Wentworth's Plane and Solid Geometry · Book I, Exs. 28–58 · Book 2: Book I: Rectilinear Figures
20. Original exercises on angles and triangles
Goal: Your student proves several original theorems about bisectors, isosceles triangles and parallels, writing each proof neatly.
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 68–70: theorem exercises 28–58, mostly on bisectors of angles, isosceles triangles, altitudes and medians. Some give a hint or a section number.
The lesson, step by step
Warm up
Ask: What are the three methods of proving theorems?
How to attack an exercise
Read the statement, draw a general figure, write "Given" and "To prove", then ask: Which triangles could be equal? Which angles are equal because of parallels? Use the hint printed after the exercise if there is one.
Together
Do one exercise together, your student writing and you asking "Why?":
Ex. 29. The bisectors of two supplementary adjacent angles are perpendicular to each other.
On their own
Your student chooses three or four more. Good ones to start: Ex. 31, 34, 38, 44, 49 and 54. For example:
Ex. 49. The lines joining the middle points of the sides of a triangle divide the triangle into four equal triangles.
Check the writing
Each proof should have a figure, "Given", "To prove", and steps with reasons. Praise a clear proof more than a fast one.
Wrap up
Your student reads one finished proof aloud, pointing at the figure.
Afterwards
The author says the exercises are "so abundant that it is not expected that any one class will work them all out." Choose some and leave the rest.
Answers to the book's problems
Hints: Ex. 29: the two halves add up to half of two right angles. Ex. 31: each half of one angle equals a half of the other (vertical angles are equal). Ex. 34: the two triangles are equal by § 143. Ex. 38: the bisector makes two triangles equal by § 139. Ex. 44: the midpoint of the hypotenuse is equidistant from all three vertices (Ex. 21). Ex. 49: each line joining midpoints is half the parallel side (§ 189), so the four triangles have equal sides. Ex. 54: triangles BEC and CDB are equal (§ 139), as the book notes.
Worked out for this site from the scan. If a number in your copy differs, trust the book.