior angles are congruent because we can rotate the angles 180 degrees around a point equidistant from both angles”
- “If we have parallel lines, alternate interior angles are congruent because you can translate one of the angles to its corresponding angle and then rotate this angle 180 degrees about the vertex; since translation and rotation preserve congruence, alternate interior angles are congruent.”

*Developing a Conceptual Understanding: A Transformations Approach to Angle Relationships*

# Angle Relationships Formed by Parallel Lines

Lesson 3 of 6

## Objective: Students will be able to correctly name types of angles and state that these angles are congruent depending on whether the lines cut by a transversal are parallel.

#### Crossing the Line

*15 min*

I have found that students often have a hard time differentiating between the names we assign to types of angles particularly because, at a quick glance, students often just see pairs of angles without noticing that their location relative to one other and to the lines and transversal matter greatly (**MP6**).

I like to have students work on the lesson opener, Crossing the Line, in pairs to create a greater sense of safety and allow more risk taking compared to groups of four. This is important for this lesson since it may hard for students to understand the vocabulary for types of angles. Students often just see pairs of angles without noticing that their location relative to one another, the boundary lines and transversal (**MP6**).

*Resource Citation: I want to acknowledge Cathy Humphreys, a colleague and mentor, who shared the original version of "crossing the line" with me.*

#### Resources

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I do a whole-class demonstration where I model for students how to use tracing paper. By this point, several students have conjectured that vertical angles are congruent. At this point, I use the document camera to show that vertical angles are congruent with tracing paper; then I prove vertical angles are congruent using linear pairs of angles in a whole-class discussion. My goal in this discussion is to begin developing students' understanding for how they can attempt to prove their conjectures true.

At this point, I ask students to draw two lines that are clearly not parallel and to cut these lines by a transversal since it is important to establish parallel lines as a special condition. I ask students to use their tracing paper to identify any pairs of angles that are congruent. We share out our findings as a whole class (only vertical pairs of congruent). Ideally, a student will ask, "But what happens if the lines are parallel? Would there be other pairs of angles that are congruent?" We then transition into the parallel lines investigation.

*expand content*

I use the document camera to project exactly how I want students to start the investigation. (We operate under the assumption that the lines on our notebook paper are parallel.) I ask students to use their tracing paper to identify pairs of angles that are congruent and to name these angles by using their work from Crossing the Line. Ultimately, students write a conjecture for the types of angles that are congruent given parallel lines.

While students work through this investigation, I circulate the room to assist them with using tracing paper and naming pairs of angles. Inevitably, some groups will finish conjecturing about the types of angle pairs that are congruent before other groups; I offer early finishers an extension: identify pairs of angles that are supplementary and prove your conjecture.

*expand content*

I debrief the Parallel Lines Investigation by calling on different groups to give an example of a pair of angles that are congruent and to name the angle relationship. Groups share out examples of congruent alternate interior angles, alternate exterior angles, and corresponding angles, which I record on the whiteboard.

After we complete the parallel lines conjecture as a whole class, I then ask groups who worked on the extension to come up to the whiteboard to share their conjecture about the type of angle pairs that is supplementary and to convince us that their reasoning makes sense. To make this presentation useful to everyone in the class, I ask questions like, "how do you know?" which requires the presenters to back up their claims. Additionally, I ask students in the audience to re-phrase or summarize what they have heard for the whole class, which adds clarity to the ideas being shared.

We formalize our understanding from the investigation by taking notes to close up the lesson.

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- UNIT 1: Creating Classroom Culture to Develop the Math Practices
- UNIT 2: Introducing Geometry
- UNIT 3: Transformations
- UNIT 4: Discovering and Proving Angle Relationships
- UNIT 5: Constructions
- UNIT 6: Midterm Exam Review
- UNIT 7: Discovering and Proving Triangle Properties
- UNIT 8: Discovering and Proving Polygon Properties
- UNIT 9: Discovering and Proving Circles Properties
- UNIT 10: Geometric Measurement and Dimension
- UNIT 11: The Pythagorean Theorem
- UNIT 12: Triangle Similarity and Trigonometric Ratios
- UNIT 13: Final Exam Review