Wentworth's Plane and Solid Geometry · Book I, §§ 117–137 (Prop. XVIII) · Book 2: Book I: Rectilinear Figures

10. Triangles and the sum of their angles

Goal: Your student can classify triangles by sides and by angles and prove that the angles of a triangle add to two right angles.

⏱ 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 protractor
  • Scissors and scrap paper

What's in the book

Definitions of triangles (§§ 117–128), then Proposition XVIII (§ 129) with its eight corollaries.

The lesson, step by step

  1. Warm up

    Ask: If two parallels are cut by a transversal, which angles are equal? Which are supplementary?

  2. The new idea

    First read §§ 117–128 and have your student sketch: scalene, isosceles, equilateral; right, obtuse, acute; and mark an exterior angle and an altitude.

    Then the great theorem:

    The sum of the three angles of a triangle is equal to two right angles.

  3. Draw the figure

    Draw a large triangle ABC. Through C draw CE parallel to AB, and extend AC beyond C to F.

  4. Walk through the proof

    The three angles at C on one side of the line AF (angle ECF, angle ECB and angle BCA) make two right angles (§ 89).

    Angle A equals angle ECF (exterior-interior angles of parallels), and angle B equals angle BCE (alternate-interior angles).

    Put A and B in place of those two angles: A + B + BCA = 2 right angles.

  5. Your student proves it back

    Close the book. Your student redraws the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("because the radii are equal", "by § 143"). Point to each line or angle as it is named.

    If a step is missing, ask "How do we know that?" rather than supplying it.

    Then go through the corollaries §§ 130–137 aloud. Ask, for each, "Why does it follow?" § 137, the exterior angle, is used often later.

  6. Exercises

    Cut a paper triangle, tear off the three corners and lay them side by side: they make a straight line. That is a check, not a proof.

  7. Wrap up

    Ask: Can a triangle have two obtuse angles? Why not?

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