Wentworth's Plane and Solid Geometry · Book II, Exs. 101–117 · Book 3: Book II: The Circle
30. Review of Book II and theorems on the circle
Goal: Your student answers the review questions on Book II and proves several original theorems about circles.
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
The Principle of Continuity (§ 299) and the Review Questions on Book II (pages 107–108), then theorem exercises 101–117.
The lesson, step by step
Warm up
Ask: What is an angle formed by two chords measured by? By two secants outside the circle?
Continuity
Read § 299. One statement, "half the sum of the intercepted arcs", covers all the angle theorems if an arc can count as negative. Your student explains each of the four cases with a sketch.
Review questions
Use the 18 Review Questions on pages 107–108 as an oral quiz. For example:
9. What is an inscribed angle ? What is its measure ?
Theorems together
Prove one together:
Ex. 103. The sum of two opposite sides of a circumscribed quadrilateral is equal to the sum of the other two sides.
Hint: the two tangents from each corner are equal.
On their own
Your student chooses two or three more from Exercises 101–117, for example Ex. 101, 105, 109 or 114.
Ex. 114. A parallelogram inscribed in a circle is a rectangle.
Wrap up
Your student reads one proof aloud and explains which earlier theorem made it work.
Answers to the book's problems
Ex. 101: both angles are measured by half the same arc (§§ 289, 295). Ex. 103: split each side at its point of contact; the eight tangent pieces pair off equally (Prop. XII), and each pair of opposite sides gets one piece from each pair. Ex. 112: the book's hint: triangle OEF is equilateral, so AF = FO = OD. Ex. 114: opposite angles are equal (parallelogram) and supplementary (Ex. 97), so each is a right angle.
Worked out for this site from the scan. If a number in your copy differs, trust the book.