Wentworth's Plane and Solid Geometry · Book VI, Props. XI–XIII · Book 7: Book VI: Lines and Planes in Space
84. Parallel planes
Goal: Your student can prove the basic facts about parallel planes: they cut a third plane in parallel lines, are everywhere equally distant, and share perpendiculars.
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 XI–XIII (§§ 528–533) with corollaries, and Exercises 606–609.
The lesson, step by step
Warm up
Ask: Two planes perpendicular to the same line are ... ? (Parallel.)
The new idea
The intersections of two parallel planes by a third plane are parallel lines.
Parallel lines included between parallel planes are equal.
A straight line perpendicular to one of two parallel planes is perpendicular to the other also.
If two intersecting straight lines are each parallel to a plane, the plane of these lines is parallel to that plane.
Draw the figure
Hold two cards parallel, one above the other, and slide a third card through both at a slant. Mark the two lines where it meets them: they are parallel.
Shelves in a bookcase show § 530: parallel planes are everywhere equally distant.
Walk through the proof
XI: the two lines are in one plane and in two planes that never meet, so they never meet (§ 103).
XII: pass two planes through AB; they cut the parallel planes in parallel lines (§ 528). AB is perpendicular to the lines in MN, so to their parallels in PQ (§ 107), so to PQ (§ 507).
XIII: drop AB perpendicular to PQ; the lines BE and BF where planes through AB cut PQ are parallel to AC and AD (§ 525). So AB is perpendicular to MN as well, and planes perpendicular to one line are parallel (§ 527).
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
Discuss the loci together:
Ex. 606. Find the locus of all lines drawn through a given point, parallel to a given plane.
Ex. 608. Find the locus of a point in space equidistant from three given points not in a straight line.
Wrap up
Ask: Why are the rungs of a straight ladder leaning on a wall all parallel to the floor? (Each rung is parallel to a line on the floor; think of § 522.)
Answers to the book's problems
Ex. 606: the plane through the point parallel to the given plane (§ 532). Ex. 607: the line in which the given plane meets the plane perpendicular to the line joining the two points at its middle point. Ex. 608: the straight line through the centre of the circle passing through the three points, perpendicular to their plane (§ 516). Ex. 609: reflect one point in the plane (go the same distance through to the other side); join the image to the other point; where this line crosses the plane is the point.
Worked out for this site from the scan. If a number in your copy differs, trust the book.