Wentworth's Plane and Solid Geometry · Book VI, Props. XVII–XIX · Book 7: Book VI: Lines and Planes in Space
87. Measuring dihedral angles; perpendicular planes
Goal: Your student can explain why the plane angle measures a dihedral angle, and prove the theorems about perpendicular planes.
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 XVII–XIX (§§ 549–555): the ratio of dihedral angles (with the commensurable and incommensurable cases), and perpendicular planes.
The lesson, step by step
Warm up
Ask: What is the plane angle of a dihedral angle?
The new idea
Two dihedral angles have the same ratio as their plane angles.
If two planes are perpendicular to each other, a straight line drawn in one of them perpendicular to their intersection is perpendicular to the other plane.
If a straight line is perpendicular to a plane, every plane passed through this line is perpendicular to the plane.
§ 550: so the plane angle may be taken as the measure of the dihedral angle.
Draw the figure
Stand a card upright on the table (a plane perpendicular to the table). Draw a line on the card at right angles to its bottom edge: it stands straight up from the table.
Walk through the proof
XVII: as in Book II, split the plane angles by a common measure; planes through the lines of division and the edge split the dihedral angles into the same numbers of equal parts. The incommensurable case uses limits, as in § 549.
XVIII: in MN draw DE perpendicular to AB; angle CDE measures the right dihedral angle, so it is a right angle, and CD is perpendicular to two lines of MN (§ 507).
XIX: draw DE perpendicular to AB in MN; CD is perpendicular to AB and DE, so angle CDE is right, and it measures the dihedral angle (§ 550).
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.
For Prop. XVII, it is enough to explain the commensurable case and say that the other follows by limits, as in Book II.
Wrap up
Ask: Why does a door stay upright however far it swings? (Its hinge line is upright, and every plane through an upright line is upright, Prop. XIX.)