Wentworth's Plane and Solid Geometry · Book I, §§ 208–213 (Props. XLI–XLII) · Book 2: Book I: Rectilinear Figures
18. Symmetry, and a review of Book I
Goal: Your student can define symmetry about a point and about a line, prove the two symmetry theorems, and answer the review questions on Book I.
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)
- Scissors and scrap paper
What's in the book
Definitions of symmetry (§§ 208–211), Propositions XLI and XLII (§§ 212–213), and the 28 Review Questions on Book I (page 63).
The lesson, step by step
Warm up
Ask: What is the sum of the exterior angles of any polygon?
The new idea
Read §§ 208–211: symmetry with respect to a centre and with respect to an axis.
If a figure is symmetrical with respect to two axes perpendicular to each other, it is symmetrical with respect to their intersection as a centre.
Prop. XLI (§ 212) is about a kite: a quadrilateral with two pairs of equal adjacent sides is symmetrical about one diagonal, and the diagonals are perpendicular.
Draw the figure
Draw a kite ABCD with AB = AD and CB = CD (use the compasses from A and from C). Draw both diagonals. Fold a paper copy along AC to see the symmetry.
Walk through the proof
XLI: triangles ABC and ADC are equal by three sides, so the angles at A and at C are bisected by AC. Turn ABC over on AC and it falls exactly on ADC, so AC is an axis of symmetry and is perpendicular to BD.
XLII: for any point N on the figure, the symmetrical points about the two axes form parallelograms whose sides show that O bisects the line from N through O.
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.
Exercises
Use the Review Questions on page 63 as an oral quiz. For example:
17. State the three cases in which two triangles are equal.
19. What is meant by a locus of points ?
Wrap up
Mark any review questions that were hard, and go back to those pages before the next session.
Tip
The author advises reviewing each Book: drawing the figures freehand and stating and proving the propositions orally, pointing to each line and angle.