G.GPE.5

[Donna Young]

What is the depth to which this standard should be understood?  Does the standard mean that students should be able to take a given of two lines that are parallel and use generic coordinates to prove that their slopes are the same?

[Bill McCallum]

Here is the standard:

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).

I’m not sure exactly what you mean by “using generic coordinates” but here’s the way I see this working. Take two parallel lines and cut them with a vertical transversal and a horizontal transversal (let’s assume the lines themselves are not vertical or horizontal). These lines form a “slope triangle” with each line: a right triangle whose ratio of rise over run gives the slope of the line. (It would be easier if I drew a picture, but I’m lazy.) Using the theorems about corresponding angles and AAA similarity, you can show these two triangles are similar, and that means the slopes are the same.

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