I think back to David Joyce, who criticized a certain Geometry text. He writes:

**Chapter 6**is on surface areas and volumes of solids. It is strange that surface areas and volumes are treated while the basics of solid geometry are ignored.

*In summary, there is little mathematics in chapter 6. Most of the results require more than what's possible in a first course in geometry. Surface areas and volumes should only be treated after the basics of solid geometry are covered.*

And so Chapter 6 of the Prentice-Hall text is just like Chapter 10 of the U of Chicago text. Joyce laments that students don't learn "the basics of solid geometry" before surface area and volume. But we can't fault Prentice-Hall for this. Even before the Common Core, most states' standards expected students to learn the 3D measurement formulas and hardly anything else about 3D solids.

We can't quite be sure what Joyce means by "the basics of solid geometry." But it's possible that some of what he wants to see actually appears in Chapter 9 of the U of Chicago text. Thus, by teaching Chapter 9, we are slightly closer to Joyce's ideal Geometry course.

And incidentally, there is one Common Core Standard in which 3D solids are mentioned, but not surface area of volume. We'll look at this standard in more detail next week, in Lesson 9-4.

Lesson 9-1 of the U of Chicago text is called "Points, Lines, and Planes in Space." The first three sections of Chapter 9 are the same in both the old Second and modern Third Editions. (As it turns out, the new Third Edition squeezes in surface area in Chapter 9, saving only volume for Chapter 10.)

The heart of this lesson is the Point-Line-Plane Postulate. We first see this postulate in Lesson 1-7, but now it includes parts e-g:

Point-Line-Plane Postulate:

a. Given a line in a plane, there exists a point in the plane not on the line. Given a plane in space, there exists a point in space not on the plane.

b. Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1.

c. Through any two points, there is exactly one line.

d. On a number line, there is a unique distance between two points.

e. If two points lie in a plane, the line containing them lies in the plane.

f. Through three noncollinear points, there is exactly one plane.

g. If two different planes have a point in common, then their intersection is a line.

There are several terms defined in this lesson --

*intersecting planes, parallel planes, perpendicular planes*, and

*a line perpendicular to a plane.*

*Actually, I'm still thinking about Joyce's "basics of solid geometry." I know that his website also links to Euclid's*

*Elements*. So Book XI of Euclid is a reasonable guess as to what Joyce wants to see taught in class:

https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html

Let's look at some of the definitions and propositions (theorems) here and compare them to the contents of Lesson 9-1. We'll start with Definition 3, since Definitions 1 and 2 will actually appear in tomorrow's Lesson 9-2.

**Definition 3.**

*at right angles*to a plane when it makes right angles with all the straight lines which meet it and are in the plane.

**Definition 4.**

*at right angles*to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane.

**Definition 5.**

*inclination*of a straight line to a plane is, assuming a perpendicular drawn from the end of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the end of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up.

Definition 3 appears in Lesson 9-1 as a line perpendicular to a plane. Definition 4 appears in this lesson as perpendicular planes. But Definition 5, the angle between a line and a plane, is only briefly mentioned in the U of Chicago text.

**Definition 6.**

*inclination*of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the intersection at the same point, one in each of the planes.

**Definition 7.**

*similarly inclined*to a plane as another is to another when the said angles of the inclinations equal one another.

**Definition 8.**

*Parallel planes*are those which do not meet.

Again, Definitions 6 and 7 are about the angle between two planes, which is not discussed in our text at all. Definition 8, of course, appears in today's lesson -- but just as with lines, the U of Chicago uses an "inclusive" definition of parallel where a line or plane can be parallel to itself. Intersecting planes (our remaining term) are implied in Definition 8 as planes that are not parallel.

Let's look at the propositions (theorems) now:

**Proposition 1.**

This is essentially part e of our Point-Line-Plane Postulate. Euclid calls it a proposition (or theorem) and even provides a proof, but Joyce argues that the proof is unclear. Thus we might as well consider it to be a postulate.

**Proposition 2.**

This is essentially part f of our Point-Line-Plane Postulate. If

*A*,

*B*, and

*C*are the three noncollinear points mentioned in part f, then we can take lines

*AB*and

*AC*to be the two intersecting lines that appear in Proposition 2, and triangle

*ABC*to be the triangle mentioned in this proposition.

**Proposition 3.**

This is very obviously part g of the Point-Line-Plane Postulate. Joyce points out that this is yet another postulate, and that it holds only in 3D, not 4D and above.

**Proposition 4.**

According to Joyce, this is the first true theorem in Book XI. It asserts that if a line intersects a plane and is perpendicular to two lines in the plane, then the line is perpendicular to the whole plane. Joyce points out that the proof is a bit long, but it works. Theoretically, our students can prove it using the new Point-Line-Postulate and theorems from the first semester of the U of Chicago text.

https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/propXI4.html

Here is a modern rendering of this proof. The idea is that line

*l*is perpendicular to each of two lines

*m*,

*n*, in plane

*P*, with all lines concurrent at point

*E*. Our goal is to prove that line

*l*is perpendicular to the entire plane

*P*by showing that, if

*o*is any other line in plane

*P*with

*E*on

*o*, then

*l*must be perpendicular to

*o*as well.

Given:

*l*perp.

*m*,

*l*perp.

*n*with

*l*,

*m*,

*n*all intersecting at

*E*

Prove: Line

*l*is perpendicular to plane that contains

*m*and

*n*.

Proof:

Statements Reasons

1. bla, bla, bla 1. Given

2. Choose

*A*,

*B*on

*m*and 2. Point-Line-Plane part b (Ruler Postulate)

*C*,

*D*on

*n*so that

*AE*=

*EB*=

*CE*=

*ED*

3. Exists plane

*P*containing

*m*,

*n*3. Point-Line-Plane part f (3 noncollinear

*A*,

*C*,

*E*)

4. Choose

*F*on

*l*, 4. Planes contain lines and lines contain points.

and

*o*in plane

*P*s.t.

*E*on

*o*

5. Lines

*AD*,

*o*intersect at

*G*, 5. Line Intersection Theorem

Lines

*BC*,

*o*intersect at

*H*

6. Angle

*AED*= Angle

*CEB*6. Vertical Angle Theorem

7. Triangle

*AED*= Triangle

*CEB*7. SAS Congruence Theorem [steps 2,6,2]

8.

*AD*=

*CB*, Angle

*DAE*=

*EBC*8. CPCTC

9. Angle

*AEG*= Angle

*BEH*9. Vertical Angle Theorem

10. Triangle

*AEG*= Triangle

*BEH*10. ASA Congruence Theorem [steps 8,2,9]

11.

*GE*=

*EH*,

*AG*=

*BH*11. CPCTC

12.

*FE*=

*FE*12. Reflexive Property of Congruence

13. Triangle

*AEF*= Triangle

*BEF*13. SAS Congruence Theorem [steps 2,1,12]

14.

*FA*=

*FB*14. CPCTC

15. Triangle

*CEF*= Triangle

*DEF*15. SAS Congruence Theorem [steps 2,1,12]

16.

*FC*=

*FD*16. CPCTC

17. Triangle

*FAD*= Triangle

*FBC*17. SSS Congruence Theorem [steps 8,14,16]

18. Angle

*FAD*= Angle

*FBC*18. CPCTC

19. Triangle

*FAG*= Triangle

*FBH*19. SAS Congruence Theorem [steps 11,18,14]

20.

*FG*=

*FH*20. CPCTC

21. Triangle

*GEF*= Triangle

*HEF*21. SSS Congruence Theorem [steps 11,12,20]

22. Angle

*GEF*= Angle

*HEF*22. CPCTC

23.

*EF*perp.

*GH*(i.e.,

*l*perp.

*o*) 23.

*GEF*,

*HEF*are congruent and a Linear Pair 24. Line

*l*perpendicular to plane

*P*24. Definition of line perpendicular to plane

So this is probably what Joyce wants to see more of. Propositions 5 through 19 aren't very much different from this one. But as I wrote above, our students will find such proofs difficult -- we had to prove

*seven*different pairs of triangles congruent above, in three dimensions to boot. No modern text teaches such theorems, since no state standards -- pre- or post-Core -- require them.

The proof works -- the definition in Step 24 is satisfied because

*o*is arbitrary. But notice that in the drawing at the above link, Euclid assumes that

*G*, the point where lines

*AD*and

*o*intersect, is between

*A*and

*D*. But this is irrelevant for the proof -- all the congruence theorems used in the proof still work even if

*G*isn't between

*A*and

*D*.

What's worse, of course, is if

*o*is

*parallel*to

*AD*. Notice that

*AD*| |

*BC*(since

*DAE*and

*EBC*, the angles proved congruent in Step 8, are alternate interior angles), so

*o*could be parallel to both. But that's no problem -- just switch points

*C*and

*D*in that rare case, and the proof still works.

Since this is a brand-new chapter for the blog, I have no worksheets for it. And so here is a newly created worksheet for Lesson 9-1.