Wentworth's Plane and Solid Geometry · Book VI, Props. VIII–X · Book 7: Book VI: Lines and Planes in Space
83. Lines parallel to planes; planes perpendicular to one line
Goal: Your student can prove when a line is parallel to a plane and that two planes perpendicular to the same line are parallel.
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
Propositions VIII–X (§§ 522–527) and Exercises 604–605.
The lesson, step by step
Warm up
Ask: State Prop. VII. Two lines perpendicular to the same plane are ...?
The new idea
If two straight lines are parallel, every plane containing one of the lines, and only one, is parallel to the other line.
If a straight line is parallel to a plane, the intersection of the plane with any plane passed through the given line is parallel to that line.
Two planes perpendicular to the same straight line are parallel.
Draw the figure
Lay a pencil CD on the table and hold another pencil AB above it, parallel. The table (a plane containing CD) is parallel to AB.
For Prop. IX, tilt a card so that it contains AB; where the card meets the table is a line parallel to AB.
Walk through the proof
VIII: AB and CD lie in one plane, which meets MN only along CD. If AB met MN at all, it would have to meet CD; but it is parallel to CD.
IX: AB and CD are in the same plane; CD lies in MN, and AB cannot meet MN, so AB cannot meet CD.
X: if the two planes met, from a point of their meeting there would be two planes perpendicular to the same line, which § 510 forbids.
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
Ex. 604. Find the locus of points in space equidistant from two given parallel planes.
Ex. 605. Find the locus of points in space equidistant from two given points and also equidistant from two given parallel planes.
Wrap up
Ask: Name two planes in this room that are parallel. What line is perpendicular to both? (The floor and the ceiling; any upright edge.)
Answers to the book's problems
Ex. 604: the plane parallel to both, halfway between them. Ex. 605: the plane of Ex. 604 meets the plane perpendicular to the line joining the two points at its middle point (§ 517); the locus is their line of intersection (if those two planes happen to be parallel there is no such point, and if they coincide the locus is the whole plane).
Worked out for this site from the scan. If a number in your copy differs, trust the book.