Angle as a measure of turn

Angles in parallel lines is a topic that doesn’t usually feel too tricky to teach, but I often feel that I am just telling them “this is how it is” without giving a good explanation.

I use Geogebra a lot. It’s a powerful tool which somehow seems more powerful when you start with a blank page rather than something that has already been created. And angles in parallel lines is quick and easy enough to create on Geogebra:

Whilst doing this, I would want to move things around a bit to show that the two lines can move but stay parallel whereas the third line can move by moving the points.

This is the point where I previously might have just started measuring angles and showing what stays the same and changes as I moved the lines.

But it has struck me that this is a good opportunity to reinforce the idea of degrees as a measure of turn.  I often use a simple “Guess the angle” game like this.

I ask students to estimate the angle and show me on mini-whiteboards. Something as simple as this can cause great excitement when someone gets it exactly correct!  But it also reinforces the idea of degrees as a measure of turn from one line to another.

So, back to parallel lines, I am trying to show that the reason that alternate angles and corresponding angles are equal is because after turning one way an then back the other I end up pointing in the same direction.  And the reason co-interior angles sum to 180° is because I end up pointing in the opposite direction.

So, here is one that I did make earlier.  I have set it up so you can see the actual lines turning with the degrees increasing as they do so. I need to convince my learners that you can move from the first arrow to the second arrow by moving down the transverse line without changing direction.  Because of that, I will need to turn through the same angle to land back onto the parallel line:

And here is the same idea for Corresponding Angles:

I’d be really interested to hear any thoughts on this as a pedagogy – please leave comments below.