Wentworth's Plane and Solid Geometry · Book VI, Props. XX–XXII · Book 7: Book VI: Lines and Planes in Space
88. Planes perpendicular to a third plane; bisecting planes
Goal: Your student can prove that two planes perpendicular to a third meet in a perpendicular line, that the bisecting plane is the locus of equal distances, and that one perpendicular plane can be passed through a 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
Propositions XX–XXII (§§ 556–562) with corollaries, and the definitions of the projection of a point and of a line on a plane.
The lesson, step by step
Warm up
Ask: State Prop. XIX.
The new idea
If two intersecting planes are each perpendicular to a third plane, their intersection is also perpendicular to that plane.
Every point in a plane which bisects a dihedral angle is equidistant from the faces of the angle.
Through a given straight line not perpendicular to a plane, one plane, and only one, can be passed perpendicular to the given plane.
Draw the figure
Stand two cards upright on the table, touching along an edge: the edge where they meet is upright. This is the corner of a room (§ 558).
Shine a flashlight straight down on a slanting pencil: its shadow is its projection (§ 562).
Walk through the proof
XX: a perpendicular to PQ at B lies in both planes (§ 552), so it is their intersection.
XXI: pass a plane through PE and PF. It is perpendicular to both faces, so to their edge (§ 557); the angles POE and POF measure the equal halves of the dihedral angle, so the right triangles POE and POF are equal (§ 141).
XXII: from a point of AB drop a perpendicular HK to MN; the plane of AB and HK is perpendicular to MN (§ 554). Two such planes would make AB perpendicular to MN (§ 556), which it is not.
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.
Wrap up
Ask: In the corner of a room, how many lines are perpendicular to the floor? to each wall? (§ 558: each of the three edges is perpendicular to the other two.)