Angles on Parallel Lines

As discussed in the presentation, various pairs of angles seem to be related to each other, and these pairs are given special names. Identify what the relationships are between the various labelled pairs with the check-boxes through modifying the position of the grey dots, and considering the resulting angle values. For a few different transversal slope values, use a protractor to measure the angles. Is the computer correctly labelling the values? What is the relationship between your assigned angle pair? Can you identify a potential general rule that applies? In developing your rule, consider the following: What effect does altering the slope of the parallel lines have on the angle relationships? What effect does altering the transversal distance between the parallel lines have on the angle relationships? What effect does altering the transversal slope relative to the parallel lines have on the angle relationships? Once you are certain of the rule for the assigned angle pair, consider the other angle pairs.

Once you have identified as many relationships as you can, consider potential approaches you could use to prove these results with minimal assumptions. As a hint, think about any line arrangements which correspond to special angle values, and whether these give interesting results. For example,  is a special angle that gives a slope of  an integer value. This particular special angle is not useful for proving the parallel line result, but other special values might be. This will be explored in detail in another lesson, but any special points of interest identified through explorations here could be useful in contributing to the proofs. Keep in mind that proof through checking every possible value for an angle pair would require checking an infinite number of values (e.g. checking doesn't prove that  will work), and is not a valid method of proof.