Wentworth's Plane and Solid Geometry · Book VIII, §§ 791–798 (Props. XIII–XV) · Book 9: Book VIII: The Sphere

122. Polar triangles and the angle sum of a spherical triangle

Goal: Your student can construct the polar triangle, prove that each angle of one is the supplement of the opposite side of the other, and prove that the angles of a spherical triangle total between 180° and 540°.

⏱ About 40 minutes · 7 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

What's in the book

The polar triangle (§ 791), Propositions XIII–XV (§§ 792–796), and the definitions of bi-rectangular and tri-rectangular triangles and spherical excess (§§ 797–798).

The lesson, step by step

  1. Warm up

    Ask: What is the polar distance of a great circle?

  2. The new idea

    If A'B'C' is the polar triangle of ABC, then, reciprocally, ABC is the polar triangle of A'B'C'.

    In two polar triangles each angle of the one is the supplement of the opposite side in the other.

    The sum of the angles of a spherical triangle is greater than 180° and less than 540°.

  3. Draw the figure

    On the ball, draw a triangle ABC. With A as pole, draw a great circle (90° from A); do the same for B and C. The three great circles form the polar triangle.

    Show a triangle with three right angles: from the north pole down two meridians 90° apart to the equator.

  4. Walk through the proof

    XIII: B' is a quadrant from A and from C, so B' is the pole of AC (§ 762); likewise for the others.

    XIV: B'E and C'D are quadrants, so B'E + C'D = 180°, that is, DE + B'C' = 180°; and DE measures angle A.

    XV: A + B + C = 540° − (a' + b' + c'), and the sides of the polar triangle total more than 0° and less than 360°.

  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. Exercises

    Ask: Can a spherical triangle have two right angles? three obtuse angles? (Yes to both, § 796.)

  7. Wrap up

    Ask: What is the spherical excess of the triangle with three right angles? (270° − 180° = 90°.)

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