Wentworth's Plane and Solid Geometry · Book I, §§ 163–182 (Props. XXXII–XXXIV) · Book 2: Book I: Rectilinear Figures

15. Quadrilaterals and parallelograms

Goal: Your student knows the kinds of quadrilaterals and can prove that the opposite sides of a parallelogram are equal, and the converse.

⏱ 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 protractor

What's in the book

Definitions of quadrilaterals (§§ 163–175), and Propositions XXXII–XXXIV (§§ 176–182) with corollaries.

The lesson, step by step

  1. Warm up

    Ask: What is a locus? State the two locus theorems.

  2. The new idea

    Read §§ 163–175 and sketch a trapezium, a trapezoid, a parallelogram, a rectangle, a square, a rhomboid and a rhombus.

    Two angles whose sides are parallel, each to each, are either equal or supplementary.

    The opposite sides of a parallelogram are equal.

    If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

  3. Draw the figure

    Draw parallelogram ABCE: two parallel lines, then two parallel crossing lines (slide a set square). Draw the diagonal AC.

  4. Walk through the proof

    XXXII: extend sides until they meet and compare the angles with exterior-interior angles of parallels (§ 112).

    XXXIII: the diagonal AC makes two triangles with AC common and two pairs of equal alternate-interior angles, so they are equal (Prop. XX) and the opposite sides match.

    XXXIV: the diagonal makes two triangles equal by three sides, so the alternate-interior angles are equal, and the sides are parallel (Prop. XIII).

  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 read the corollaries §§ 179–181: parallels are everywhere equally distant.

  6. Wrap up

    Your student explains the difference between a rhombus and a square, and between a trapezoid and a trapezium.

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