In many ways this is a light chapter. Whereas the modern Third Edition includes surface area in Chapter 9 (Lessons 9-9 and 9-10), my old Second Edition stops after Lesson 9-8. Then again, students who have trouble visualizing three dimensions will struggle on tomorrow's test.

Well, at least our students won't have any Euclid on their test. Let's return to his next proposition:

**Proposition 12.**

Yesterday we construct the perpendicular from a point

*not on*the plane, and now we construct the perpendicular from a point

*on*the plane. This construction uses yesterday's as a subroutine.

This construction is even sillier than yesterday's to perform inside a classroom. This time, we have a point

*A*on the floor and we wish to find a point directly above it. First, we label point

*B*on the ceiling and use yesterday's construction to find a point

*C*directly below it. Then we construct the line through

*A*that is parallel to

*A*,

*B*, and

*C*. This plane is neither the floor nor the ceiling, but an invisible vertical plane that isn't even necessarily parallel to a wall. There is no reasonable way to perform this construction in the classroom.

And so there's no way that our students can physically perform this construction. There will be nothing like this on tomorrow's test, even though ironically, it would be easier to answer test questions about this construction than physically perform it.

Notice that the U of Chicago text doesn't actually provide the construction for drawing a parallel to a line through a point not on the line (which is a simple plane construction). The only way implied in the text to perform this construction is to make two perpendicular constructions (which we did in yesterday's post).

Many texts that teach the construction of parallel lines use copying an angle (as in corresponding or alternate interior angles). Lesson 7-10 of the Third Edition is on constructions, and duplication of an angle is given, but still no parallel lines. (I also see some DGS "constructions" mentioned there -- I wonder whether this is similar to Euclid the Game, as alluded to in yesterday's post.)

If you must, here is a modernization of the proof of Proposition 11:

Given: the segments and angles in the above construction.

Prove:

*P*

*Proof:*

Statements Reasons

1. bla, bla, bla 1. Given

2.

*P*2. Perpendicular to Parallels (spatial, last week's Prop 8)

The proof is trivial since both

*P*and

*if*) we were to prove this in the classroom, it would be more instructive to show Euclid's proof of both yesterday's and especially today's propositions directly, than to attempt to convert the proofs to two columns.

Anyway, here is the Review for Chapter 9 Test.

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