Wentworth's Plane and Solid Geometry · Book VIII, Props. XVI–XVII · Book 9: Book VIII: The Sphere

123. Bi-rectangular and symmetrical spherical triangles

Goal: Your student can prove the properties of bi-rectangular triangles and that symmetrical spherical triangles are equivalent.

⏱ About 40 minutes · 6 steps

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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
  • An orange and a knife (with an adult)

What's in the book

Propositions XVI–XVII (§§ 799–807), with symmetrical spherical triangles (§§ 803–806).

The lesson, step by step

  1. Warm up

    Ask: What is a polar triangle? State Prop. XIV.

  2. The new idea

    In a bi-rectangular spherical triangle the sides opposite the right angles are quadrants, and the side opposite the third angle measures that angle.

    Two symmetrical spherical triangles are equivalent.

    § 802: three planes through the centre, each perpendicular to the other two, divide the surface into eight equal tri-rectangular triangles.

  3. Draw the figure

    Cut an orange in half, then each half in half the other way, then across the middle: eight equal pieces, each with a tri-rectangular triangle of peel.

    For symmetrical triangles (§ 803), draw ABC on the ball and the triangle A'B'C' at the opposite ends of the diameters.

  4. Walk through the proof

    XVI: the arcs PA and PB are perpendicular to AB, so they pass through its pole (§ 754); so P is the pole, PA and PB are quadrants, and angle P is measured by AB.

    XVII: let P be the pole of the small circle through A, B, C. The triangle splits into three isosceles triangles PAB, PBC, PAC; their symmetrical triangles are superposable (§ 806). Add them up.

  5. 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.

  6. Wrap up

    Ask: Why can't you slide a left glove into the shape of a right glove? How is that like two symmetrical spherical triangles?

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