## 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:

1. “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”
2. “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

Unit 4: Discovering and Proving Angle Relationships
Lesson 3 of 6

## Big Idea: A tracing paper activity allows students to see [through] that that corresponding angles and alternate interior angles are congruent only if the transversal cuts across parallel lines.

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80 minutes

### Jessica Uy

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