Wentworth's Plane and Solid Geometry · Book I, Exs. 59–90 · Book 2: Book I: Rectilinear Figures
21. Original exercises on quadrilaterals
Goal: Your student proves original theorems about parallelograms, rectangles, rhombuses and trapezoids.
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)
What's in the book
Pages 71–74: exercises 59–90, many on parallelograms and trapezoids, with hints. This finishes Book I.
The lesson, step by step
Warm up
Ask: What are the tests for a parallelogram? (§§ 182, 183, and opposite sides parallel by definition.)
Together
Do this one together:
Ex. 69. The diagonals of a rectangle are equal.
The book's hint: prove triangle ABC equal to triangle BAD.
On their own
Your student picks three or four from Exercises 68–88. For example:
Ex. 74. The lines joining the middle points of the sides of any quadrilateral, taken in order, enclose a parallelogram.
Ex. 87. The diagonals of an isosceles trapezoid are equal.
Draw and check
Before proving Ex. 74, draw a very lopsided quadrilateral, join the midpoints and measure: the middle figure is still a parallelogram. That surprise is worth a minute.
Review Book I
Your student states five propositions from Book I chosen by you at random, and proves one of them.
Wrap up
Book I is finished. Let your student pick their favorite proof and explain it to someone else in the family.
Answers to the book's problems
Hints: Ex. 68: prove triangles AOB and COD equal, then use Prop. XIII. Ex. 69: triangles ABC and BAD are equal by § 143. Ex. 71: the diagonals cut the rhombus into four triangles equal by three sides. Ex. 74: each pair of opposite midpoint lines is parallel to a diagonal and equal to half of it (§ 189). Ex. 87: triangles ACD and BDC are equal (§ 143), using Ex. 84 for the equal angles.
Worked out for this site from the scan. If a number in your copy differs, trust the book.