Wentworth's Plane and Solid Geometry · Book VI, §§ 569–580 (Prop. XXVI) · Book 7: Book VI: Lines and Planes in Space
90. Polyhedral angles
Goal: Your student can name the parts of a polyhedral angle, explain symmetrical polyhedral angles, and prove that any two face angles of a trihedral angle together exceed the third.
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)
- A protractor, scissors and a card
What's in the book
Definitions of polyhedral angles (§§ 569–578), symmetrical polyhedral angles (§ 579), and Proposition XXVI.
The lesson, step by step
Warm up
Ask: What is a dihedral angle? How is it measured?
The new idea
The opening of three or more planes which meet at a common point is called a polyhedral angle.
The sum of any two face angles of a trihedral angle is greater than the third face angle.
Draw the figure
Look at the corner of a box: three faces meet at a point, a trihedral angle. The tip of a square pyramid (or a folded paper cone with four creases) is a tetrahedral angle.
For symmetry (§ 579), hold your two hands up: they have the same parts in reverse order and cannot be laid one on the other.
Walk through the proof
XXVI: in face ASC draw SD making angle ASD equal to ASB, and take SB = SD. Triangles ASD and ASB are equal (§ 143), so AD = AB. In triangle ABC, AB + BC > AC, so BC > DC. Triangles BSC and DSC have two sides equal and BC > DC, so angle BSC > angle DSC (§ 155). Add angle ASB = angle ASD to both.
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
Make a trihedral angle from paper: cut three angles of 40°, 50° and 60° side by side from a card and fold them up so the outer edges meet. Then try 30°, 40° and 80°: it will not close up. Why? (30 + 40 is less than 80.)
Wrap up
Ask: Why are two symmetrical trihedral angles usually not equal, though all their parts are equal? (The parts come in reverse order, like left and right hands.)