Parallel Line Theorems

This year, I started to use a more visual aspect of teaching parallel line theorems. In the past, I have just given very technical location comparisons that are not intuitive and are hard to see – especially when there are ‘trick’ questions where the angles actually have no relationship.

Before:

• Vertical angles: “angles on opposite sides of intersecting lines”
• Alternate Interior Angles: “Interior angles, non-adjacent, and on opposite sides of the transversal”
• Alternate Exterior Angles: “Exterior angles, non-adjacent, and on opposite sides of the transversal”
• Corresponding Angles: “Angles in the same ‘position’ at each intersection”
  • I give them a reference to “news correspondents”, who are doing the “same” job, just in a different location
• Consecutive Interior Angles: “Interior angles, non-adjacent, and on the same side of the transversal

Now:

• Vertical Angles: Make an X shape
• Alternate Interior Angles: Make a Z shape
• Corresponding Angles: Make an F shape
• Co-interior Angles: Make a C shape
• Linear Pair: Make a T shape

To be clear, I still offer the technical definition for students who work better with exact specifications like that, and to ensure that they still understand that this isn’t simply made up relationships. Unfortunately, alternate exterior does not offer a good letter shape, but the students came up with ‘Reversed L’ shape as a supplement.

My students are really taking to this, and are using their diagrams more closely than past years. They are also less thrown off by ‘trick’ questions, because angles without a relationship (excluding alternate exterior angles) do not make a nice letter shape.

This visualization will come in handy when we move to triangles, because they can still see the shapes without the extended lines.

I also switched to the UK version of ‘consecutive interior’. I don’t know if that’s a problem but the students seem to remember it easier than ‘consecutive-interior’ or ‘same-side interior’, even though they still have that listed in their chapter dictionaries.