Wentworth's Plane and Solid Geometry · Book VI, Props. XXVII–XXVIII · Book 7: Book VI: Lines and Planes in Space
91. Face angles of a polyhedral angle; equal trihedral angles
Goal: Your student can prove that the face angles of a convex polyhedral angle total less than four right angles, and that trihedral angles with equal face angles are equal or symmetrical.
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 XXVII–XXVIII (§§ 581–583).
The lesson, step by step
Warm up
Ask: State Prop. XXVI.
The new idea
The sum of the face angles of any convex polyhedral angle is less than four right angles.
Two trihedral angles are equal or symmetrical when the three face angles of the one are respectively equal to the three face angles of the other.
Draw the figure
Cut a square of paper, mark its centre, and fold toward the centre: if you cut out a wedge and close the gap, the paper rises into a point. A complete 360° lies flat; a corner needs less than 360°.
For Prop. XXVII, draw a five-sided polyhedral angle S cut by a plane in the pentagon ABCDE, with a point O inside the pentagon.
Walk through the proof
XXVII: the triangles with vertex S and those with vertex O are equal in number, so their angles have the same total. At each corner of the base, the two angles of the S-triangles exceed the angle of the polygon (Prop. XXVI). So the S-triangles have more at their bases and less at the top. The angles at O make four right angles, so the angles at S make less.
XXVIII: take equal lengths on all six edges; the isosceles triangles are equal, so the bases ABC and A'B'C' are equal. Perpendiculars to SA at D make triangles that prove the plane angles of the dihedral angles equal, so the dihedral angles are equal.
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: Could three regular hexagons meet at the corner of a solid? (No: 3 × 120° = 360°, not less.) We will use this in Book VII.