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The standards expect students to prove the slope criteria:
G-GPE 5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
Most of the proofs I’ve seen involve similarity; however, I was thinking that one could also prove the slope criteria using (just) rigid motions. But then I recalled that another standard expects students to
G-CO 4 Develop definitions of rotations, reflections, and translations in terms of …parallel lines…
Which seems to beg the question. If the criterion for the slope of parallel lines is proven using transformations, but transformations are themselves defined in terms of parallel lines, there seems to be some circularity going on.
I am hoping someone here could shed some light on: (A) How this isn’t circular and (B) What a good proof of the slope criterion for parallel lines would look like. Wu doesn’t address the topic, so far as I know.
To sketch the answer to (B) first: given two non-vertical parallel lines, draw a vertical transversal and a horizontal transversal so that they intersect at a point not on either of the lines. The transversals form a right triangle with each of the lines, and the slope of each line is the quotient of the lengths of the vertical and horizontal sides. Using the fact that alternate interior angles are congruent, you can use the AAA criterion to show that these two triangles are similar. That means the corresponding sides are related by the same scale factor, so the quotients of the lengths of the horizontal and vertical sides are the same.
As to (A), I agree there would be a danger of circularity of you defined the notion of parallel lines in terms of slope. So it would be a good idea not to do that! A standard definition is to say that two lines are parallel if they are either identical or do not intersect at all.
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