Wentworth's Plane and Solid Geometry · Book VI, §§ 492–506 (Prop. I) · Book 7: Book VI: Lines and Planes in Space
80. Planes, and what determines a plane
Goal: Your student can define a plane, explain what it means for lines or points to determine a plane, and prove that two planes meet in a straight line.
You'll need
- The book, open to these pages
- A ruler (straightedge) and a pair of compasses
- Sharp pencil, eraser and plain paper (or a geometry notebook)
- Two or three stiff cards (to stand for planes) and a few pencils or straws (to stand for lines)
What's in the book
The opening of Solid Geometry: the definition of a plane, the five ways a plane is determined (§§ 494–498), definitions of a line perpendicular, parallel or oblique to a plane, and Proposition I.
The lesson, step by step
Warm up
Ask: In plane geometry, how many straight lines can be drawn through two points? (One.) Today we leave the flat page and work in space.
The new idea
A plane is a surface such that a straight line joining any two points in it lies wholly in the surface.
Three points not in a straight line determine a plane.
If two planes cut each other, their intersection is a straight line.
The five corollaries (§§ 494–498): one line does not determine a plane; a line and a point outside it, three points, two intersecting lines, or two parallel lines each do.
Draw the figure
Hold a card on one pencil and let it spin: one line does not fix it. Add a second pencil crossing the first, and the card can no longer move.
Then show why a three-legged stool never wobbles but a four-legged one can: three points determine a plane.
The book draws a plane as a parallelogram, a flat sheet seen at a slant. Practice drawing two such planes cutting each other.
Walk through the proof
Prop. I: take two points A and B common to both planes. The line AB lies in both (§ 492). No point outside AB can be in both, for a line and a point outside it determine only one plane (§ 495). So the intersection is exactly the line AB.
Your student proves it back
Close the book. Your student sketches the figure from memory (a simple perspective drawing is fine), letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("by § 501", "by § 507"). Point to each line and plane as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Exercises
Read the definitions in §§ 499–505 aloud. Ask your student to point to a line perpendicular to the floor, one parallel to it, and one oblique to it, somewhere in the room.
Wrap up
Ask: Why does a door swing on its hinges but stop being movable once it touches the doorframe? (Hinge line plus one more point fixes the plane.)
Tip
Solid geometry is much easier with real objects. Keep the cards and pencils on the table for the whole of Book VI and build every figure before drawing it.