Wentworth's Plane and Solid Geometry · Book VIII, Props. XVIII–XXII · Book 9: Book VIII: The Sphere
124. Equal and symmetrical spherical triangles
Goal: Your student can state when two spherical triangles are equal or symmetrical, and prove that the base angles of an isosceles spherical triangle are equal.
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)
- A ball (an orange or a rubber ball) and a rubber band or string
- A calculator (optional)
What's in the book
Propositions XVIII–XXII (§§ 808–813), the note after § 811, and Exercises 737–745.
The lesson, step by step
Warm up
Ask: What are the ways to prove two plane triangles equal?
The new idea
Two mutually equilateral triangles on the same sphere or equal spheres are mutually equiangular, and are equal or symmetrical.
Two mutually equiangular triangles on the same sphere or equal spheres are mutually equilateral, and are either equal or symmetrical.
In an isosceles spherical triangle, the angles opposite the equal sides are equal.
Notice the surprise of Prop. XXI: on a sphere, triangles with equal angles must have equal sides. There are no similar spherical triangles of different sizes on the same sphere.
Draw the figure
Draw an isosceles spherical triangle ABC on the ball and the arc AD to the middle of BC.
Walk through the proof
XVIII and XIX: superposition, as for plane triangles; when the parts come in reverse order, use the symmetrical triangle.
XX: equal sides mean equal face angles at the centre, so equal dihedral angles (§ 583), so equal angles.
XXI: pass to the polar triangles: they are mutually equilateral, so mutually equiangular, so the original triangles are mutually equilateral (§ 793).
XXII: ABD and ACD are mutually equilateral, so mutually equiangular.
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
Ex. 743. Given a spherical triangle whose sides are 60°, 80°, and 100°. Find the angles of its polar triangle.
Ex. 744. Given a spherical triangle whose angles are 70°, 75°, and 95°. Find the sides of its polar triangle.
Ex. 739. Find the diameter of the section of a sphere 10 inches in diameter made by a plane 3 inches from the centre.
Wrap up
Ask: Why do plane triangles have similar copies of every size but spherical triangles on one sphere do not?
Answers to the book's problems
Ex. 737: the circle's radius is r = 3√55/8 ≈ 2.78 in (from the right triangle with the diameter); area 495π/64 ≈ 24.3 sq in. Ex. 738: R = a√6/12 (inscribed), R' = a√6/4 (circumscribed). Ex. 739: 2 × √(25 − 9) = 8 in. Ex. 743: 120°, 100°, 80°. Ex. 744: 110°, 105°, 85°. Ex. 745: 12 : 20 = 3 : 5.
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