Monday, October 6, 2014

The Parallel Tests, Continued (Day 42)

The Parallel Tests are obviously one of the most important concepts in high school geometry. We recall that David Joyce suggests that the Parallel Tests (and Consequences) and the Triangle Congruence Theorems (ASA, SAS, etc.) should form the bulk of the course. Also, the New York State Lesson 18 recommends that teachers allow more than one day to cover this lesson. And so I will devote several days to this lesson as well.

But as I mentioned before, I won't include what typical geometry books do at this point, and write in contrived questions such as "What value must x have in order to make the lines parallel?" Such questions end up only blurring the distinction between a statement and its converse -- between the Parallel Tests and their respective Parallel Consequences. The Parallel Tests are used to prove lines parallel -- that is, they test two lines to determine whether or not they are parallel. Therefore, the emphasis should be on which lines are parallel -- not the measures of the corresponding (or whatever type of) angles. Including "What value must x have in order to make the lines parallel?" in the section on Parallel Tests is like asking what hypothesis implies the conclusion, rather than what conclusion can one derive from the hypothesis.

For the most part, Dr. Franklin Mason (the one who came up with the Parallel Tests vs. Parallel Consequences distinction) avoids this type of question. Dr. M does include the question "What relation between those two [corresponding -- dw] angles will guarantee that lines a and b are parallel?" at the end of his Section 4-1 -- but clearly as a preview for the next day's lesson where he introduces the Corresponding Angles Postulate/Test. And besides -- he only asks for the relation between the two angles 1 and 2, rather than the measure of one of the angles given that of the other.

In Dr. M's Section 4-2, his exercises begin with proofs of all of the various Parallel Tests, and then, somewhat similar to last Friday's activity, asks whether there is sufficient information to conclude that two lines are parallel. All of the angle measures are given, so the students don't have to determine what angle measures will make the lines parallel. And just as in last week's activity for advanced students, his Question 4 gives three angles in such a way as to make both pairs of line parallel.

For my worksheet, I include proofs of the Alternate Exterior and Same-Side Interior Angles Tests, since the others have already been proved in this class. Then I include Questions 1-4 just as in Dr. M, and a few more questions that involve the Two Perpendiculars Theorem ("Perpendicular Transversal Test"), since Dr. M doesn't include any such questions. (Notice that theoretically, we could have same-side exterior angles. Like same-side interior angles, if the exterior angles are supplementary, then the lines are parallel.)

That final question, the one with the impossible triangle, is an interesting one. Dr. M writes the parenthetical remark, "In your explanation, use only what we've already proven. We haven’t yet proven that the sum of the angles of a triangle is 180 degrees." Notice that a true proof would be an indirect proof:

Assume towards a contradiction that a triangle with angle measures 60 and 120 exists. By taking the side adjacent to these two angles to be a transversal for the other two sides, we see that these two angles become same-side interior angles, and since these two angles add up to 180, those other two sides become parallel. But this is a contradiction because they are given to intersect at the third vertex of the triangle, a contradiction. Notice that in hyperbolic non-Euclidean geometry, the angle sum of a triangle is less than 180 degrees, so we still can't have a triangle with both a 60 and a 120-degree angle, so this statement is provable without a Parallel Postulate. The point is that some students may remember the Triangle-Sum Theorem from middle school, but they aren't supposed to use it in this above proof. QED

Because I'm not keen on using indirect proof in class yet, I decided to include this Question 5 only as a Bonus Exercise.

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