Now let's return to geometry. It's now time for the slopes of parallel and perpendicular lines. In Lesson 3-4 of the U of Chicago text, we see the following theorem:
Parallel Lines and Slopes Theorem:
Two nonvertical lines are parallel if and only if they have the same slope.
Of course, Common Core now expects us to prove this theorem. The text writes:
"A proof requires quite a bit of algebra, and is omitted."
Actually, I don't see that much algebra needed for the proof at all. In fact, the proof of the theorem is very similar to yesterday's proof of the well-definition of slope. This time, instead of the four points (x_1, y_1), (x_2, y_2), (x_3, y_3), and (x_4, y_4) all being on the same line, we place the first two points on the first of two parallel lines and the other two on the other line.
But now we're given the two parallel lines, not the points so we are allowed to choose any points on the line that we wish. Since we want to set up corresponding angles, we begin by selecting a line to be the transversal and then let (x_1, y_1) and (x_3, y_3) both lie on this transversal. Any line may be chosen as the transversal as long as it intersects both given parallel lines.
Well, since we're given that the two parallel lines aren't vertical (as vertical lines have no slope), we can let any vertical line be the transversal. In fact, why don't we choose the easiest vertical line to work with -- the y-axis? These means that our two points (x_1, y_1) and (x_3, y_3) are now the two points (0, y_1) and (0, y_3). Notice that these two chosen points are where the two lines intersect the y-axis -- that is, they are the y-intercepts. So let's change those x's to b's -- and so the two points in our proof will be labeled (0, b_1) on the first line and (0, b_2) on the second line.
Notice that by letting the transversal be the y-axis and two of our points have 0 as their first coordinate, we're demonstrating the hallmarks of a good coordinate proof. We want to place as many key lines and segments along the coordinate axes as possible.
Now let's consider our second points on each line. Since our first points have 0 as the x-coordinate, let's let the second points have 1 as the x-coordinate. So these points are (1, y_1) and (1, y_2).
Now just as we did for the Distance Formula and Slope Well-Definition proofs, we add the points E(0, y_1) and F(0, y_2) to form two right triangles. Notice that these two right triangles aren't merely similar, as they were in the Slope Well-Definition proof -- they are in fact congruent.
Given: k | | l, A(0, b_1), B(1, y_1) on k, C(0, b_2), D(1, y_2) on l
Prove: slope of k = slope of l
Statements Reasons
1. k | | l, etc. 1. Given
2. Angle BAE = DCF 2. Corresponding Angles Consequence
3. Angle AEB = CFD 3. All right angles are congruent.
4. BE = DF 4. All segments of unit length are congruent.
5. Tri. AEB, CFD cong. 5. AAS Congruence Theorem
6. AE = CF 6. CPCTC
7. slope k = AE/1 = AE, 7. Definition of slope
slope l = CF/1 = CF
8. slope k = slope l 8. Substitution
We see that there are several cop-outs in the above proof to avoid excessive formalism. Technically speaking, "all segments of unit length (that is, of length 1) are congruent" is not a valid reason in a formal proof. We actually need four steps here -- the first to prove that BE = 1, the second to prove that DF = 1, the third to prove that BE = DF, and final to prove that these equal lengths imply that the segments are congruent. Arguably, we'd need to do the same with the right angles. Also, several steps are skipped in finding the slopes of both lines -- it may be instructive to calculate the slope by plugging the values into the formula to show that the slope of k is y_1 - b_1, while the slope of l is y_2 - b_2, and these are the values proved equal in the CPCTC step. (Notice that the theorem to be proved is stated as "if and only if," so we need to prove the converse -- that is, if the slopes are equal, then the lines are parallel. A proof of the converse requires SAS rather than AAS. For the converse, our given lines have the same slope -- so we can choose the familiar letter m to denote that slope!)
But there is one more subtlety that we need to discuss. We assumed that AE = y_1 - b_1 in the proof above, but actually by the definition of 1D distance, AE = |y_1 - b_1|, not y_1 - b_1. That is, we've ignored the sign of the slope in the above proof! Without this, it could be the fact that parallel lines have slopes with opposite sign. Even the proof of the well-definition of slope from last week isn't really complete since there was no mention of any sign.
What we actually need here is that extra postulate that Dr. Wu gave us -- Plane Separation. Our lines have constant slope because we showed that, if we start at a point on the line and can move v units up and w units right to arrive back on the line, then the same is true no matter where we start. There is no point where we can move w units right, but we have to go v units down instead of up in order to get back on the line.
The reason is that set of all points w units right of the line is a translation image of the line, and as a translation image, it is parallel to its pre-image. So it can't cross from one half-plane defined by the original line to the other. In particular, it can't cross from the half-plane containing the line v units above the starting line to the half-plane containing the line v units below the starting line. And so it's the Plane Separation Postulate that ultimately guarantees that slope is well-defined and that parallel lines have equal slope.
As Wu often points out, this is a subtle point that is ignored in most math classes. The problem, of course, is that we're about to jump to the case where the sign of the slope makes a huge difference: the slope of perpendicular lines.
Now here's a mini-activity that I like to begin with to illustrate the slopes of parallel and perpendicular lines. I choose two students and ask them to draw a line that is parallel to my given line, which has a predetermined slope of, let's say, 2/3. I inspect the two lines and declare the student whose slope more clearly represents going up 2 and right 3 units to be the winner.
Then I ask the students to draw a line that is perpendicular to my given line. This will be harder for someone who hasn't learned the formula yet, and so it's likely that both lines will be far away from a truly perpendicular line. And so I come in and draw my own perpendicular line. To do this, I point out that since perpendicular means 90 degrees, I rotate my paper 90 degrees and then redraw the original line on the rotated paper. I make sure that my new line also has rise of 2 and run of 3. Then I rotate the paper -90 degrees to retain the original position, and notice that the rise of 2 rotates to a run of 2, and the run of 3 rotates to a rise of -3. So the new slope is -3/2 -- the opposite reciprocal of the original slope. And then I declare myself to be the winner!
I don't include a formal proof of the Perpendicular Lines and Slopes Theorem on the worksheet, but we can figure it out -- instead of a translation, we use a rotation. We rotate the given line AB 90 degrees about some point (say the origin) to obtain a new line A'B', and we see that its slope is the opposite reciprocal of the original slope, since rise and run become run and negative rise. Then, by the Two Perpendiculars Theorem, any other line CD that is perpendicular to AB must be parallel to A'B', and so has that same slope as A'B' (which is the opposite reciprocal of that of AB). It may be possible to skip a step here and simply rotate about the point where the two given lines AB and CD intersect -- then the image of line AB would be line CD itself (but A doesn't necessarily rotate into either C or D).
By the way, since I posted this lesson last year, I've learned more about this kind of proof. You see, the triangles ABC and DEF have a special name -- slope triangles. The slope triangle of any line is a right triangle whose height equals the rise of the line and whose base equals the run of the line. Some authors requires the height to be the right leg of the triangle, while others allow the height to be on either the right or the left. The slope triangles in the above proof have their heights on the left side.
Then the three main properties of slope are determined by actions on the slope triangles:
-- Slope is well-defined: a dilation maps a slope triangle of a line to another slope triangle of the line.
-- Parallel lines have the same slope: a translation maps a slope triangle of a line to a slope triangle of a parallel line.
-- Perpendicular lines have opposite reciprocal slopes: a rotation maps a slope triangle of a line to a slope triangle of a perpendicular line.
One last thing I want to mention in this post -- when I was young, I remember being fascinated by an old college chemistry text. Some of the problems in this text required algebra, and one of them was a slope question which used the notation slope = delta-y / delta-x -- where the symbol "delta" stands for something like "difference" or "the change in." So when my Algebra I teacher assigned some slope problems, I started writing "delta" in all of my answers.
I still remember her response: "Delta is one of my favorite Greek letters." Believe it or not, I've since noticed that nowadays, some Algebra I teachers actually use "delta" when teaching slope!
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